summaryrefslogtreecommitdiff
path: root/Build/source/utils/asymptote/base/geometry.asy
blob: bbbc823fc54dc0a156eaf8660a63db652e3e7620 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
1640
1641
1642
1643
1644
1645
1646
1647
1648
1649
1650
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
1664
1665
1666
1667
1668
1669
1670
1671
1672
1673
1674
1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685
1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
1697
1698
1699
1700
1701
1702
1703
1704
1705
1706
1707
1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
1718
1719
1720
1721
1722
1723
1724
1725
1726
1727
1728
1729
1730
1731
1732
1733
1734
1735
1736
1737
1738
1739
1740
1741
1742
1743
1744
1745
1746
1747
1748
1749
1750
1751
1752
1753
1754
1755
1756
1757
1758
1759
1760
1761
1762
1763
1764
1765
1766
1767
1768
1769
1770
1771
1772
1773
1774
1775
1776
1777
1778
1779
1780
1781
1782
1783
1784
1785
1786
1787
1788
1789
1790
1791
1792
1793
1794
1795
1796
1797
1798
1799
1800
1801
1802
1803
1804
1805
1806
1807
1808
1809
1810
1811
1812
1813
1814
1815
1816
1817
1818
1819
1820
1821
1822
1823
1824
1825
1826
1827
1828
1829
1830
1831
1832
1833
1834
1835
1836
1837
1838
1839
1840
1841
1842
1843
1844
1845
1846
1847
1848
1849
1850
1851
1852
1853
1854
1855
1856
1857
1858
1859
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
1885
1886
1887
1888
1889
1890
1891
1892
1893
1894
1895
1896
1897
1898
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
2033
2034
2035
2036
2037
2038
2039
2040
2041
2042
2043
2044
2045
2046
2047
2048
2049
2050
2051
2052
2053
2054
2055
2056
2057
2058
2059
2060
2061
2062
2063
2064
2065
2066
2067
2068
2069
2070
2071
2072
2073
2074
2075
2076
2077
2078
2079
2080
2081
2082
2083
2084
2085
2086
2087
2088
2089
2090
2091
2092
2093
2094
2095
2096
2097
2098
2099
2100
2101
2102
2103
2104
2105
2106
2107
2108
2109
2110
2111
2112
2113
2114
2115
2116
2117
2118
2119
2120
2121
2122
2123
2124
2125
2126
2127
2128
2129
2130
2131
2132
2133
2134
2135
2136
2137
2138
2139
2140
2141
2142
2143
2144
2145
2146
2147
2148
2149
2150
2151
2152
2153
2154
2155
2156
2157
2158
2159
2160
2161
2162
2163
2164
2165
2166
2167
2168
2169
2170
2171
2172
2173
2174
2175
2176
2177
2178
2179
2180
2181
2182
2183
2184
2185
2186
2187
2188
2189
2190
2191
2192
2193
2194
2195
2196
2197
2198
2199
2200
2201
2202
2203
2204
2205
2206
2207
2208
2209
2210
2211
2212
2213
2214
2215
2216
2217
2218
2219
2220
2221
2222
2223
2224
2225
2226
2227
2228
2229
2230
2231
2232
2233
2234
2235
2236
2237
2238
2239
2240
2241
2242
2243
2244
2245
2246
2247
2248
2249
2250
2251
2252
2253
2254
2255
2256
2257
2258
2259
2260
2261
2262
2263
2264
2265
2266
2267
2268
2269
2270
2271
2272
2273
2274
2275
2276
2277
2278
2279
2280
2281
2282
2283
2284
2285
2286
2287
2288
2289
2290
2291
2292
2293
2294
2295
2296
2297
2298
2299
2300
2301
2302
2303
2304
2305
2306
2307
2308
2309
2310
2311
2312
2313
2314
2315
2316
2317
2318
2319
2320
2321
2322
2323
2324
2325
2326
2327
2328
2329
2330
2331
2332
2333
2334
2335
2336
2337
2338
2339
2340
2341
2342
2343
2344
2345
2346
2347
2348
2349
2350
2351
2352
2353
2354
2355
2356
2357
2358
2359
2360
2361
2362
2363
2364
2365
2366
2367
2368
2369
2370
2371
2372
2373
2374
2375
2376
2377
2378
2379
2380
2381
2382
2383
2384
2385
2386
2387
2388
2389
2390
2391
2392
2393
2394
2395
2396
2397
2398
2399
2400
2401
2402
2403
2404
2405
2406
2407
2408
2409
2410
2411
2412
2413
2414
2415
2416
2417
2418
2419
2420
2421
2422
2423
2424
2425
2426
2427
2428
2429
2430
2431
2432
2433
2434
2435
2436
2437
2438
2439
2440
2441
2442
2443
2444
2445
2446
2447
2448
2449
2450
2451
2452
2453
2454
2455
2456
2457
2458
2459
2460
2461
2462
2463
2464
2465
2466
2467
2468
2469
2470
2471
2472
2473
2474
2475
2476
2477
2478
2479
2480
2481
2482
2483
2484
2485
2486
2487
2488
2489
2490
2491
2492
2493
2494
2495
2496
2497
2498
2499
2500
2501
2502
2503
2504
2505
2506
2507
2508
2509
2510
2511
2512
2513
2514
2515
2516
2517
2518
2519
2520
2521
2522
2523
2524
2525
2526
2527
2528
2529
2530
2531
2532
2533
2534
2535
2536
2537
2538
2539
2540
2541
2542
2543
2544
2545
2546
2547
2548
2549
2550
2551
2552
2553
2554
2555
2556
2557
2558
2559
2560
2561
2562
2563
2564
2565
2566
2567
2568
2569
2570
2571
2572
2573
2574
2575
2576
2577
2578
2579
2580
2581
2582
2583
2584
2585
2586
2587
2588
2589
2590
2591
2592
2593
2594
2595
2596
2597
2598
2599
2600
2601
2602
2603
2604
2605
2606
2607
2608
2609
2610
2611
2612
2613
2614
2615
2616
2617
2618
2619
2620
2621
2622
2623
2624
2625
2626
2627
2628
2629
2630
2631
2632
2633
2634
2635
2636
2637
2638
2639
2640
2641
2642
2643
2644
2645
2646
2647
2648
2649
2650
2651
2652
2653
2654
2655
2656
2657
2658
2659
2660
2661
2662
2663
2664
2665
2666
2667
2668
2669
2670
2671
2672
2673
2674
2675
2676
2677
2678
2679
2680
2681
2682
2683
2684
2685
2686
2687
2688
2689
2690
2691
2692
2693
2694
2695
2696
2697
2698
2699
2700
2701
2702
2703
2704
2705
2706
2707
2708
2709
2710
2711
2712
2713
2714
2715
2716
2717
2718
2719
2720
2721
2722
2723
2724
2725
2726
2727
2728
2729
2730
2731
2732
2733
2734
2735
2736
2737
2738
2739
2740
2741
2742
2743
2744
2745
2746
2747
2748
2749
2750
2751
2752
2753
2754
2755
2756
2757
2758
2759
2760
2761
2762
2763
2764
2765
2766
2767
2768
2769
2770
2771
2772
2773
2774
2775
2776
2777
2778
2779
2780
2781
2782
2783
2784
2785
2786
2787
2788
2789
2790
2791
2792
2793
2794
2795
2796
2797
2798
2799
2800
2801
2802
2803
2804
2805
2806
2807
2808
2809
2810
2811
2812
2813
2814
2815
2816
2817
2818
2819
2820
2821
2822
2823
2824
2825
2826
2827
2828
2829
2830
2831
2832
2833
2834
2835
2836
2837
2838
2839
2840
2841
2842
2843
2844
2845
2846
2847
2848
2849
2850
2851
2852
2853
2854
2855
2856
2857
2858
2859
2860
2861
2862
2863
2864
2865
2866
2867
2868
2869
2870
2871
2872
2873
2874
2875
2876
2877
2878
2879
2880
2881
2882
2883
2884
2885
2886
2887
2888
2889
2890
2891
2892
2893
2894
2895
2896
2897
2898
2899
2900
2901
2902
2903
2904
2905
2906
2907
2908
2909
2910
2911
2912
2913
2914
2915
2916
2917
2918
2919
2920
2921
2922
2923
2924
2925
2926
2927
2928
2929
2930
2931
2932
2933
2934
2935
2936
2937
2938
2939
2940
2941
2942
2943
2944
2945
2946
2947
2948
2949
2950
2951
2952
2953
2954
2955
2956
2957
2958
2959
2960
2961
2962
2963
2964
2965
2966
2967
2968
2969
2970
2971
2972
2973
2974
2975
2976
2977
2978
2979
2980
2981
2982
2983
2984
2985
2986
2987
2988
2989
2990
2991
2992
2993
2994
2995
2996
2997
2998
2999
3000
3001
3002
3003
3004
3005
3006
3007
3008
3009
3010
3011
3012
3013
3014
3015
3016
3017
3018
3019
3020
3021
3022
3023
3024
3025
3026
3027
3028
3029
3030
3031
3032
3033
3034
3035
3036
3037
3038
3039
3040
3041
3042
3043
3044
3045
3046
3047
3048
3049
3050
3051
3052
3053
3054
3055
3056
3057
3058
3059
3060
3061
3062
3063
3064
3065
3066
3067
3068
3069
3070
3071
3072
3073
3074
3075
3076
3077
3078
3079
3080
3081
3082
3083
3084
3085
3086
3087
3088
3089
3090
3091
3092
3093
3094
3095
3096
3097
3098
3099
3100
3101
3102
3103
3104
3105
3106
3107
3108
3109
3110
3111
3112
3113
3114
3115
3116
3117
3118
3119
3120
3121
3122
3123
3124
3125
3126
3127
3128
3129
3130
3131
3132
3133
3134
3135
3136
3137
3138
3139
3140
3141
3142
3143
3144
3145
3146
3147
3148
3149
3150
3151
3152
3153
3154
3155
3156
3157
3158
3159
3160
3161
3162
3163
3164
3165
3166
3167
3168
3169
3170
3171
3172
3173
3174
3175
3176
3177
3178
3179
3180
3181
3182
3183
3184
3185
3186
3187
3188
3189
3190
3191
3192
3193
3194
3195
3196
3197
3198
3199
3200
3201
3202
3203
3204
3205
3206
3207
3208
3209
3210
3211
3212
3213
3214
3215
3216
3217
3218
3219
3220
3221
3222
3223
3224
3225
3226
3227
3228
3229
3230
3231
3232
3233
3234
3235
3236
3237
3238
3239
3240
3241
3242
3243
3244
3245
3246
3247
3248
3249
3250
3251
3252
3253
3254
3255
3256
3257
3258
3259
3260
3261
3262
3263
3264
3265
3266
3267
3268
3269
3270
3271
3272
3273
3274
3275
3276
3277
3278
3279
3280
3281
3282
3283
3284
3285
3286
3287
3288
3289
3290
3291
3292
3293
3294
3295
3296
3297
3298
3299
3300
3301
3302
3303
3304
3305
3306
3307
3308
3309
3310
3311
3312
3313
3314
3315
3316
3317
3318
3319
3320
3321
3322
3323
3324
3325
3326
3327
3328
3329
3330
3331
3332
3333
3334
3335
3336
3337
3338
3339
3340
3341
3342
3343
3344
3345
3346
3347
3348
3349
3350
3351
3352
3353
3354
3355
3356
3357
3358
3359
3360
3361
3362
3363
3364
3365
3366
3367
3368
3369
3370
3371
3372
3373
3374
3375
3376
3377
3378
3379
3380
3381
3382
3383
3384
3385
3386
3387
3388
3389
3390
3391
3392
3393
3394
3395
3396
3397
3398
3399
3400
3401
3402
3403
3404
3405
3406
3407
3408
3409
3410
3411
3412
3413
3414
3415
3416
3417
3418
3419
3420
3421
3422
3423
3424
3425
3426
3427
3428
3429
3430
3431
3432
3433
3434
3435
3436
3437
3438
3439
3440
3441
3442
3443
3444
3445
3446
3447
3448
3449
3450
3451
3452
3453
3454
3455
3456
3457
3458
3459
3460
3461
3462
3463
3464
3465
3466
3467
3468
3469
3470
3471
3472
3473
3474
3475
3476
3477
3478
3479
3480
3481
3482
3483
3484
3485
3486
3487
3488
3489
3490
3491
3492
3493
3494
3495
3496
3497
3498
3499
3500
3501
3502
3503
3504
3505
3506
3507
3508
3509
3510
3511
3512
3513
3514
3515
3516
3517
3518
3519
3520
3521
3522
3523
3524
3525
3526
3527
3528
3529
3530
3531
3532
3533
3534
3535
3536
3537
3538
3539
3540
3541
3542
3543
3544
3545
3546
3547
3548
3549
3550
3551
3552
3553
3554
3555
3556
3557
3558
3559
3560
3561
3562
3563
3564
3565
3566
3567
3568
3569
3570
3571
3572
3573
3574
3575
3576
3577
3578
3579
3580
3581
3582
3583
3584
3585
3586
3587
3588
3589
3590
3591
3592
3593
3594
3595
3596
3597
3598
3599
3600
3601
3602
3603
3604
3605
3606
3607
3608
3609
3610
3611
3612
3613
3614
3615
3616
3617
3618
3619
3620
3621
3622
3623
3624
3625
3626
3627
3628
3629
3630
3631
3632
3633
3634
3635
3636
3637
3638
3639
3640
3641
3642
3643
3644
3645
3646
3647
3648
3649
3650
3651
3652
3653
3654
3655
3656
3657
3658
3659
3660
3661
3662
3663
3664
3665
3666
3667
3668
3669
3670
3671
3672
3673
3674
3675
3676
3677
3678
3679
3680
3681
3682
3683
3684
3685
3686
3687
3688
3689
3690
3691
3692
3693
3694
3695
3696
3697
3698
3699
3700
3701
3702
3703
3704
3705
3706
3707
3708
3709
3710
3711
3712
3713
3714
3715
3716
3717
3718
3719
3720
3721
3722
3723
3724
3725
3726
3727
3728
3729
3730
3731
3732
3733
3734
3735
3736
3737
3738
3739
3740
3741
3742
3743
3744
3745
3746
3747
3748
3749
3750
3751
3752
3753
3754
3755
3756
3757
3758
3759
3760
3761
3762
3763
3764
3765
3766
3767
3768
3769
3770
3771
3772
3773
3774
3775
3776
3777
3778
3779
3780
3781
3782
3783
3784
3785
3786
3787
3788
3789
3790
3791
3792
3793
3794
3795
3796
3797
3798
3799
3800
3801
3802
3803
3804
3805
3806
3807
3808
3809
3810
3811
3812
3813
3814
3815
3816
3817
3818
3819
3820
3821
3822
3823
3824
3825
3826
3827
3828
3829
3830
3831
3832
3833
3834
3835
3836
3837
3838
3839
3840
3841
3842
3843
3844
3845
3846
3847
3848
3849
3850
3851
3852
3853
3854
3855
3856
3857
3858
3859
3860
3861
3862
3863
3864
3865
3866
3867
3868
3869
3870
3871
3872
3873
3874
3875
3876
3877
3878
3879
3880
3881
3882
3883
3884
3885
3886
3887
3888
3889
3890
3891
3892
3893
3894
3895
3896
3897
3898
3899
3900
3901
3902
3903
3904
3905
3906
3907
3908
3909
3910
3911
3912
3913
3914
3915
3916
3917
3918
3919
3920
3921
3922
3923
3924
3925
3926
3927
3928
3929
3930
3931
3932
3933
3934
3935
3936
3937
3938
3939
3940
3941
3942
3943
3944
3945
3946
3947
3948
3949
3950
3951
3952
3953
3954
3955
3956
3957
3958
3959
3960
3961
3962
3963
3964
3965
3966
3967
3968
3969
3970
3971
3972
3973
3974
3975
3976
3977
3978
3979
3980
3981
3982
3983
3984
3985
3986
3987
3988
3989
3990
3991
3992
3993
3994
3995
3996
3997
3998
3999
4000
4001
4002
4003
4004
4005
4006
4007
4008
4009
4010
4011
4012
4013
4014
4015
4016
4017
4018
4019
4020
4021
4022
4023
4024
4025
4026
4027
4028
4029
4030
4031
4032
4033
4034
4035
4036
4037
4038
4039
4040
4041
4042
4043
4044
4045
4046
4047
4048
4049
4050
4051
4052
4053
4054
4055
4056
4057
4058
4059
4060
4061
4062
4063
4064
4065
4066
4067
4068
4069
4070
4071
4072
4073
4074
4075
4076
4077
4078
4079
4080
4081
4082
4083
4084
4085
4086
4087
4088
4089
4090
4091
4092
4093
4094
4095
4096
4097
4098
4099
4100
4101
4102
4103
4104
4105
4106
4107
4108
4109
4110
4111
4112
4113
4114
4115
4116
4117
4118
4119
4120
4121
4122
4123
4124
4125
4126
4127
4128
4129
4130
4131
4132
4133
4134
4135
4136
4137
4138
4139
4140
4141
4142
4143
4144
4145
4146
4147
4148
4149
4150
4151
4152
4153
4154
4155
4156
4157
4158
4159
4160
4161
4162
4163
4164
4165
4166
4167
4168
4169
4170
4171
4172
4173
4174
4175
4176
4177
4178
4179
4180
4181
4182
4183
4184
4185
4186
4187
4188
4189
4190
4191
4192
4193
4194
4195
4196
4197
4198
4199
4200
4201
4202
4203
4204
4205
4206
4207
4208
4209
4210
4211
4212
4213
4214
4215
4216
4217
4218
4219
4220
4221
4222
4223
4224
4225
4226
4227
4228
4229
4230
4231
4232
4233
4234
4235
4236
4237
4238
4239
4240
4241
4242
4243
4244
4245
4246
4247
4248
4249
4250
4251
4252
4253
4254
4255
4256
4257
4258
4259
4260
4261
4262
4263
4264
4265
4266
4267
4268
4269
4270
4271
4272
4273
4274
4275
4276
4277
4278
4279
4280
4281
4282
4283
4284
4285
4286
4287
4288
4289
4290
4291
4292
4293
4294
4295
4296
4297
4298
4299
4300
4301
4302
4303
4304
4305
4306
4307
4308
4309
4310
4311
4312
4313
4314
4315
4316
4317
4318
4319
4320
4321
4322
4323
4324
4325
4326
4327
4328
4329
4330
4331
4332
4333
4334
4335
4336
4337
4338
4339
4340
4341
4342
4343
4344
4345
4346
4347
4348
4349
4350
4351
4352
4353
4354
4355
4356
4357
4358
4359
4360
4361
4362
4363
4364
4365
4366
4367
4368
4369
4370
4371
4372
4373
4374
4375
4376
4377
4378
4379
4380
4381
4382
4383
4384
4385
4386
4387
4388
4389
4390
4391
4392
4393
4394
4395
4396
4397
4398
4399
4400
4401
4402
4403
4404
4405
4406
4407
4408
4409
4410
4411
4412
4413
4414
4415
4416
4417
4418
4419
4420
4421
4422
4423
4424
4425
4426
4427
4428
4429
4430
4431
4432
4433
4434
4435
4436
4437
4438
4439
4440
4441
4442
4443
4444
4445
4446
4447
4448
4449
4450
4451
4452
4453
4454
4455
4456
4457
4458
4459
4460
4461
4462
4463
4464
4465
4466
4467
4468
4469
4470
4471
4472
4473
4474
4475
4476
4477
4478
4479
4480
4481
4482
4483
4484
4485
4486
4487
4488
4489
4490
4491
4492
4493
4494
4495
4496
4497
4498
4499
4500
4501
4502
4503
4504
4505
4506
4507
4508
4509
4510
4511
4512
4513
4514
4515
4516
4517
4518
4519
4520
4521
4522
4523
4524
4525
4526
4527
4528
4529
4530
4531
4532
4533
4534
4535
4536
4537
4538
4539
4540
4541
4542
4543
4544
4545
4546
4547
4548
4549
4550
4551
4552
4553
4554
4555
4556
4557
4558
4559
4560
4561
4562
4563
4564
4565
4566
4567
4568
4569
4570
4571
4572
4573
4574
4575
4576
4577
4578
4579
4580
4581
4582
4583
4584
4585
4586
4587
4588
4589
4590
4591
4592
4593
4594
4595
4596
4597
4598
4599
4600
4601
4602
4603
4604
4605
4606
4607
4608
4609
4610
4611
4612
4613
4614
4615
4616
4617
4618
4619
4620
4621
4622
4623
4624
4625
4626
4627
4628
4629
4630
4631
4632
4633
4634
4635
4636
4637
4638
4639
4640
4641
4642
4643
4644
4645
4646
4647
4648
4649
4650
4651
4652
4653
4654
4655
4656
4657
4658
4659
4660
4661
4662
4663
4664
4665
4666
4667
4668
4669
4670
4671
4672
4673
4674
4675
4676
4677
4678
4679
4680
4681
4682
4683
4684
4685
4686
4687
4688
4689
4690
4691
4692
4693
4694
4695
4696
4697
4698
4699
4700
4701
4702
4703
4704
4705
4706
4707
4708
4709
4710
4711
4712
4713
4714
4715
4716
4717
4718
4719
4720
4721
4722
4723
4724
4725
4726
4727
4728
4729
4730
4731
4732
4733
4734
4735
4736
4737
4738
4739
4740
4741
4742
4743
4744
4745
4746
4747
4748
4749
4750
4751
4752
4753
4754
4755
4756
4757
4758
4759
4760
4761
4762
4763
4764
4765
4766
4767
4768
4769
4770
4771
4772
4773
4774
4775
4776
4777
4778
4779
4780
4781
4782
4783
4784
4785
4786
4787
4788
4789
4790
4791
4792
4793
4794
4795
4796
4797
4798
4799
4800
4801
4802
4803
4804
4805
4806
4807
4808
4809
4810
4811
4812
4813
4814
4815
4816
4817
4818
4819
4820
4821
4822
4823
4824
4825
4826
4827
4828
4829
4830
4831
4832
4833
4834
4835
4836
4837
4838
4839
4840
4841
4842
4843
4844
4845
4846
4847
4848
4849
4850
4851
4852
4853
4854
4855
4856
4857
4858
4859
4860
4861
4862
4863
4864
4865
4866
4867
4868
4869
4870
4871
4872
4873
4874
4875
4876
4877
4878
4879
4880
4881
4882
4883
4884
4885
4886
4887
4888
4889
4890
4891
4892
4893
4894
4895
4896
4897
4898
4899
4900
4901
4902
4903
4904
4905
4906
4907
4908
4909
4910
4911
4912
4913
4914
4915
4916
4917
4918
4919
4920
4921
4922
4923
4924
4925
4926
4927
4928
4929
4930
4931
4932
4933
4934
4935
4936
4937
4938
4939
4940
4941
4942
4943
4944
4945
4946
4947
4948
4949
4950
4951
4952
4953
4954
4955
4956
4957
4958
4959
4960
4961
4962
4963
4964
4965
4966
4967
4968
4969
4970
4971
4972
4973
4974
4975
4976
4977
4978
4979
4980
4981
4982
4983
4984
4985
4986
4987
4988
4989
4990
4991
4992
4993
4994
4995
4996
4997
4998
4999
5000
5001
5002
5003
5004
5005
5006
5007
5008
5009
5010
5011
5012
5013
5014
5015
5016
5017
5018
5019
5020
5021
5022
5023
5024
5025
5026
5027
5028
5029
5030
5031
5032
5033
5034
5035
5036
5037
5038
5039
5040
5041
5042
5043
5044
5045
5046
5047
5048
5049
5050
5051
5052
5053
5054
5055
5056
5057
5058
5059
5060
5061
5062
5063
5064
5065
5066
5067
5068
5069
5070
5071
5072
5073
5074
5075
5076
5077
5078
5079
5080
5081
5082
5083
5084
5085
5086
5087
5088
5089
5090
5091
5092
5093
5094
5095
5096
5097
5098
5099
5100
5101
5102
5103
5104
5105
5106
5107
5108
5109
5110
5111
5112
5113
5114
5115
5116
5117
5118
5119
5120
5121
5122
5123
5124
5125
5126
5127
5128
5129
5130
5131
5132
5133
5134
5135
5136
5137
5138
5139
5140
5141
5142
5143
5144
5145
5146
5147
5148
5149
5150
5151
5152
5153
5154
5155
5156
5157
5158
5159
5160
5161
5162
5163
5164
5165
5166
5167
5168
5169
5170
5171
5172
5173
5174
5175
5176
5177
5178
5179
5180
5181
5182
5183
5184
5185
5186
5187
5188
5189
5190
5191
5192
5193
5194
5195
5196
5197
5198
5199
5200
5201
5202
5203
5204
5205
5206
5207
5208
5209
5210
5211
5212
5213
5214
5215
5216
5217
5218
5219
5220
5221
5222
5223
5224
5225
5226
5227
5228
5229
5230
5231
5232
5233
5234
5235
5236
5237
5238
5239
5240
5241
5242
5243
5244
5245
5246
5247
5248
5249
5250
5251
5252
5253
5254
5255
5256
5257
5258
5259
5260
5261
5262
5263
5264
5265
5266
5267
5268
5269
5270
5271
5272
5273
5274
5275
5276
5277
5278
5279
5280
5281
5282
5283
5284
5285
5286
5287
5288
5289
5290
5291
5292
5293
5294
5295
5296
5297
5298
5299
5300
5301
5302
5303
5304
5305
5306
5307
5308
5309
5310
5311
5312
5313
5314
5315
5316
5317
5318
5319
5320
5321
5322
5323
5324
5325
5326
5327
5328
5329
5330
5331
5332
5333
5334
5335
5336
5337
5338
5339
5340
5341
5342
5343
5344
5345
5346
5347
5348
5349
5350
5351
5352
5353
5354
5355
5356
5357
5358
5359
5360
5361
5362
5363
5364
5365
5366
5367
5368
5369
5370
5371
5372
5373
5374
5375
5376
5377
5378
5379
5380
5381
5382
5383
5384
5385
5386
5387
5388
5389
5390
5391
5392
5393
5394
5395
5396
5397
5398
5399
5400
5401
5402
5403
5404
5405
5406
5407
5408
5409
5410
5411
5412
5413
5414
5415
5416
5417
5418
5419
5420
5421
5422
5423
5424
5425
5426
5427
5428
5429
5430
5431
5432
5433
5434
5435
5436
5437
5438
5439
5440
5441
5442
5443
5444
5445
5446
5447
5448
5449
5450
5451
5452
5453
5454
5455
5456
5457
5458
5459
5460
5461
5462
5463
5464
5465
5466
5467
5468
5469
5470
5471
5472
5473
5474
5475
5476
5477
5478
5479
5480
5481
5482
5483
5484
5485
5486
5487
5488
5489
5490
5491
5492
5493
5494
5495
5496
5497
5498
5499
5500
5501
5502
5503
5504
5505
5506
5507
5508
5509
5510
5511
5512
5513
5514
5515
5516
5517
5518
5519
5520
5521
5522
5523
5524
5525
5526
5527
5528
5529
5530
5531
5532
5533
5534
5535
5536
5537
5538
5539
5540
5541
5542
5543
5544
5545
5546
5547
5548
5549
5550
5551
5552
5553
5554
5555
5556
5557
5558
5559
5560
5561
5562
5563
5564
5565
5566
5567
5568
5569
5570
5571
5572
5573
5574
5575
5576
5577
5578
5579
5580
5581
5582
5583
5584
5585
5586
5587
5588
5589
5590
5591
5592
5593
5594
5595
5596
5597
5598
5599
5600
5601
5602
5603
5604
5605
5606
5607
5608
5609
5610
5611
5612
5613
5614
5615
5616
5617
5618
5619
5620
5621
5622
5623
5624
5625
5626
5627
5628
5629
5630
5631
5632
5633
5634
5635
5636
5637
5638
5639
5640
5641
5642
5643
5644
5645
5646
5647
5648
5649
5650
5651
5652
5653
5654
5655
5656
5657
5658
5659
5660
5661
5662
5663
5664
5665
5666
5667
5668
5669
5670
5671
5672
5673
5674
5675
5676
5677
5678
5679
5680
5681
5682
5683
5684
5685
5686
5687
5688
5689
5690
5691
5692
5693
5694
5695
5696
5697
5698
5699
5700
5701
5702
5703
5704
5705
5706
5707
5708
5709
5710
5711
5712
5713
5714
5715
5716
5717
5718
5719
5720
5721
5722
5723
5724
5725
5726
5727
5728
5729
5730
5731
5732
5733
5734
5735
5736
5737
5738
5739
5740
5741
5742
5743
5744
5745
5746
5747
5748
5749
5750
5751
5752
5753
5754
5755
5756
5757
5758
5759
5760
5761
5762
5763
5764
5765
5766
5767
5768
5769
5770
5771
5772
5773
5774
5775
5776
5777
5778
5779
5780
5781
5782
5783
5784
5785
5786
5787
5788
5789
5790
5791
5792
5793
5794
5795
5796
5797
5798
5799
5800
5801
5802
5803
5804
5805
5806
5807
5808
5809
5810
5811
5812
5813
5814
5815
5816
5817
5818
5819
5820
5821
5822
5823
5824
5825
5826
5827
5828
5829
5830
5831
5832
5833
5834
5835
5836
5837
5838
5839
5840
5841
5842
5843
5844
5845
5846
5847
5848
5849
5850
5851
5852
5853
5854
5855
5856
5857
5858
5859
5860
5861
5862
5863
5864
5865
5866
5867
5868
5869
5870
5871
5872
5873
5874
5875
5876
5877
5878
5879
5880
5881
5882
5883
5884
5885
5886
5887
5888
5889
5890
5891
5892
5893
5894
5895
5896
5897
5898
5899
5900
5901
5902
5903
5904
5905
5906
5907
5908
5909
5910
5911
5912
5913
5914
5915
5916
5917
5918
5919
5920
5921
5922
5923
5924
5925
5926
5927
5928
5929
5930
5931
5932
5933
5934
5935
5936
5937
5938
5939
5940
5941
5942
5943
5944
5945
5946
5947
5948
5949
5950
5951
5952
5953
5954
5955
5956
5957
5958
5959
5960
5961
5962
5963
5964
5965
5966
5967
5968
5969
5970
5971
5972
5973
5974
5975
5976
5977
5978
5979
5980
5981
5982
5983
5984
5985
5986
5987
5988
5989
5990
5991
5992
5993
5994
5995
5996
5997
5998
5999
6000
6001
6002
6003
6004
6005
6006
6007
6008
6009
6010
6011
6012
6013
6014
6015
6016
6017
6018
6019
6020
6021
6022
6023
6024
6025
6026
6027
6028
6029
6030
6031
6032
6033
6034
6035
6036
6037
6038
6039
6040
6041
6042
6043
6044
6045
6046
6047
6048
6049
6050
6051
6052
6053
6054
6055
6056
6057
6058
6059
6060
6061
6062
6063
6064
6065
6066
6067
6068
6069
6070
6071
6072
6073
6074
6075
6076
6077
6078
6079
6080
6081
6082
6083
6084
6085
6086
6087
6088
6089
6090
6091
6092
6093
6094
6095
6096
6097
6098
6099
6100
6101
6102
6103
6104
6105
6106
6107
6108
6109
6110
6111
6112
6113
6114
6115
6116
6117
6118
6119
6120
6121
6122
6123
6124
6125
6126
6127
6128
6129
6130
6131
6132
6133
6134
6135
6136
6137
6138
6139
6140
6141
6142
6143
6144
6145
6146
6147
6148
6149
6150
6151
6152
6153
6154
6155
6156
6157
6158
6159
6160
6161
6162
6163
6164
6165
6166
6167
6168
6169
6170
6171
6172
6173
6174
6175
6176
6177
6178
6179
6180
6181
6182
6183
6184
6185
6186
6187
6188
6189
6190
6191
6192
6193
6194
6195
6196
6197
6198
6199
6200
6201
6202
6203
6204
6205
6206
6207
6208
6209
6210
6211
6212
6213
6214
6215
6216
6217
6218
6219
6220
6221
6222
6223
6224
6225
6226
6227
6228
6229
6230
6231
6232
6233
6234
6235
6236
6237
6238
6239
6240
6241
6242
6243
6244
6245
6246
6247
6248
6249
6250
6251
6252
6253
6254
6255
6256
6257
6258
6259
6260
6261
6262
6263
6264
6265
6266
6267
6268
6269
6270
6271
6272
6273
6274
6275
6276
6277
6278
6279
6280
6281
6282
6283
6284
6285
6286
6287
6288
6289
6290
6291
6292
6293
6294
6295
6296
6297
6298
6299
6300
6301
6302
6303
6304
6305
6306
6307
6308
6309
6310
6311
6312
6313
6314
6315
6316
6317
6318
6319
6320
6321
6322
6323
6324
6325
6326
6327
6328
6329
6330
6331
6332
6333
6334
6335
6336
6337
6338
6339
6340
6341
6342
6343
6344
6345
6346
6347
6348
6349
6350
6351
6352
6353
6354
6355
6356
6357
6358
6359
6360
6361
6362
6363
6364
6365
6366
6367
6368
6369
6370
6371
6372
6373
6374
6375
6376
6377
6378
6379
6380
6381
6382
6383
6384
6385
6386
6387
6388
6389
6390
6391
6392
6393
6394
6395
6396
6397
6398
6399
6400
6401
6402
6403
6404
6405
6406
6407
6408
6409
6410
6411
6412
6413
6414
6415
6416
6417
6418
6419
6420
6421
6422
6423
6424
6425
6426
6427
6428
6429
6430
6431
6432
6433
6434
6435
6436
6437
6438
6439
6440
6441
6442
6443
6444
6445
6446
6447
6448
6449
6450
6451
6452
6453
6454
6455
6456
6457
6458
6459
6460
6461
6462
6463
6464
6465
6466
6467
6468
6469
6470
6471
6472
6473
6474
6475
6476
6477
6478
6479
6480
6481
6482
6483
6484
6485
6486
6487
6488
6489
6490
6491
6492
6493
6494
6495
6496
6497
6498
6499
6500
6501
6502
6503
6504
6505
6506
6507
6508
6509
6510
6511
6512
6513
6514
6515
6516
6517
6518
6519
6520
6521
6522
6523
6524
6525
6526
6527
6528
6529
6530
6531
6532
6533
6534
6535
6536
6537
6538
6539
6540
6541
6542
6543
6544
6545
6546
6547
6548
6549
6550
6551
6552
6553
6554
6555
6556
6557
6558
6559
6560
6561
6562
6563
6564
6565
6566
6567
6568
6569
6570
6571
6572
6573
6574
6575
6576
6577
6578
6579
6580
6581
6582
6583
6584
6585
6586
6587
6588
6589
6590
6591
6592
6593
6594
6595
6596
6597
6598
6599
6600
6601
6602
6603
6604
6605
6606
6607
6608
6609
6610
6611
6612
6613
6614
6615
6616
6617
6618
6619
6620
6621
6622
6623
6624
6625
6626
6627
6628
6629
6630
6631
6632
6633
6634
6635
6636
6637
6638
6639
6640
6641
6642
6643
6644
6645
6646
6647
6648
6649
6650
6651
6652
6653
6654
6655
6656
6657
6658
6659
6660
6661
6662
6663
6664
6665
6666
6667
6668
6669
6670
6671
6672
6673
6674
6675
6676
6677
6678
6679
6680
6681
6682
6683
6684
6685
6686
6687
6688
6689
6690
6691
6692
6693
6694
6695
6696
6697
6698
6699
6700
6701
6702
6703
6704
6705
6706
6707
6708
6709
6710
6711
6712
6713
6714
6715
6716
6717
6718
6719
6720
6721
6722
6723
6724
6725
6726
6727
6728
6729
6730
6731
6732
6733
6734
6735
6736
6737
6738
6739
6740
6741
6742
6743
6744
6745
6746
6747
6748
6749
6750
6751
6752
6753
6754
6755
6756
6757
6758
6759
6760
6761
6762
6763
6764
6765
6766
6767
6768
6769
6770
6771
6772
6773
6774
6775
6776
6777
6778
6779
6780
6781
6782
6783
6784
6785
6786
6787
6788
6789
6790
6791
6792
6793
6794
6795
6796
6797
6798
6799
6800
6801
6802
6803
6804
6805
6806
6807
6808
6809
6810
6811
6812
6813
6814
6815
6816
6817
6818
6819
6820
6821
6822
6823
6824
6825
6826
6827
6828
6829
6830
6831
6832
6833
6834
6835
6836
6837
6838
6839
6840
6841
6842
6843
6844
6845
6846
6847
6848
6849
6850
6851
6852
6853
6854
6855
6856
6857
6858
6859
6860
6861
6862
6863
6864
6865
6866
6867
6868
6869
6870
6871
6872
6873
6874
6875
6876
6877
6878
6879
6880
6881
6882
6883
6884
6885
6886
6887
6888
6889
6890
6891
6892
6893
6894
6895
6896
6897
6898
6899
6900
6901
6902
6903
6904
6905
6906
6907
6908
6909
6910
6911
6912
6913
6914
6915
6916
6917
6918
6919
6920
6921
6922
6923
6924
6925
6926
6927
6928
6929
6930
6931
6932
6933
6934
6935
6936
6937
6938
6939
6940
6941
6942
6943
6944
6945
6946
6947
6948
6949
6950
6951
6952
6953
6954
6955
6956
6957
6958
6959
6960
6961
6962
6963
6964
6965
6966
6967
6968
6969
6970
6971
6972
6973
6974
6975
6976
6977
6978
6979
6980
6981
6982
6983
6984
6985
6986
6987
6988
6989
6990
6991
6992
6993
6994
6995
6996
6997
6998
6999
7000
7001
7002
7003
7004
7005
7006
7007
7008
7009
7010
7011
7012
7013
7014
7015
7016
7017
7018
7019
7020
7021
7022
7023
7024
7025
7026
7027
7028
7029
7030
7031
7032
7033
7034
7035
7036
7037
7038
7039
7040
7041
7042
7043
7044
7045
7046
7047
7048
7049
7050
7051
7052
7053
7054
7055
7056
7057
7058
7059
7060
7061
7062
7063
7064
7065
7066
7067
7068
7069
7070
7071
7072
7073
7074
7075
7076
7077
7078
7079
7080
7081
7082
7083
7084
7085
7086
7087
7088
7089
7090
7091
7092
7093
7094
7095
7096
7097
7098
7099
7100
7101
7102
7103
7104
7105
7106
7107
7108
7109
7110
7111
7112
7113
7114
7115
7116
7117
7118
7119
7120
7121
7122
7123
7124
7125
7126
7127
7128
7129
7130
7131
7132
7133
7134
7135
7136
7137
7138
7139
7140
7141
7142
7143
7144
7145
7146
7147
7148
7149
7150
7151
7152
7153
7154
7155
7156
7157
7158
7159
7160
7161
7162
7163
7164
7165
7166
7167
7168
7169
7170
7171
7172
7173
7174
7175
7176
7177
7178
7179
7180
7181
7182
7183
7184
7185
7186
7187
7188
7189
7190
7191
7192
// geometry.asy

// Copyright (C) 2007
// Author: Philippe IVALDI 2007/09/01
// http://www.piprime.fr/

// This program is free software ; you can redistribute it and/or modify
// it under the terms of the GNU Lesser General Public License as published by
// the Free Software Foundation ; either version 3 of the License, or
// (at your option) any later version.

// This program is distributed in the hope that it will be useful, but
// WITHOUT ANY WARRANTY ; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// Lesser General Public License for more details.

// You should have received a copy of the GNU Lesser General Public License
// along with this program ; if not, write to the Free Software
// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA

// COMMENTARY:
// An Asymptote geometry module.

// THANKS:
// Special thanks to Olivier Guibé for his help in mathematical issues.

// BUGS:

// CODE:

import math;
import markers;

// A rotation in the direction dir limited to [-90,90]
// This is useful for rotating text along a line in the direction dir.
private transform rotate(explicit pair dir)
{
  real angle=degrees(dir);
  if(angle > 90 && angle < 270) angle -= 180;
  return rotate(angle);
} 

// *=======================================================*
// *........................HEADER.........................*
/*<asyxml><variable type="real" signature="epsgeo"><code></asyxml>*/
real epsgeo = 10 * sqrt(realEpsilon);/*<asyxml></code><documentation>Variable used in the approximate calculations.</documentation></variable></asyxml>*/

/*<asyxml><function type="void" signature="addMargins(picture,real,real,real,real)"><code></asyxml>*/
void addMargins(picture pic = currentpicture,
                real lmargin = 0, real bmargin = 0,
                real rmargin = lmargin, real tmargin = bmargin,
                bool rigid = true, bool allObject = true)
{/*<asyxml></code><documentation>Add margins to 'pic' with respect to
   the current bounding box of 'pic'.
   If 'rigid' is false, margins are added iff an infinite curve will
   be prolonged on the margin.
   If 'allObject' is false, fixed - size objects (such as labels and
   arrowheads) will be ignored.</documentation></function></asyxml>*/
  pair m = allObject ? truepoint(pic, SW) : point(pic, SW);
  pair M = allObject ? truepoint(pic, NE) : point(pic, NE);
  if(rigid) {
    draw(m - inverse(pic.calculateTransform()) * (lmargin, bmargin), invisible);
    draw(M + inverse(pic.calculateTransform()) * (rmargin, tmargin), invisible);
  } else pic.addBox(m, M, -(lmargin, bmargin), (rmargin, tmargin));
}

real approximate(real t)
{
  real ot = t;
  if(abs(t - ceil(t)) < epsgeo) ot = ceil(t);
  else if(abs(t - floor(t)) < epsgeo) ot = floor(t);
  return ot;
}

real[] approximate(real[] T)
{
  return map(approximate, T);
}

/*<asyxml><function type="real" signature="binomial(real,real)"><code></asyxml>*/
real binomial(real n, real k)
{/*<asyxml></code><documentation>Return n!/((n - k)!*k!)</documentation></function></asyxml>*/
  return gamma(n + 1)/(gamma(n - k + 1) * gamma(k + 1));
}

/*<asyxml><function type="real" signature="rf(real,real,real)"><code></asyxml>*/
real rf(real x, real y, real z)
{/*<asyxml></code><documentation>Computes Carlson's elliptic integral of the first kind.
   x, y, and z must be non negative, and at most one can be zero.</documentation></function></asyxml>*/
  real ERRTOL = 0.0025,
    TINY = 1.5e-38,
    BIG = 3e37,
    THIRD = 1/3,
    C1 = 1/24,
    C2 = 0.1,
    C3 = 3/44,
    C4 = 1/14;
  real alamb, ave, delx, dely, delz, e2, e3, sqrtx, sqrty, sqrtz, xt, yt, zt;
  if(min(x, y, z) < 0 || min(x + y, x + z, y + z) < TINY ||
     max(x, y, z) > BIG) abort("rf: invalid arguments.");
  xt = x;
  yt = y;
  zt = z;
  do {
    sqrtx = sqrt(xt);
    sqrty = sqrt(yt);
    sqrtz = sqrt(zt);
    alamb = sqrtx * (sqrty + sqrtz) + sqrty * sqrtz;
    xt = 0.25 * (xt + alamb);
    yt = 0.25 * (yt + alamb);
    zt = 0.25 * (zt + alamb);
    ave = THIRD * (xt + yt + zt);
    delx = (ave - xt)/ave;
    dely = (ave - yt)/ave;
    delz = (ave - zt)/ave;
  } while(max(fabs(delx), fabs(dely), fabs(delz)) > ERRTOL);
  e2 = delx * dely - delz * delz;
  e3 = delx * dely * delz;
  return (1.0 + (C1 * e2 - C2 - C3 * e3) * e2 + C4 * e3)/sqrt(ave);
}

/*<asyxml><function type="real" signature="rd(real,real,real)"><code></asyxml>*/
real rd(real x, real y, real z)
{/*<asyxml></code><documentation>Computes Carlson's elliptic integral of the second kind.
   x and y must be positive, and at most one can be zero.
   z must be non negative.</documentation></function></asyxml>*/
  real ERRTOL = 0.0015,
    TINY = 1e-25,
    BIG = 4.5 * 10.0^21,
    C1 = (3/14),
    C2 = (1/6),
    C3 = (9/22),
    C4 = (3/26),
    C5 = (0.25 * C3),
    C6 = (1.5 * C4);
  real alamb, ave, delx, dely, delz, ea, eb, ec, ed, ee, fac, sqrtx, sqrty,
    sqrtz, sum, xt, yt, zt;
  if (min(x, y) < 0 || min(x + y, z) < TINY || max(x, y, z) > BIG)
    abort("rd: invalid arguments");
  xt = x;
  yt = y;
  zt = z;
  sum = 0;
  fac = 1;
  do {
    sqrtx = sqrt(xt);
    sqrty = sqrt(yt);
    sqrtz = sqrt(zt);
    alamb = sqrtx * (sqrty + sqrtz) + sqrty * sqrtz;
    sum += fac/(sqrtz * (zt + alamb));
    fac = 0.25 * fac;
    xt = 0.25 * (xt + alamb);
    yt = 0.25 * (yt + alamb);
    zt = 0.25 * (zt + alamb);
    ave = 0.2 * (xt + yt + 3.0 * zt);
    delx = (ave - xt)/ave;
    dely = (ave - yt)/ave;
    delz = (ave - zt)/ave;
  } while (max(fabs(delx), fabs(dely), fabs(delz)) > ERRTOL);
  ea = delx * dely;
  eb = delz * delz;
  ec = ea - eb;
  ed = ea - 6 * eb;
  ee = ed + ec + ec;
  return 3 * sum + fac * (1.0 + ed * (-C1 + C5 * ed - C6 * delz * ee)
                    +delz * (C2 * ee + delz * (-C3 * ec + delz * C4 * ea)))/(ave * sqrt(ave));
}

/*<asyxml><function type="real" signature="elle(real,real)"><code></asyxml>*/
real elle(real phi, real k)
{/*<asyxml></code><documentation>Legendre elliptic integral of the 2nd kind,
   evaluated using Carlson's functions RD and RF.
   The argument ranges are -infinity < phi < +infinity, 0 <= k * sin(phi) <= 1.</documentation></function></asyxml>*/
  real result;
  if (phi >= 0 && phi <= pi/2) {
    real cc, q, s;
    s = sin(phi);
    cc = cos(phi)^2;
    q = (1 - s * k) * (1 + s * k);
    result = s * (rf(cc, q, 1) - (s * k)^2 * rd(cc, q, 1)/3);
  } else
    if (phi <= pi && phi >= 0) {
      result = 2 * elle(pi/2, k) - elle(pi - phi, k);
    } else
      if (phi <= 3 * pi/2 && phi >= 0) {
        result = 2 * elle(pi/2, k) + elle(phi - pi, k);
      } else
        if (phi <= 2 * pi && phi >= 0) {
          result = 4 * elle(pi/2, k) - elle(2 * pi - phi, k);
        } else
          if (phi >= 0) {
            int nb = floor(0.5 * phi/pi);
            result = nb * elle(2 * pi, k) + elle(phi%(2 * pi), k);
          } else result = -elle(-phi, k);
  return result;
}

/*<asyxml><function type="pair[]" signature="intersectionpoints(pair,pair,real,real,real,real,real,real)"><code></asyxml>*/
pair[] intersectionpoints(pair A, pair B,
                          real a, real b, real c, real d, real f, real g)
{/*<asyxml></code><documentation>Intersection points with the line (AB) and the quadric curve
   a * x^2 + b * x * y + c * y^2 + d * x + f * y + g = 0 given in the default coordinate system</documentation></function></asyxml>*/
  pair[] op;
  real ap = B.y - A.y,
    bpp = A.x - B.x,
    cp = A.y * B.x - A.x * B.y;
  real sol[];
  if (abs(ap) > epsgeo) {
    real aa = ap * c + a * bpp^2/ap - b * bpp,
      bb = ap * f - bpp * d + 2 * a * bpp * cp/ap - b * cp,
      cc = ap * g - cp * d + a * cp^2/ap;
    sol = quadraticroots(aa, bb, cc);
    for (int i = 0; i < sol.length; ++i) {
      op.push((-bpp * sol[i]/ap - cp/ap, sol[i]));
    }
  } else {
    real aa = a * bpp,
      bb = d * bpp - b * cp,
      cc = g * bpp - cp * f + c * cp^2/bpp;
    sol = quadraticroots(aa, bb, cc);
    for (int i = 0; i < sol.length; ++i) {
      op.push((sol[i], -cp/bpp));
    }
  }
  return op;
}

/*<asyxml><function type="pair[]" signature="intersectionpoints(pair,pair,real[])"><code></asyxml>*/
pair[] intersectionpoints(pair A, pair B, real[] equation)
{/*<asyxml></code><documentation>Return the intersection points of the line AB with
   the conic whose an equation is
   equation[0] * x^2 + equation[1] * x * y + equation[2] * y^2 + equation[3] * x + equation[4] * y + equation[5] = 0</documentation></function></asyxml>*/
  if(equation.length != 6) abort("intersectionpoints: bad length of array for a conic equation.");
  return intersectionpoints(A, B, equation[0], equation[1], equation[2],
                            equation[3], equation[4], equation[5]);
}
// *........................HEADER.........................*
// *=======================================================*

// *=======================================================*
// *......................COORDINATES......................*

real EPS = sqrt(realEpsilon);

/*<asyxml><typedef type = "convert" return = "pair" params = "pair"><code></asyxml>*/
typedef pair convert(pair);/*<asyxml></code><documentation>Function type to convert pair in an other coordinate system.</documentation></typedef></asyxml>*/
/*<asyxml><typedef type = "abs" return = "real" params = "pair"><code></asyxml>*/
typedef real abs(pair);/*<asyxml></code><documentation>Function type to calculate modulus of pair.</documentation></typedef></asyxml>*/
/*<asyxml><typedef type = "dot" return = "real" params = "pair, pair"><code></asyxml>*/
typedef real dot(pair, pair);/*<asyxml></code><documentation>Function type to calculate dot product.</documentation></typedef></asyxml>*/
/*<asyxml><typedef type = "polar" return = "pair" params = "real, real"><code></asyxml>*/
typedef pair polar(real, real);/*<asyxml></code><documentation>Function type to calculate the coordinates from the polar coordinates.</documentation></typedef></asyxml>*/

/*<asyxml><struct signature="coordsys"><code></asyxml>*/
struct coordsys
{/*<asyxml></code><documentation>This structure represents a coordinate system in the plane.</documentation></asyxml>*/
  /*<asyxml><method type = "pair" signature="relativetodefault(pair)"><code></asyxml>*/
  restricted convert relativetodefault = new pair(pair m){return m;};/*<asyxml></code><documentation>Convert a pair given relatively to this coordinate system to
                                                                     the pair relatively to the default coordinate system.</documentation></method></asyxml>*/
  /*<asyxml><method type = "pair" signature="defaulttorelativet(pair)"><code></asyxml>*/
  restricted convert defaulttorelative = new pair(pair m){return m;};/*<asyxml></code><documentation>Convert a pair given relatively to the default coordinate system to
                                                                     the pair relatively to this coordinate system.</documentation></method></asyxml>*/
  /*<asyxml><method type = "real" signature="dot(pair,pair)"><code></asyxml>*/
  restricted dot dot = new real(pair m, pair n){return dot(m, n);};/*<asyxml></code><documentation>Return the dot product of this coordinate system.</documentation></method></asyxml>*/
  /*<asyxml><method type = "real" signature="abs(pair)"><code></asyxml>*/
  restricted abs abs = new real(pair m){return abs(m);};/*<asyxml></code><documentation>Return the modulus of a pair in this coordinate system.</documentation></method></asyxml>*/
  /*<asyxml><method type = "pair" signature="polar(real,real)"><code></asyxml>*/
  restricted polar polar = new pair(real r, real a){return (r * cos(a), r * sin(a));};/*<asyxml></code><documentation>Polar coordinates routine of this coordinate system.</documentation></method></asyxml>*/
  /*<asyxml><property type = "pair" signature="O,i,j"><code></asyxml>*/
  restricted pair O = (0, 0), i = (1, 0), j = (0, 1);/*<asyxml></code><documentation>Origin and units vector.</documentation></property></asyxml>*/
  /*<asyxml><method type = "void" signature="init(convert,convert,polar,dot)"><code></asyxml>*/
  void init(convert rtd, convert dtr,
            polar polar, dot dot)
  {/*<asyxml></code><documentation>The default constructor of the coordinate system.</documentation></method></asyxml>*/
    this.relativetodefault = rtd;
    this.defaulttorelative = dtr;
    this.polar = polar;
    this.dot = dot;
    this.abs = new real(pair m){return sqrt(dot(m, m));};;
    this.O = rtd((0, 0));
    this.i = rtd((1, 0)) - O;
    this.j = rtd((0, 1)) - O;
  }
}/*<asyxml></struct></asyxml>*/

/*<asyxml><operator type = "bool" signature="==(coordsys,coordsys)"><code></asyxml>*/
bool operator ==(coordsys c1, coordsys c2)
{/*<asyxml></code><documentation>Return true iff the coordinate system have the same origin and units vector.</documentation></operator></asyxml>*/
  return c1.O == c2.O && c1.i == c2.i && c1.j == c2.j;
}

/*<asyxml><function type="coordsys" signature="cartesiansystem(pair,pair,pair)"><code></asyxml>*/
coordsys cartesiansystem(pair O = (0, 0), pair i, pair j)
{/*<asyxml></code><documentation>Return the Cartesian coordinate system (O, i, j).</documentation></function></asyxml>*/
  coordsys R;
  real[][] P = {{0, 0}, {0, 0}};
  real[][] iP;
  P[0][0] = i.x;
  P[0][1] = j.x;
  P[1][0] = i.y;
  P[1][1] = j.y;
  iP = inverse(P);
  real ni = abs(i);
  real nj = abs(j);
  real ij = angle(j) - angle(i);

  pair rtd(pair m)
  {
    return O + (P[0][0] * m.x + P[0][1] * m.y, P[1][0] * m.x + P[1][1] * m.y);
  }

  pair dtr(pair m)
  {
    m-=O;
    return (iP[0][0] * m.x + iP[0][1] * m.y, iP[1][0] * m.x + iP[1][1] * m.y);
  }

  pair polar(real r, real a)
  {
    real ca = sin(ij - a)/(ni * sin(ij));
    real sa = sin(a)/(nj * sin(ij));
    return r * (ca, sa);
  }

  real tdot(pair m, pair n)
  {
    return m.x * n.x * ni^2 + m.y * n.y * nj^2 + (m.x * n.y + n.x * m.y) * dot(i, j);
  }

  R.init(rtd, dtr, polar, tdot);
  return R;
}


/*<asyxml><function type="void" signature="show(picture,Label,Label,Label,coordsys,pen,pen,pen,pen,pen)"><code></asyxml>*/
void show(picture pic = currentpicture, Label lo = "$O$",
          Label li = "$\vec{\imath}$",
          Label lj = "$\vec{\jmath}$",
          coordsys R,
          pen dotpen = currentpen, pen xpen = currentpen, pen ypen = xpen,
          pen ipen = red,
          pen jpen = ipen,
          arrowbar arrow = Arrow)
{/*<asyxml></code><documentation>Draw the components (O, i, j, x - axis, y - axis) of 'R'.</documentation></function></asyxml>*/
  unravel R;
  dot(pic, O, dotpen);
  drawline(pic, O, O + i, xpen);
  drawline(pic, O, O + j, ypen);
  draw(pic, li, O--(O + i), ipen, arrow);
  Label lj = lj.copy();
  lj.align(lj.align, unit(I * j));
  draw(pic, lj, O--(O + j), jpen, arrow);
  draw(pic, lj, O--(O + j), jpen, arrow);
  Label lo = lo.copy();
  lo.align(lo.align, -2 * dir(O--O + i, O--O + j));
  lo.p(dotpen);
  label(pic, lo, O);
}

/*<asyxml><operator type = "pair" signature="/(pair,coordsys)"><code></asyxml>*/
pair operator /(pair p, coordsys R)
{/*<asyxml></code><documentation>Return the xy - coordinates of 'p' relatively to
   the coordinate system 'R'.
   For example, if R = cartesiansystem((1, 2), (1, 0), (0, 1)), (0, 0)/R is (-1, -2).</documentation></operator></asyxml>*/
  return R.defaulttorelative(p);
}

/*<asyxml><operator type = "pair" signature="*(coordsys,pair)"><code></asyxml>*/
pair operator *(coordsys R, pair p)
{/*<asyxml></code><documentation>Return the coordinates of 'p' given in the
   xy - coordinates 'R'.
   For example, if R = cartesiansystem((1, 2), (1, 0), (0, 1)), R * (0, 0) is (1, 2).</documentation></operator></asyxml>*/
  return R.relativetodefault(p);
}

/*<asyxml><operator type = "path" signature="*(coordsys,path)"><code></asyxml>*/
path operator *(coordsys R, path g)
{/*<asyxml></code><documentation>Return the reconstructed path applying R * pair to each node, pre and post control point of 'g'.</documentation></operator></asyxml>*/
  guide og = R * point(g, 0);
  real l = length(g);
  for(int i = 1; i <= l; ++i)
    {
      pair P = R * point(g, i);
      pair post = R * postcontrol(g, i - 1);
      pair pre = R * precontrol(g, i);
      if(i == l && (cyclic(g)))
        og = og..controls post and pre..cycle;
      else
        og = og..controls post and pre..P;
    }
  return og;
}

/*<asyxml><operator type = "coordsys" signature="*(transform,coordsys)"><code></asyxml>*/
coordsys operator *(transform t,coordsys R)
{/*<asyxml></code><documentation>Provide transform * coordsys.
   Note that shiftless(t) is applied to R.i and R.j.</documentation></operator></asyxml>*/
  coordsys oc;
  oc = cartesiansystem(t * R.O, shiftless(t) * R.i, shiftless(t) * R.j);
  return oc;
}

/*<asyxml><constant type = "coordsys" signature="defaultcoordsys"><code></asyxml>*/
restricted coordsys defaultcoordsys = cartesiansystem(0, (1, 0), (0, 1));/*<asyxml></code><documentation>One can always refer to the default coordinate system using this constant.</documentation></constant></asyxml>*/
/*<asyxml><variable type="coordsys" signature="currentcoordsys"><code></asyxml>*/
coordsys currentcoordsys = defaultcoordsys;/*<asyxml></code><documentation>The coordinate system used by default.</documentation></variable></asyxml>*/

/*<asyxml><struct signature="point"><code></asyxml>*/
struct point
{/*<asyxml></code><documentation>This structure replaces the pair to embed its coordinate system.
   For example, if 'P = point(cartesiansystem((1, 2), i, j), (0, 0))',
   P is equal to the pair (1, 2).</documentation></asyxml>*/
  /*<asyxml><property type = "coordsys" signature="coordsys"><code></asyxml>*/
  coordsys coordsys;/*<asyxml></code><documentation>The coordinate system of this point.</documentation></property><property type = "pair" signature="coordinates"><code></asyxml>*/
  restricted pair coordinates;/*<asyxml></code><documentation>The coordinates of this point relatively to the coordinate system 'coordsys'.</documentation></property><property type = "real" signature="x, y"><code></asyxml>*/
  restricted real x, y;/*<asyxml></code><documentation>The xpart and the ypart of 'coordinates'.</documentation></property></asyxml>*/
  /*<asyxml><method type = "" signature="init(coordsys,pair)"><code><property type = "real" signature="m"><code></asyxml>*/
  real m = 1;/*<asyxml></code><documentation>Used to cast mass<->point.</documentation></property></asyxml>*/
  void init(coordsys R, pair coordinates, real mass)
  {/*<asyxml></code><documentation>The constructor.</documentation></method></asyxml>*/
    this.coordsys = R;
    this.coordinates = coordinates;
    this.x = coordinates.x;
    this.y = coordinates.y;
    this.m = mass;
  }
}/*<asyxml></struct></asyxml>*/

/*<asyxml><function type="point" signature="point(coordsys,pair,real)"><code></asyxml>*/
point point(coordsys R, pair p, real m = 1)
{/*<asyxml></code><documentation>Return the point which has the coodinates 'p' in the
   coordinate system 'R' and the mass 'm'.</documentation></function></asyxml>*/
  point op;
  op.init(R, p, m);
  return op;
}

/*<asyxml><function type="point" signature="point(explicit pair,real)"><code></asyxml>*/
point point(explicit pair p, real m)
{/*<asyxml></code><documentation>Return the point which has the coodinates 'p' in the current
   coordinate system and the mass 'm'.</documentation></function></asyxml>*/
  point op;
  op.init(currentcoordsys, p, m);
  return op;
}

/*<asyxml><function type="point" signature="point(coordsys,explicit point,real)"><code></asyxml>*/
point point(coordsys R, explicit point M, real m = M.m)
{/*<asyxml></code><documentation>Return the point of 'R' which has the coordinates of 'M' and the mass 'm'.
   Do not confuse this routine with the further routine 'changecoordsys'.</documentation></function></asyxml>*/
  point op;
  op.init(R, M.coordinates, M.m);
  return op;
}

/*<asyxml><function type="point" signature="changecoordsys(coordsys,point)"><code></asyxml>*/
point changecoordsys(coordsys R, point M)
{/*<asyxml></code><documentation>Return the point 'M' in the coordinate system 'coordsys'.
   In other words, the returned point marks the same plot as 'M' does.</documentation></function></asyxml>*/
  point op;
  coordsys mco = M.coordsys;
  op.init(R, R.defaulttorelative(mco.relativetodefault(M.coordinates)), M.m);
  return op;
}

/*<asyxml><function type="pair" signature="pair coordinates(point)"><code></asyxml>*/
pair coordinates(point M)
{/*<asyxml></code><documentation>Return the coordinates of 'M' in its coordinate system.</documentation></function></asyxml>*/
  return M.coordinates;
}

/*<asyxml><function type="bool" signature="bool samecoordsys(bool...point[])"><code></asyxml>*/
bool samecoordsys(bool warn = true ... point[] M)
{/*<asyxml></code><documentation>Return true iff all the points have the same coordinate system.
   If 'warn' is true and the coordinate systems are different, a warning is sent.</documentation></function></asyxml>*/
  bool ret = true;
  coordsys t = M[0].coordsys;
  for (int i = 1; i < M.length; ++i) {
    ret = (t == M[i].coordsys);
    if(!ret) break;
    t = M[i].coordsys;
  }
  if(warn && !ret)
    warning("coodinatesystem",
            "the coordinate system of two objects are not the same.
The operation will be done relative to the default coordinate system.");
  return ret;
}

/*<asyxml><function type="point[]" signature="standardizecoordsys(coordsys,bool...point[])"><code></asyxml>*/
point[] standardizecoordsys(coordsys R = currentcoordsys,
                            bool warn = true ... point[] M)
{/*<asyxml></code><documentation>Return the points with the same coordinate system 'R'.
   If 'warn' is true and the coordinate systems are different, a warning is sent.</documentation></function></asyxml>*/
  point[] op = new point[];
  op = M;
  if(!samecoordsys(warn ... M))
    for (int i = 1; i < M.length; ++i)
      op[i] = changecoordsys(R, M[i]);
  return op;
}

/*<asyxml><operator type = "pair" signature="cast(point)"><code></asyxml>*/
pair operator cast(point P)
{/*<asyxml></code><documentation>Cast point to pair.</documentation></operator></asyxml>*/
  return P.coordsys.relativetodefault(P.coordinates);
}

/*<asyxml><operator type = "pair[]" signature="cast(point[])"><code></asyxml>*/
pair[] operator cast(point[] P)
{/*<asyxml></code><documentation>Cast point[] to pair[].</documentation></operator></asyxml>*/
  pair[] op;
  for (int i = 0; i < P.length; ++i) {
    op.push((pair)P[i]);
  }
  return op;
}

/*<asyxml><operator type = "point" signature="cast(pair)"><code></asyxml>*/
point operator cast(pair p)
{/*<asyxml></code><documentation>Cast pair to point relatively to the current coordinate
   system 'currentcoordsys'.</documentation></operator></asyxml>*/
  return point(currentcoordsys, p);
}

/*<asyxml><operator type = "point[]" signature="cast(pair[])"><code></asyxml>*/
point[] operator cast(pair[] p)
{/*<asyxml></code><documentation>Cast pair[] to point[] relatively to the current coordinate
   system 'currentcoordsys'.</documentation></operator></asyxml>*/
  pair[] op;
  for (int i = 0; i < p.length; ++i) {
    op.push((point)p[i]);
  }
  return op;
}

/*<asyxml><function type="pair" signature="locate(point)"><code></asyxml>*/
pair locate(point P)
{/*<asyxml></code><documentation>Return the coordinates of 'P' in the default coordinate system.</documentation></function></asyxml>*/
  return P.coordsys * P.coordinates;
}

/*<asyxml><function type="point" signature="locate(pair)"><code></asyxml>*/
point locate(pair p)
{/*<asyxml></code><documentation>Return the point in the current coordinate system 'currentcoordsys'.</documentation></function></asyxml>*/
  return p; //automatic casting 'pair to point'.
}

/*<asyxml><operator type = "point" signature="*(real,explicit point)"><code></asyxml>*/
point operator *(real x, explicit point P)
{/*<asyxml></code><documentation>Multiply the coordinates (not the mass) of 'P' by 'x'.</documentation></operator></asyxml>*/
  return point(P.coordsys, x * P.coordinates, P.m);
}

/*<asyxml><operator type = "point" signature="/(explicit point,real)"><code></asyxml>*/
point operator /(explicit point P, real x)
{/*<asyxml></code><documentation>Divide the coordinates (not the mass) of 'P' by 'x'.</documentation></operator></asyxml>*/
  return point(P.coordsys, P.coordinates/x, P.m);
}

/*<asyxml><operator type = "point" signature="/(real,explicit point)"><code></asyxml>*/
point operator /(real x, explicit point P)
{/*<asyxml></code><documentation></documentation></operator></asyxml>*/
  return point(P.coordsys, x/P.coordinates, P.m);
}

/*<asyxml><operator type = "point" signature="-(explicit point)"><code></asyxml>*/
point operator -(explicit point P)
{/*<asyxml></code><documentation>-P. The mass is inchanged.</documentation></operator></asyxml>*/
  return point(P.coordsys, -P.coordinates, P.m);
}

/*<asyxml><operator type = "point" signature="+(explicit point,explicit point)"><code></asyxml>*/
point operator +(explicit point P1, explicit point P2)
{/*<asyxml></code><documentation>Provide 'point + point'.
   If the two points haven't the same coordinate system, a warning is sent and the
   returned point has the default coordinate system 'defaultcoordsys'.
   The masses are added.</documentation></operator></asyxml>*/
  point[] P = standardizecoordsys(P1, P2);
  coordsys R = P[0].coordsys;
  return point(R, P[0].coordinates + P[1].coordinates, P1.m + P2.m);
}

/*<asyxml><operator type = "point" signature="+(explicit point,explicit pair)"><code></asyxml>*/
point operator +(explicit point P1, explicit pair p2)
{/*<asyxml></code><documentation>Provide 'point + pair'.
   The pair 'p2' is supposed to be coordinates relatively to the coordinates system of 'P1'.
   The mass is not changed.</documentation></operator></asyxml>*/
  coordsys R = currentcoordsys;
  return point(R, P1.coordinates + point(R, p2).coordinates, P1.m);
}
point operator +(explicit pair p1, explicit point p2)
{
  return p2 + p1;
}

/*<asyxml><operator type = "point" signature="-(explicit point,explicit point)"><code></asyxml>*/
point operator -(explicit point P1, explicit point P2)
{/*<asyxml></code><documentation>Provide 'point - point'.</documentation></operator></asyxml>*/
  return P1 + (-P2);
}

/*<asyxml><operator type = "point" signature="-(explicit point,explicit pair)"><code></asyxml>*/
point operator -(explicit point P1, explicit pair p2)
{/*<asyxml></code><documentation>Provide 'point - pair'.
   The pair 'p2' is supposed to be coordinates relatively to the coordinates system of 'P1'.</documentation></operator></asyxml>*/
  return P1 + (-p2);
}
point operator -(explicit pair p1, explicit point P2)
{
  return p1 + (-P2);
}

/*<asyxml><operator type = "point" signature="*(transform,explicit point)"><code></asyxml>*/
point operator *(transform t, explicit point P)
{/*<asyxml></code><documentation>Provide 'transform * point'.
   Note that the transforms scale, xscale, yscale and rotate are carried out relatively
   the default coordinate system 'defaultcoordsys' which is not desired for point
   defined in an other coordinate system.
   On can use scale(real, point), xscale(real, point), yscale(real, point), rotate(real, point),
   scaleO(real), xscaleO(real), yscaleO(real) and rotateO(real) (described further)
   to change the coordinate system of reference.</documentation></operator></asyxml>*/
  coordsys R = P.coordsys;
  return point(R, (t * locate(P))/R, P.m);
}

/*<asyxml><operator type = "point" signature="*(explicit point,explicit point)"><code></asyxml>*/
point operator *(explicit point P1, explicit point P2)
{/*<asyxml></code><documentation>Provide 'point * point'.
   The resulted mass is the mass of P2</documentation></operator></asyxml>*/
  point[] P = standardizecoordsys(P1, P2);
  coordsys R = P[0].coordsys;
  return point(R, P[0].coordinates * P[1].coordinates, P2.m);
}

/*<asyxml><operator type = "point" signature="*(explicit point,explicit pair)"><code></asyxml>*/
point operator *(explicit point P1, explicit pair p2)
{/*<asyxml></code><documentation>Provide 'point * pair'.
   The pair 'p2' is supposed to be the coordinates of
   the point in the coordinates system of 'P1'.
   'pair * point' is also defined.</documentation></operator></asyxml>*/
  point P = point(P1.coordsys, p2, P1.m);
  return P1 * P;
}
point operator *(explicit pair p1, explicit point p2)
{
  return p2 * p1;
}

/*<asyxml><operator type = "bool" signature="==(explicit point,explicit point)"><code></asyxml>*/
bool operator ==(explicit point M, explicit point N)
{/*<asyxml></code><documentation>Provide the test 'M == N' wish returns true iff MN < EPS</documentation></operator></asyxml>*/
  return abs(locate(M) - locate(N)) < EPS;
}

/*<asyxml><operator type = "bool" signature="!=(explicit point,explicit point)"><code></asyxml>*/
bool operator !=(explicit point M, explicit point N)
{/*<asyxml></code><documentation>Provide the test 'M != N' wish return true iff MN >= EPS</documentation></operator></asyxml>*/
  return !(M == N);
}

/*<asyxml><operator type = "guide" signature="cast(point)"><code></asyxml>*/
guide operator cast(point p)
{/*<asyxml></code><documentation>Cast point to guide.</documentation></operator></asyxml>*/
  return locate(p);
}

/*<asyxml><operator type = "path" signature="cast(point)"><code></asyxml>*/
path operator cast(point p)
{/*<asyxml></code><documentation>Cast point to path.</documentation></operator></asyxml>*/
  return locate(p);
}

/*<asyxml><function type="void" signature="dot(picture,Label,explicit point,align,string,pen)"><code></asyxml>*/
void dot(picture pic = currentpicture, Label L, explicit point Z,
         align align = NoAlign,
         string format = defaultformat, pen p = currentpen)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  Label L = L.copy();
  L.position(locate(Z));
  if(L.s == "") {
    if(format == "") format = defaultformat;
    L.s = "("+format(format, Z.x)+", "+format(format, Z.y)+")";
  }
  L.align(align, E);
  L.p(p);
  dot(pic, locate(Z), p);
  add(pic, L);
}

/*<asyxml><function type="real" signature="abs(coordsys,pair)"><code></asyxml>*/
real abs(coordsys R, pair m)
{/*<asyxml></code><documentation>Return the modulus |m| in the coordinate system 'R'.</documentation></function></asyxml>*/
  return R.abs(m);
}

/*<asyxml><function type="real" signature="abs(explicit point)"><code></asyxml>*/
real abs(explicit point M)
{/*<asyxml></code><documentation>Return the modulus |M| in its coordinate system.</documentation></function></asyxml>*/
  return M.coordsys.abs(M.coordinates);
}

/*<asyxml><function type="real" signature="length(explicit point)"><code></asyxml>*/
real length(explicit point M)
{/*<asyxml></code><documentation>Return the modulus |M| in its coordinate system (same as 'abs').</documentation></function></asyxml>*/
  return M.coordsys.abs(M.coordinates);
}

/*<asyxml><function type="point" signature="conj(explicit point)"><code></asyxml>*/
point conj(explicit point M)
{/*<asyxml></code><documentation>Conjugate.</documentation></function></asyxml>*/
  return point(M.coordsys, conj(M.coordinates), M.m);
}

/*<asyxml><function type="real" signature="degrees(explicit point,coordsys,bool)"><code></asyxml>*/
real degrees(explicit point M, coordsys R = M.coordsys, bool warn = true)
{/*<asyxml></code><documentation>Return the angle of M (in degrees) relatively to 'R'.</documentation></function></asyxml>*/
  return (degrees(locate(M) - R.O, warn) - degrees(R.i))%360;
}

/*<asyxml><function type="real" signature="angle(explicit point,coordsys,bool)"><code></asyxml>*/
real angle(explicit point M, coordsys R = M.coordsys, bool warn = true)
{/*<asyxml></code><documentation>Return the angle of M (in radians) relatively to 'R'.</documentation></function></asyxml>*/
  return radians(degrees(M, R, warn));
}

/*<asyxml><function type="bool" signature="finite(explicit point)"><code></asyxml>*/
bool finite(explicit point p)
{/*<asyxml></code><documentation>Avoid to compute 'finite((pair)(infinite_point))'.</documentation></function></asyxml>*/
  return finite(p.coordinates);
}

/*<asyxml><function type="real" signature="dot(point,point)"><code></asyxml>*/
real dot(point A, point B)
{/*<asyxml></code><documentation>Return the dot product in the coordinate system of 'A'.</documentation></function></asyxml>*/
  point[] P = standardizecoordsys(A.coordsys, A, B);
  return P[0].coordsys.dot(P[0].coordinates, P[1].coordinates);
}

/*<asyxml><function type="real" signature="dot(point,explicit pair)"><code></asyxml>*/
real dot(point A, explicit pair B)
{/*<asyxml></code><documentation>Return the dot product in the default coordinate system.
   dot(explicit pair, point) is also defined.</documentation></function></asyxml>*/
  return dot(locate(A), B);
}
real dot(explicit pair A, point B)
{
  return dot(A, locate(B));
}

/*<asyxml><function type="transforms" signature="rotateO(real)"><code></asyxml>*/
transform rotateO(real a)
{/*<asyxml></code><documentation>Rotation around the origin of the current coordinate system.</documentation></function></asyxml>*/
  return rotate(a, currentcoordsys.O);
};

/*<asyxml><function type="transform" signature="projection(point,point)"><code></asyxml>*/
transform projection(point A, point B)
{/*<asyxml></code><documentation>Return the orthogonal projection on the line (AB).</documentation></function></asyxml>*/
  pair dir = unit(locate(A) - locate(B));
  pair a = locate(A);
  real cof = dir.x * a.x + dir.y * a.y;
  real tx = a.x - dir.x * cof;
  real txx = dir.x^2;
  real txy = dir.x * dir.y;
  real ty = a.y - dir.y * cof;
  real tyx = txy;
  real tyy = dir.y^2;
  transform t = (tx, ty, txx, txy, tyx, tyy);
  return t;
}

/*<asyxml><function type="transform" signature="projection(point,point,point,point,bool)"><code></asyxml>*/
transform projection(point A, point B, point C, point D, bool safe = false)
{/*<asyxml></code><documentation>Return the (CD) parallel projection on (AB).
   If 'safe = true' and (AB)//(CD) return the identity.
   If 'safe = false' and (AB)//(CD) return an infinity scaling.</documentation></function></asyxml>*/
  pair a = locate(A);
  pair u = unit(locate(B) - locate(A));
  pair v = unit(locate(D) - locate(C));
  real c = u.x * a.y - u.y * a.x;
  real d = (conj(u) * v).y;
  if (abs(d) < epsgeo) {
    return safe ? identity() : scale(infinity);
  }
  real tx = c * v.x/d;
  real ty = c * v.y/d;
  real txx = u.x * v.y/d;
  real txy = -u.x * v.x/d;
  real tyx = u.y * v.y/d;
  real tyy = -u.y * v.x/d;
  transform t = (tx, ty, txx, txy, tyx, tyy);
  return t;
}

/*<asyxml><function type="transform" signature="scale(real,point)"><code></asyxml>*/
transform scale(real k, point M)
{/*<asyxml></code><documentation>Homothety.</documentation></function></asyxml>*/
  pair P = locate(M);
  return shift(P) * scale(k) * shift(-P);
}

/*<asyxml><function type="transform" signature="xscale(real,point)"><code></asyxml>*/
transform xscale(real k, point M)
{/*<asyxml></code><documentation>xscale from 'M' relatively to the x - axis of the coordinate system of 'M'.</documentation></function></asyxml>*/
  pair P = locate(M);
  real a = degrees(M.coordsys.i);
  return (shift(P) * rotate(a)) * xscale(k) * (rotate(-a) * shift(-P));
}

/*<asyxml><function type="transform" signature="yscale(real,point)"><code></asyxml>*/
transform yscale(real k, point M)
{/*<asyxml></code><documentation>yscale from 'M' relatively to the y - axis of the coordinate system of 'M'.</documentation></function></asyxml>*/
  pair P = locate(M);
  real a = degrees(M.coordsys.j) - 90;
  return (shift(P) * rotate(a)) * yscale(k) * (rotate(-a) * shift(-P));
}

/*<asyxml><function type="transform" signature="scale(real,point,point,point,point,bool)"><code></asyxml>*/
transform scale(real k, point A, point B, point C, point D, bool safe = false)
{/*<asyxml></code><documentation><url href = "http://fr.wikipedia.org/wiki/Affinit%C3%A9_%28math%C3%A9matiques%29"/>
   (help me for English translation...)
   If 'safe = true' and (AB)//(CD) return the identity.
   If 'safe = false' and (AB)//(CD) return a infinity scaling.</documentation></function></asyxml>*/
  pair a = locate(A);
  pair u = unit(locate(B) - locate(A));
  pair v = unit(locate(D) - locate(C));
  real c = u.x * a.y - u.y * a.x;
  real d = (conj(u) * v).y;
  real d = (conj(u) * v).y;
  if (abs(d) < epsgeo) {
    return safe ? identity() : scale(infinity);
  }
  real tx = (1 - k) * c * v.x/d;
  real ty = (1 - k) * c * v.y/d;
  real txx = (1 - k) * u.x * v.y/d + k;
  real txy = (k - 1) * u.x * v.x/d;
  real tyx = (1 - k) * u.y * v.y/d;
  real tyy = (k - 1) * u.y * v.x/d + k;
  transform t = (tx, ty, txx, txy, tyx, tyy);
  return t;
}

/*<asyxml><function type="transform" signature="scaleO(real)"><code></asyxml>*/
transform scaleO(real x)
{/*<asyxml></code><documentation>Homothety from the origin of the current coordinate system.</documentation></function></asyxml>*/
  return scale(x, (0, 0));
}

/*<asyxml><function type="transform" signature="xscaleO(real)"><code></asyxml>*/
transform xscaleO(real x)
{/*<asyxml></code><documentation>xscale from the origin and relatively to the current coordinate system.</documentation></function></asyxml>*/
  return scale(x, (0, 0), (0, 1), (0, 0), (1, 0));
}

/*<asyxml><function type="transform" signature="yscaleO(real)"><code></asyxml>*/
transform yscaleO(real x)
{/*<asyxml></code><documentation>yscale from the origin and relatively to the current coordinate system.</documentation></function></asyxml>*/
  return scale(x, (0, 0), (1, 0), (0, 0), (0, 1));
}

/*<asyxml><struct signature="vector"><code></asyxml>*/
struct vector
{/*<asyxml></code><documentation>Like a point but casting to pair, adding etc does not take account
   of the origin of the coordinate system.</documentation><property type = "point" signature="v"><code></asyxml>*/
  point v;/*<asyxml></code><documentation>Coordinates as a point (embed coordinate system and pair).</documentation></property></asyxml>*/
}/*<asyxml></struct></asyxml>*/

/*<asyxml><operator type = "point" signature="cast(vector)"><code></asyxml>*/
point operator cast(vector v)
{/*<asyxml></code><documentation>Cast vector 'v' to point 'M' so that OM = v.</documentation></operator></asyxml>*/
  return v.v;
}

/*<asyxml><operator type = "vector" signature="cast(pair)"><code></asyxml>*/
vector operator cast(pair v)
{/*<asyxml></code><documentation>Cast pair to vector relatively to the current coordinate
   system 'currentcoordsys'.</documentation></operator></asyxml>*/
  vector ov;
  ov.v = point(currentcoordsys, v);
  return ov;
}

/*<asyxml><operator type = "vector" signature="cast(explicit point)"><code></asyxml>*/
vector operator cast(explicit point v)
{/*<asyxml></code><documentation>A point can be interpreted like a vector using the code
   '(vector)a_point'.</documentation></operator></asyxml>*/
  vector ov;
  ov.v = v;
  return ov;
}

/*<asyxml><operator type = "pair" signature="cast(explicit vector)"><code></asyxml>*/
pair operator cast(explicit vector v)
{/*<asyxml></code><documentation>Cast vector to pair (the coordinates of 'v' in the default coordinate system).</documentation></operator></asyxml>*/
  return locate(v.v) - v.v.coordsys.O;
}

/*<asyxml><operator type = "align" signature="cast(vector)"><code></asyxml>*/
align operator cast(vector v)
{/*<asyxml></code><documentation>Cast vector to align.</documentation></operator></asyxml>*/
  return (pair)v;
}

/*<asyxml><function type="vector" signature="vector(coordsys, pair)"><code></asyxml>*/
vector vector(coordsys R = currentcoordsys, pair v)
{/*<asyxml></code><documentation>Return the vector of 'R' which has the coordinates 'v'.</documentation></function></asyxml>*/
  vector ov;
  ov.v = point(R, v);
  return ov;
}

/*<asyxml><function type="vector" signature="vector(point)"><code></asyxml>*/
vector vector(point M)
{/*<asyxml></code><documentation>Return the vector OM, where O is the origin of the coordinate system of 'M'.
   Useful to write 'vector(P - M);' instead of '(vector)(P - M)'.</documentation></function></asyxml>*/
  return M;
}

/*<asyxml><function type="point" signature="point(explicit vector)"><code></asyxml>*/
point point(explicit vector u)
{/*<asyxml></code><documentation>Return the point M so that OM = u, where O is the origin of the coordinate system of 'u'.</documentation></function></asyxml>*/
  return u.v;
}

/*<asyxml><function type="pair" signature="locate(explicit vector)"><code></asyxml>*/
pair locate(explicit vector v)
{/*<asyxml></code><documentation>Return the coordinates of 'v' in the default coordinate system (like casting vector to pair).</documentation></function></asyxml>*/
  return (pair)v;
}

/*<asyxml><function type="void" signature="show(Label,pen,arrowbar)"><code></asyxml>*/
void show(Label L, vector v, pen p = currentpen, arrowbar arrow = Arrow)
{/*<asyxml></code><documentation>Draw the vector v (from the origin of its coordinate system).</documentation></function></asyxml>*/
  coordsys R = v.v.coordsys;
  draw(L, R.O--v.v, p, arrow);
}

/*<asyxml><function type="vector" signature="changecoordsys(coordsys,vector)"><code></asyxml>*/
vector changecoordsys(coordsys R, vector v)
{/*<asyxml></code><documentation>Return the vector 'v' relatively to coordinate system 'R'.</documentation></function></asyxml>*/
  vector ov;
  ov.v = point(R, (locate(v) + R.O)/R);
  return ov;
}

/*<asyxml><operator type = "vector" signature="*(real,explicit vector)"><code></asyxml>*/
vector operator *(real x, explicit vector v)
{/*<asyxml></code><documentation>Provide real * vector.</documentation></operator></asyxml>*/
  return x * v.v;
}

/*<asyxml><operator type = "vector" signature="/(explicit vector,real)"><code></asyxml>*/
vector operator /(explicit vector v, real x)
{/*<asyxml></code><documentation>Provide vector/real</documentation></operator></asyxml>*/
  return v.v/x;
}

/*<asyxml><operator type = "vector" signature="*(transform t,explicit vector)"><code></asyxml>*/
vector operator *(transform t, explicit vector v)
{/*<asyxml></code><documentation>Provide transform * vector.</documentation></operator></asyxml>*/
  return t * v.v;
}

/*<asyxml><operator type = "vector" signature="*(explicit point,explicit vector)"><code></asyxml>*/
vector operator *(explicit point M, explicit vector v)
{/*<asyxml></code><documentation>Provide point * vector</documentation></operator></asyxml>*/
  return M * v.v;
}

/*<asyxml><operator type = "point" signature="+(explicit point,explicit vector)"><code></asyxml>*/
point operator +(point M, explicit vector v)
{/*<asyxml></code><documentation>Return 'M' shifted by 'v'.</documentation></operator></asyxml>*/
  return shift(locate(v)) * M;
}

/*<asyxml><operator type = "point" signature="-(explicit point,explicit vector)"><code></asyxml>*/
point operator -(point M, explicit vector v)
{/*<asyxml></code><documentation>Return 'M' shifted by '-v'.</documentation></operator></asyxml>*/
  return shift(-locate(v)) * M;
}

/*<asyxml><operator type = "vector" signature="-(explicit vector)"><code></asyxml>*/
vector operator -(explicit vector v)
{/*<asyxml></code><documentation>Provide -v.</documentation></operator></asyxml>*/
  return -v.v;
}

/*<asyxml><operator type = "point" signature="+(explicit pair,explicit vector)"><code></asyxml>*/
point operator +(explicit pair m, explicit vector v)
{/*<asyxml></code><documentation>The pair 'm' is supposed to be the coordinates of
   a point in the current coordinates system 'currentcoordsys'.
   Return this point shifted by the vector 'v'.</documentation></operator></asyxml>*/
  return locate(m) + v;
}

/*<asyxml><operator type = "point" signature="-(explicit pair,explicit vector)"><code></asyxml>*/
point operator -(explicit pair m, explicit vector v)
{/*<asyxml></code><documentation>The pair 'm' is supposed to be the coordinates of
   a point in the current coordinates system 'currentcoordsys'.
   Return this point shifted by the vector '-v'.</documentation></operator></asyxml>*/
  return m + (-v);
}

/*<asyxml><operator type = "vector" signature="+(explicit vector,explicit vector)"><code></asyxml>*/
vector operator +(explicit vector v1, explicit vector v2)
{/*<asyxml></code><documentation>Provide vector + vector.
   If the two vector haven't the same coordinate system, the returned
   vector is relative to the default coordinate system (without warning).</documentation></operator></asyxml>*/
  coordsys R = v1.v.coordsys;
  if(samecoordsys(false, v1, v2)){R = defaultcoordsys;}
  return vector(R, (locate(v1) + locate(v2))/R);
}

/*<asyxml><operator type = "vector" signature="-(explicit vector, explicit vector)"><code></asyxml>*/
vector operator -(explicit vector v1, explicit vector v2)
{/*<asyxml></code><documentation>Provide vector - vector.
   If the two vector haven't the same coordinate system, the returned
   vector is relative to the default coordinate system (without warning).</documentation></operator></asyxml>*/
  return v1 + (-v2);
}

/*<asyxml><operator type = "bool" signature="==(explicit vector,explicit vector)"><code></asyxml>*/
bool operator ==(explicit vector u, explicit vector v)
{/*<asyxml></code><documentation>Return true iff |u - v|<EPS.</documentation></operator></asyxml>*/
  return abs(u - v) < EPS;
}

/*<asyxml><function type="bool" signature="collinear(vector,vector)"><code></asyxml>*/
bool collinear(vector u, vector v)
{/*<asyxml></code><documentation>Return 'true' iff the vectors 'u' and 'v' are collinear.</documentation></function></asyxml>*/
  return abs(ypart((conj((pair)u) * (pair)v))) < EPS;
}

/*<asyxml><function type="vector" signature="unit(point)"><code></asyxml>*/
vector unit(point M)
{/*<asyxml></code><documentation>Return the unit vector according to the modulus of its coordinate system.</documentation></function></asyxml>*/
  return M/abs(M);
}

/*<asyxml><function type="vector" signature="unit(vector)"><code></asyxml>*/
vector unit(vector u)
{/*<asyxml></code><documentation>Return the unit vector according to the modulus of its coordinate system.</documentation></function></asyxml>*/
  return u.v/abs(u.v);
}

/*<asyxml><function type="real" signature="degrees(vector,coordsys,bool)"><code></asyxml>*/
real degrees(vector v,
             coordsys R = v.v.coordsys,
             bool warn = true)
{/*<asyxml></code><documentation>Return the angle of 'v' (in degrees) relatively to 'R'.</documentation></function></asyxml>*/
  return (degrees(locate(v), warn) - degrees(R.i))%360;
}

/*<asyxml><function type="real" signature="angle(vector,coordsys,bool)"><code></asyxml>*/
real angle(explicit vector v,
           coordsys R = v.v.coordsys,
           bool warn = true)
{/*<asyxml></code><documentation>Return the angle of 'v' (in radians) relatively to 'R'.</documentation></function></asyxml>*/
  return radians(degrees(v, R, warn));
}

/*<asyxml><function type="vector" signature="conj(explicit vector)"><code></asyxml>*/
vector conj(explicit vector u)
{/*<asyxml></code><documentation>Conjugate.</documentation></function></asyxml>*/
  return conj(u.v);
}

/*<asyxml><function type="transform" signature="rotate(explicit vector)"><code></asyxml>*/
transform rotate(explicit vector dir)
{/*<asyxml></code><documentation>A rotation in the direction 'dir' limited to [-90, 90]
   This is useful for rotating text along a line in the direction dir.
   rotate(explicit point dir) is also defined.
   </documentation></function></asyxml>*/
  return rotate(locate(dir));
}
transform rotate(explicit point dir){return rotate(locate(vector(dir)));}
// *......................COORDINATES......................*
// *=======================================================*

// *=======================================================*
// *.........................BASES.........................*
/*<asyxml><variable type="point" signature="origin"><code></asyxml>*/
point origin = point(defaultcoordsys, (0, 0));/*<asyxml></code><documentation>The origin of the current coordinate system.</documentation></variable></asyxml>*/

/*<asyxml><function type="point" signature="origin(coordsys)"><code></asyxml>*/
point origin(coordsys R = currentcoordsys)
{/*<asyxml></code><documentation>Return the origin of the coordinate system 'R'.</documentation></function></asyxml>*/
  return point(R, (0, 0)); //use automatic casting;
}

/*<asyxml><variable type="real" signature="linemargin"><code></asyxml>*/
real linemargin = 0;/*<asyxml></code><documentation>Margin used to draw lines.</documentation></variable></asyxml>*/
/*<asyxml><function type="real" signature="linemargin()"><code></asyxml>*/
real linemargin()
{/*<asyxml></code><documentation>Return the margin used to draw lines.</documentation></function></asyxml>*/
  return linemargin;
}

/*<asyxml><variable type="pen" signature="addpenline"><code></asyxml>*/
pen addpenline = squarecap;/*<asyxml></code><documentation>Add this property to the drawing pen of "finish" lines.</documentation></variable></asyxml>*/
pen addpenline(pen p) {
  return addpenline + p;
}

/*<asyxml><variable type="pen" signature="addpenarc"><code></asyxml>*/
pen addpenarc = squarecap;/*<asyxml></code><documentation>Add this property to the drawing pen of arcs.</documentation></variable></asyxml>*/
pen addpenarc(pen p) {return addpenarc + p;}

/*<asyxml><variable type="string" signature="defaultmassformat"><code></asyxml>*/
string defaultmassformat = "$\left(%L;%.4g\right)$";/*<asyxml></code><documentation>Format used to construct the default label of masses.</documentation></variable></asyxml>*/

/*<asyxml><function type="int" signature="sgnd(real)"><code></asyxml>*/
int sgnd(real x)
{/*<asyxml></code><documentation>Return the -1 if x < 0, 1 if x >= 0.</documentation></function></asyxml>*/
  return (x == 0) ? 1 : sgn(x);
}
int sgnd(int x)
{
  return (x == 0) ? 1 : sgn(x);
}

/*<asyxml><function type="bool" signature="defined(pair)"><code></asyxml>*/
bool defined(point P)
{/*<asyxml></code><documentation>Return true iff the coordinates of 'P' are finite.</documentation></function></asyxml>*/
  return finite(P.coordinates);
}

/*<asyxml><function type="bool" signature="onpath(picture,path,point,pen)"><code></asyxml>*/
bool onpath(picture pic = currentpicture, path g, point M, pen p = currentpen)
{/*<asyxml></code><documentation>Return true iff 'M' is on the path drawn with the pen 'p' in 'pic'.</documentation></function></asyxml>*/
  transform t = inverse(pic.calculateTransform());
  return intersect(g, shift(locate(M)) * scale(linewidth(p)/2) * t * unitcircle).length > 0;
}

/*<asyxml><function type="bool" signature="sameside(point,point,point)"><code></asyxml>*/
bool sameside(point M, point N, point O)
{/*<asyxml></code><documentation>Return 'true' iff 'M' and 'N' are same side of the point 'O'.</documentation></function></asyxml>*/
  pair m = M, n = N, o = O;
  return dot(m - o, n - o) >= -epsgeo;
}

/*<asyxml><function type="bool" signature="between(point,point,point)"><code></asyxml>*/
bool between(point M, point O, point N)
{/*<asyxml></code><documentation>Return 'true' iff 'O' is between 'M' and 'N'.</documentation></function></asyxml>*/
  return (!sameside(N, M, O) || M == O || N == O);
}


typedef path pathModifier(path);
pathModifier NoModifier = new path(path g){return g;};

private void Drawline(picture pic = currentpicture, Label L = "", pair P, bool dirP = true, pair Q, bool dirQ = true,
                      align align = NoAlign, pen p = currentpen,
                      arrowbar arrow = None,
                      Label legend = "", marker marker = nomarker,
                      pathModifier pathModifier = NoModifier)
{/* Add the two parameters 'dirP' and 'dirQ' to the native routine
    'drawline' of the module 'math'.
    Segment [PQ] will be prolonged in direction of P if 'dirP = true', in
    direction of Q if 'dirQ = true'.
    If 'dirP = dirQ = true', the behavior is that of the native 'drawline'.
    Add all the other parameters of 'Draw'.*/
  pic.add(new void (frame f, transform t, transform T, pair m, pair M) {
      picture opic;
      // Reduce the bounds by the size of the pen.
      m -= min(p) - (linemargin(), linemargin()); M -= max(p) + (linemargin(), linemargin());

      // Calculate the points and direction vector in the transformed space.
      t = t * T;
      pair z = t * P;
      pair q = t * Q;
      pair v = q - z;
      // path g;
      pair ptp, ptq;
      real cp = dirP ? 1:0;
      real cq = dirQ ? 1:0;
      // Handle horizontal and vertical lines.
      if(v.x == 0) {
        if(m.x <= z.x && z.x <= M.x)
          if (dot(v, m - z) < 0) {
            ptp = (z.x, z.y + cp * (m.y - z.y));
            ptq = (z.x, q.y + cq * (M.y - q.y));
          } else {
            ptq = (z.x, q.y + cq * (m.y - q.y));
            ptp = (z.x, z.y + cp * (M.y - z.y));
          }
      } else if(v.y == 0) {
        if (dot(v, m - z) < 0) {
          ptp = (z.x + cp * (m.x - z.x), z.y);
          ptq = (q.x + cq * (M.x - q.x), z.y);
        } else {
          ptq = (q.x + cq * (m.x - q.x), z.y);
          ptp = (z.x + cp * (M.x - z.x), z.y);
        }
      } else {
        // Calculate the maximum and minimum t values allowed for the
        // parametric equation z + t * v
        real mx = (m.x - z.x)/v.x, Mx = (M.x - z.x)/v.x;
        real my = (m.y - z.y)/v.y, My = (M.y - z.y)/v.y;
        real tmin = max(v.x > 0 ? mx : Mx, v.y > 0 ? my : My);
        real tmax = min(v.x > 0 ? Mx : mx, v.y > 0 ? My : my);
        pair pmin = z + tmin * v;
        pair pmax = z + tmax * v;
        if(tmin <= tmax) {
          ptp = z + cp * tmin * v;
          ptq = z + (cq == 0 ? v:tmax * v);
        }
      }
      path g = ptp--ptq;
      if (length(g)>0)
        {
          if(L.s != "") {
            Label lL = L.copy();
            if(L.defaultposition) lL.position(Relative(.9));
            lL.p(p);
            lL.out(opic, g);
          }
          g = pathModifier(g);
          if(linetype(p).length == 0){
            pair m = midpoint(g);
            pen tp;
            tp = dirP ? p : addpenline(p);
            draw(opic, pathModifier(m--ptp), tp);
            tp = dirQ ? p : addpenline(p);
            draw(opic, pathModifier(m--ptq), tp);
          } else {
            draw(opic, g, p);
          }
          marker.markroutine(opic, marker.f, g);
          arrow(opic, g, p, NoMargin);
          add(f, opic.fit());
        }
    });
}

/*<asyxml><function type="void" signature="clipdraw(picture,Label,path,align,pen,arrowbar,arrowbar,real,real,Label,marker)"><code></asyxml>*/
void clipdraw(picture pic = currentpicture, Label L = "", path g,
              align align = NoAlign, pen p = currentpen,
              arrowbar arrow = None, arrowbar bar = None,
              real xmargin = 0, real ymargin = xmargin,
              Label legend = "", marker marker = nomarker)
{/*<asyxml></code><documentation>Draw the path 'g' on 'pic' clipped to the bounding box of 'pic'.</documentation></function></asyxml>*/
  if(L.s != "") {
    picture tmp;
    label(tmp, L, g, p);
    add(pic, tmp);
  }
  pic.add(new void (frame f, transform t, transform T, pair m, pair M) {
      // Reduce the bounds by the size of the pen and the margins.
      m += min(p) + (xmargin, ymargin); M -= max(p) + (xmargin, ymargin);
      path bound = box(m, M);
      picture tmp;
      draw(tmp, "", t * T * g, align, p, arrow, bar, NoMargin, legend, marker);
      clip(tmp, bound);
      add(f, tmp.fit());
    });
}

/*<asyxml><function type="void" signature="distance(picture pic,Label,point,point,bool,real,pen,pen,arrow)"><code></asyxml>*/
void distance(picture pic = currentpicture, Label L = "", point A, point B,
              bool rotated = true, real offset = 3mm,
              pen p = currentpen, pen joinpen = invisible,
              arrowbar arrow = Arrows(NoFill))
{/*<asyxml></code><documentation>Draw arrow between A and B (from FAQ).</documentation></function></asyxml>*/
  pair A = A, B = B;
  path g = A--B;
  transform Tp = shift(-offset * unit(B - A) * I);
  pic.add(new void(frame f, transform t) {
      picture opic;
      path G = Tp * t * g;
      transform id = identity();
      transform T = rotated ? rotate(B - A) : id;
      Label L = L.copy();
      L.align(L.align, Center);
      if(abs(ypart((conj(A - B) * L.align.dir))) < epsgeo && L.filltype == NoFill)
        L.filltype = UnFill(1);
      draw(opic, T * L, G, p, arrow, Bars, PenMargins);
      pair Ap = t * A, Bp = t * B;
      draw(opic, (Ap--Tp * Ap)^^(Bp--Tp * Bp), joinpen);
      add(f, opic.fit());
    }, true);
  pic.addBox(min(g), max(g), Tp * min(p), Tp * max(p));
}

/*<asyxml><variable type="real" signature="perpfactor"><code></asyxml>*/
real perpfactor = 1;/*<asyxml></code><documentation>Factor for drawing perpendicular symbol.</documentation></variable></asyxml>*/
/*<asyxml><function type="void" signature="perpendicularmark(picture,point,explicit pair,explicit pair,real,pen,margin,filltype)"><code></asyxml>*/
void perpendicularmark(picture pic = currentpicture, point z,
                       explicit pair align,
                       explicit pair dir = E, real size = 0,
                       pen p = currentpen,
                       margin margin = NoMargin,
                       filltype filltype = NoFill)
{/*<asyxml></code><documentation>Draw a perpendicular symbol at z aligned in the direction align
   relative to the path z--z + dir.
   dir(45 + n * 90), where n in N*, are common values for 'align'.</documentation></function></asyxml>*/
  p = squarecap + p;
  if(size == 0) size = perpfactor * 3mm + sqrt(1 + linewidth(p)) - 1;
  frame apic;
  pair d1 = size * align * unit(dir) * dir(-45);
  pair d2 = I * d1;
  path g = d1--d1 + d2--d2;
  g = margin(g, p).g;
  draw(apic, g, p);
  if(filltype != NoFill) filltype.fill(apic, (relpoint(g, 0) - relpoint(g, 0.5)+
                                             relpoint(g, 1))--g--cycle, p + solid);
  add(pic, apic, locate(z));
}

/*<asyxml><function type="void" signature="perpendicularmark(picture,point,vector,vector,real,pen,margin,filltype)"><code></asyxml>*/
void perpendicularmark(picture pic = currentpicture, point z,
                       vector align,
                       vector dir = E, real size = 0,
                       pen p = currentpen,
                       margin margin = NoMargin,
                       filltype filltype = NoFill)
{/*<asyxml></code><documentation>Draw a perpendicular symbol at z aligned in the direction align
   relative to the path z--z + dir.
   dir(45 + n * 90), where n in N, are common values for 'align'.</documentation></function></asyxml>*/
  perpendicularmark(pic, z, (pair)align, (pair)dir, size,
                    p, margin, filltype);
}

/*<asyxml><function type="void" signature="perpendicularmark(picture,point,explicit pair,path,real,pen,margin,filltype)"><code></asyxml>*/
void perpendicularmark(picture pic = currentpicture, point z, explicit pair align, path g,
                       real size = 0, pen p = currentpen,
                       margin margin = NoMargin,
                       filltype filltype = NoFill)
{/*<asyxml></code><documentation>Draw a perpendicular symbol at z aligned in the direction align
   relative to the path z--z + dir(g, 0).
   dir(45 + n * 90), where n in N, are common values for 'align'.</documentation></function></asyxml>*/
  perpendicularmark(pic, z, align, dir(g, 0), size, p, margin, filltype);
}

/*<asyxml><function type="void" signature="perpendicularmark(picture,point,vector,path,real,pen,margin,filltype)"><code></asyxml>*/
void perpendicularmark(picture pic = currentpicture, point z, vector align, path g,
                       real size = 0, pen p = currentpen,
                       margin margin = NoMargin,
                       filltype filltype = NoFill)
{/*<asyxml></code><documentation>Draw a perpendicular symbol at z aligned in the direction align
   relative to the path z--z + dir(g, 0).
   dir(45 + n * 90), where n in N, are common values for 'align'.</documentation></function></asyxml>*/
  perpendicularmark(pic, z, (pair)align, dir(g, 0), size, p, margin, filltype);
}

/*<asyxml><function type="void" signature="markrightangle(picture,point,point,point,real,pen,margin,filltype)"><code></asyxml>*/
void markrightangle(picture pic = currentpicture, point A, point O,
                    point B, real size = 0, pen p = currentpen,
                    margin margin = NoMargin,
                    filltype filltype = NoFill)
{/*<asyxml></code><documentation>Mark the angle AOB with a perpendicular symbol.</documentation></function></asyxml>*/
  pair Ap = A, Bp = B, Op = O;
  pair dir = Ap - Op;
  real a1 = degrees(dir);
  pair align = rotate(-a1) * unit(dir(Op--Ap, Op--Bp));
  if (margin == NoMargin)
    margin = TrueMargin(linewidth(currentpen)/2, linewidth(currentpen)/2);
  perpendicularmark(pic = pic, z = O, align = align,
                    dir = dir, size = size, p = p,
                    margin = margin, filltype = filltype);
}

/*<asyxml><function type="bool" signature="simeq(point,point,real)"><code></asyxml>*/
bool simeq(point A, point B, real fuzz = epsgeo)
{/*<asyxml></code><documentation>Return true iff abs(A - B) < fuzz.
   This routine is used internally to know if two points are equal, in particular by the operator == in 'point == point'.</documentation></function></asyxml>*/
  return (abs(A - B) < fuzz);
}
bool simeq(point a, real b, real fuzz = epsgeo)
{
  coordsys R = a.coordsys;
  return (abs(a - point(R, ((pair)b)/R)) < fuzz);
}

/*<asyxml><function type="pair" signature="attract(pair,path,real)"><code></asyxml>*/
pair attract(pair m, path g, real fuzz = 0)
{/*<asyxml></code><documentation>Return the nearest point (A PAIR) of 'm' which is on the path g.
   'fuzz' is the argument 'fuzz' of 'intersect'.</documentation></function></asyxml>*/
  if(intersect(m, g, fuzz).length > 0) return m;
  pair p;
  real step = 1, r = 0;
  real[] t;
  static real eps = sqrt(realEpsilon);
  do {// Find a radius for intersection
    r += step;
    t = intersect(shift(m) * scale(r) * unitcircle, g);
  } while(t.length <= 0);
  p = point(g, t[1]);
  real rm = 0, rM = r;
  while(rM - rm > eps) {
    r = (rm + rM)/2;
    t = intersect(shift(m) * scale(r) * unitcircle, g, fuzz);
    if(t.length <= 0) {
      rm = r;
    } else {
      rM = r;
      p = point(g, t[1]);
    }
  }
  return p;
}

/*<asyxml><function type="point" signature="attract(point,path,real)"><code></asyxml>*/
point attract(point M, path g, real fuzz = 0)
{/*<asyxml></code><documentation>Return the nearest point (A POINT) of 'M' which is on the path g.
   'fuzz' is the argument 'fuzz' of 'intersect'.</documentation></function></asyxml>*/
  return point(M.coordsys, attract(locate(M), g)/M.coordsys);
}

/*<asyxml><function type="real[]" signature="intersect(path,explicit pair)"><code></asyxml>*/
real[] intersect(path g, explicit pair p, real fuzz = 0)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  fuzz = fuzz <= 0 ? sqrt(realEpsilon) : fuzz;
  real[] or;
  real r = realEpsilon;
  do{
    or = intersect(g, shift(p) * scale(r) * unitcircle, fuzz);
    r *= 2;
  } while(or.length == 0);
  return or;
}

/*<asyxml><function type="real[]" signature="intersect(path,explicit point)"><code></asyxml>*/
real[] intersect(path g, explicit point P, real fuzz = epsgeo)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  return intersect(g, locate(P), fuzz);
}
// *.........................BASES.........................*
// *=======================================================*

// *=======================================================*
// *.........................LINES.........................*
/*<asyxml><struct signature="line"><code></asyxml>*/
struct line
{/*<asyxml></code><documentation>This structure provides the objects line, semi - line and segment oriented from A to B.
   All the calculus with this structure will be as exact as Asymptote can do.
   For a full precision, you must not cast 'line' to 'path' excepted for drawing routines.</documentation></asyxml>*/
  /*<asyxml><property type = "point" signature="A,B"><code></asyxml>*/
  restricted point A,B;/*<asyxml></code><documentation>Two line's points with same coordinate system.</documentation></property><property type = "bool" signature="extendA,extendB"><code></asyxml>*/
  bool extendA,extendB;/*<asyxml></code><documentation>If true,extend 'l' in direction of A (resp. B).</documentation></property><property type = "vector" signature="u,v"><code></asyxml>*/
  restricted vector u,v;/*<asyxml></code><documentation>u = unit(AB) = direction vector,v = normal vector.</documentation></property><property type = "real" signature="a,b,c"><code></asyxml>*/
  restricted real a,b,c;/*<asyxml></code><documentation>Coefficients of the equation ax + by + c = 0 in the coordinate system of 'A'.</documentation></property><property type = "real" signature="slope,origin"><code></asyxml>*/
  restricted real slope, origin;/*<asyxml></code><documentation>Slope and ordinate at the origin.</documentation></property></asyxml>*/
  /*<asyxml><method type = "line" signature="copy()"><code></asyxml>*/
  line copy()
  {/*<asyxml></code><documentation>Copy a line in a new instance.</documentation></method></asyxml>*/
    line l = new line;
    l.A = A;
    l.B = B;
    l.a = a;
    l.b = b;
    l.c = c;
    l.slope = slope;
    l.origin = origin;
    l.u = u;
    l.v = v;
    l.extendA = extendA;
    l.extendB = extendB;
    return l;
  }

  /*<asyxml><method type = "void" signature="init(point,bool,point,bool)"><code></asyxml>*/
  void init(point A, bool extendA = true, point B, bool extendB = true)
  {/*<asyxml></code><documentation>Initialize line.
     If 'extendA' is true, the "line" is infinite in the direction of A.</documentation></method></asyxml>*/
    point[] P = standardizecoordsys(A, B);
    this.A = P[0];
    this.B = P[1];
    this.a = B.y - A.y;
    this.b = A.x - B.x;
    this.c = A.y * B.x - A.x * B.y;
    this.slope= (this.b == 0) ? infinity : -this.a/this.b;
    this.origin = (this.b == 0) ? (this.c == 0) ? 0:infinity : -this.c/this.b;
    this.u = unit(P[1]-P[0]);
    //     int tmp = sgnd(this.slope);
    //     this.u = (dot((pair)this.u, N) >= 0) ? tmp * this.u : -tmp * this.u;
    this.v = rotate(90, point(P[0].coordsys, (0, 0))) * this.u;
    this.extendA = extendA;
    this.extendB = extendB;
  }
}/*<asyxml></struct></asyxml>*/

/*<asyxml><function type="line" signature="line(point,bool,point,bool)"><code></asyxml>*/
line line(point A, bool extendA = true, point B, bool extendB = true)
{/*<asyxml></code><documentation>Return the line passing through 'A' and 'B'.
   If 'extendA' is true, the "line" is infinite in the direction of A.
   A "line" can be half-line or segment.</documentation></function></asyxml>*/
  if (A == B) abort("line: the points must be distinct.");
  line l;
  l.init(A, extendA, B, extendB);
  return l;
}

/*<asyxml><struct signature="segment"><code></asyxml>*/
struct segment
{/*<asyxml></code><documentation><look href = "struct line"/>.</documentation></asyxml>*/
  restricted point A, B;// Extremity.
  restricted vector u, v;// u = direction vector, v = normal vector.
  restricted real a, b, c;// Coefficients of the équation ax + by + c = 0
  restricted real slope, origin;
  segment copy()
  {
    segment s = new segment;
    s.A = A;
    s.B = B;
    s.a = a;
    s.b = b;
    s.c = c;
    s.slope = slope;
    s.origin = origin;
    s.u = u;
    s.v = v;
    return s;
  }

  void init(point A, point B)
  {
    line l;
    l.init(A, B);
    this.A = l.A; this.B = l.B;
    this.a = l.a; this.b = l.b; this.c = l.c;
    this.slope = l.slope; this.origin = l.origin;
    this.u = l.u; this.v = l.v;
  }
}/*<asyxml></struct></asyxml>*/

/*<asyxml><function type="segment" signature="segment(point,point)"><code></asyxml>*/
segment segment(point A, point B)
{/*<asyxml></code><documentation>Return the segment whose the extremities are A and B.</documentation></function></asyxml>*/
  segment s;
  s.init(A, B);
  return s;
}

/*<asyxml><function type="real" signature="length(segment)"><code></asyxml>*/
real length(segment s)
{/*<asyxml></code><documentation>Return the length of 's'.</documentation></function></asyxml>*/
  return abs(s.A - s.B);
}

/*<asyxml><operator type = "line" signature="cast(segment)"><code></asyxml>*/
line operator cast(segment s)
{/*<asyxml></code><documentation>A segment is casted to a "finite line".</documentation></operator></asyxml>*/
  return line(s.A, false, s.B, false);
}

/*<asyxml><operator type = "segment" signature="cast(line)"><code></asyxml>*/
segment operator cast(line l)
{/*<asyxml></code><documentation>Cast line 'l' to segment [l.A l.B].</documentation></operator></asyxml>*/
  return segment(l.A, l.B);
}

/*<asyxml><operator type = "line" signature="*(transform,line)"><code></asyxml>*/
line operator *(transform t, line l)
{/*<asyxml></code><documentation>Provide transform * line</documentation></operator></asyxml>*/
  return line(t * l.A, l.extendA, t * l.B, l.extendB);
}
/*<asyxml><operator type = "line" signature="/(line,real)"><code></asyxml>*/
line operator /(line l, real x)
{/*<asyxml></code><documentation>Provide l/x.
   Return the line passing through l.A/x and l.B/x.</documentation></operator></asyxml>*/
  return line(l.A/x, l.extendA, l.B/x, l.extendB);
}
line operator /(line l, int x){return line(l.A/x, l.B/x);}
/*<asyxml><operator type = "line" signature="*(real,line)"><code></asyxml>*/
line operator *(real x, line l)
{/*<asyxml></code><documentation>Provide x * l.
   Return the line passing through x * l.A and x * l.B.</documentation></operator></asyxml>*/
  return line(x * l.A, l.extendA, x * l.B, l.extendB);
}
line operator *(int x, line l){return line(x * l.A, l.extendA, x * l.B, l.extendB);}

/*<asyxml><operator type = "line" signature="*(point,line)"><code></asyxml>*/
line operator *(point M, line l)
{/*<asyxml></code><documentation>Provide point * line.
   Return the line passing through unit(M) * l.A and unit(M) * l.B.</documentation></operator></asyxml>*/
  return line(unit(M) * l.A, l.extendA, unit(M) * l.B, l.extendB);
}
/*<asyxml><operator type = "line" signature="+(line,point)"><code></asyxml>*/
line operator +(line l, vector u)
{/*<asyxml></code><documentation>Provide line + vector (and so line + point).
   Return the line 'l' shifted by 'u'.</documentation></operator></asyxml>*/
  return line(l.A + u, l.extendA, l.B + u, l.extendB);
}
/*<asyxml><operator type = "line" signature="-(line,vector)"><code></asyxml>*/
line operator -(line l, vector u)
{/*<asyxml></code><documentation>Provide line - vector (and so line - point).
   Return the line 'l' shifted by '-u'.</documentation></operator></asyxml>*/
  return line(l.A - u, l.extendA, l.B - u, l.extendB);
}

/*<asyxml><operator type = "line[]" signature="^^(line,line)"><code></asyxml>*/
line[] operator ^^(line l1, line l2)
{/*<asyxml></code><documentation>Provide line^^line.
   Return the line array {l1, l2}.</documentation></operator></asyxml>*/
  line[] ol;
  ol.push(l1); ol.push(l2);
  return ol;
}

/*<asyxml><operator type = "line[]" signature="^^(line,line[])"><code></asyxml>*/
line[] operator ^^(line l1, line[] l2)
{/*<asyxml></code><documentation>Provide line^^line[].
   Return the line array {l1, l2[0], l2[1]...}.
   line[]^^line is also defined.</documentation></operator></asyxml>*/
  line[] ol;
  ol.push(l1);
  for (int i = 0; i < l2.length; ++i) {
    ol.push(l2[i]);
  }
  return ol;
}
line[] operator ^^(line[] l2, line l1)
{
  line[] ol = l2;
  ol.push(l1);
  return ol;
}

/*<asyxml><operator type = "line[]" signature="^^(line,line[])"><code></asyxml>*/
line[] operator ^^(line l1[], line[] l2)
{/*<asyxml></code><documentation>Provide line[]^^line[].
   Return the line array {l1[0], l1[1], ..., l2[0], l2[1], ...}.</documentation></operator></asyxml>*/
  line[] ol = l1;
  for (int i = 0; i < l2.length; ++i) {
    ol.push(l2[i]);
  }
  return ol;
}

/*<asyxml><function type="bool" signature="sameside(point,point,line)"><code></asyxml>*/
bool sameside(point M, point P, line l)
{/*<asyxml></code><documentation>Return 'true' iff 'M' and 'N' are same side of the line (or on the line) 'l'.</documentation></function></asyxml>*/
  pair A = l.A, B = l.B, m = M, p = P;
  pair mil = (A + B)/2;
  pair mA = rotate(90, mil) * A;
  pair mB = rotate(-90, mil) * A;
  return (abs(m - mA) <= abs(m - mB)) == (abs(p - mA) <= abs(p - mB));
  // transform proj = projection(l.A, l.B);
  // point Mp = proj * M;
  // point Pp = proj * P;
  // dot(Mp);dot(Pp);
  // return dot(locate(Mp - M), locate(Pp - P)) >= 0;
}

/*<asyxml><function type="line" signature="line(segment)"><code></asyxml>*/
line line(segment s)
{/*<asyxml></code><documentation>Return the line passing through 's.A'
   and 's.B'.</documentation></function></asyxml>*/
  return line(s.A, s.B);
}
/*<asyxml><function type="segment" signature="segment(line)"><code></asyxml>*/
segment segment(line l)
{/*<asyxml></code><documentation>Return the segment whose extremities
   are 'l.A' and 'l.B'.</documentation></function></asyxml>*/
  return segment(l.A, l.B);
}

/*<asyxml><function type="point" signature="midpoint(segment)"><code></asyxml>*/
point midpoint(segment s)
{/*<asyxml></code><documentation>Return the midpoint of 's'.</documentation></function></asyxml>*/
  return 0.5 * (s.A + s.B);
}

/*<asyxml><function type="void" signature="write(line)"><code></asyxml>*/
void write(explicit line l)
{/*<asyxml></code><documentation>Write some informations about 'l'.</documentation></function></asyxml>*/
  write("A = "+(string)((pair)l.A));
  write("Extend A = "+(l.extendA ? "true" : "false"));
  write("B = "+(string)((pair)l.B));
  write("Extend B = "+(l.extendB ? "true" : "false"));
  write("u = "+(string)((pair)l.u));
  write("v = "+(string)((pair)l.v));
  write("a = "+(string) l.a);
  write("b = "+(string) l.b);
  write("c = "+(string) l.c);
  write("slope = "+(string) l.slope);
  write("origin = "+(string) l.origin);
}

/*<asyxml><function type="void" signature="write(explicit segment)"><code></asyxml>*/
void write(explicit segment s)
{/*<asyxml></code><documentation>Write some informations about 's'.</documentation></function></asyxml>*/
  write("A = "+(string)((pair)s.A));
  write("B = "+(string)((pair)s.B));
  write("u = "+(string)((pair)s.u));
  write("v = "+(string)((pair)s.v));
  write("a = "+(string) s.a);
  write("b = "+(string) s.b);
  write("c = "+(string) s.c);
  write("slope = "+(string) s.slope);
  write("origin = "+(string) s.origin);
}

/*<asyxml><operator type = "bool" signature="==(line,line)"><code></asyxml>*/
bool operator ==(line l1, line l2)
{/*<asyxml></code><documentation>Provide the test 'line == line'.</documentation></operator></asyxml>*/
  return (collinear(l1.u, l2.u) &&
          abs(ypart((locate(l1.A) - locate(l1.B))/(locate(l1.A) - locate(l2.B)))) < epsgeo &&
          l1.extendA == l2.extendA && l1.extendB == l2.extendB);
}

/*<asyxml><operator type = "bool" signature="!=(line,line)"><code></asyxml>*/
bool operator !=(line l1, line l2)
{/*<asyxml></code><documentation>Provide the test 'line != line'.</documentation></operator></asyxml>*/
  return !(l1 == l2);
}

/*<asyxml><operator type = "bool" signature="@(point,line)"><code></asyxml>*/
bool operator @(point m, line l)
{/*<asyxml></code><documentation>Provide the test 'point @ line'.
   Return true iff 'm' is on the 'l'.</documentation></operator></asyxml>*/
  point M = changecoordsys(l.A.coordsys, m);
  if (abs(l.a * M.x + l.b * M.y + l.c) >= epsgeo) return false;
  if (l.extendA && l.extendB) return true;
  if (!l.extendA && !l.extendB) return between(l.A, M, l.B);
  if (l.extendA) return sameside(M, l.A, l.B);
  return sameside(M, l.B, l.A);
}

/*<asyxml><function type="coordsys" signature="coordsys(line)"><code></asyxml>*/
coordsys coordsys(line l)
{/*<asyxml></code><documentation>Return the coordinate system in which 'l' is defined.</documentation></function></asyxml>*/
  return l.A.coordsys;
}

/*<asyxml><function type="line" signature="reverse(line)"><code></asyxml>*/
line reverse(line l)
{/*<asyxml></code><documentation>Permute the points 'A' and 'B' of 'l' and so its orientation.</documentation></function></asyxml>*/
  return line(l.B, l.extendB, l.A, l.extendA);
}

/*<asyxml><function type="line" signature="extend(line)"><code></asyxml>*/
line extend(line l)
{/*<asyxml></code><documentation>Return the infinite line passing through 'l.A' and 'l.B'.</documentation></function></asyxml>*/
  line ol = l.copy();
  ol.extendA = true;
  ol.extendB = true;
  return ol;
}

/*<asyxml><function type="line" signature="complementary(explicit line)"><code></asyxml>*/
line complementary(explicit line l)
{/*<asyxml></code><documentation>Return the complementary of a half-line with respect of
   the full line 'l'.</documentation></function></asyxml>*/
  if (l.extendA && l.extendB)
    abort("complementary: the parameter is not a half-line.");
  point origin = l.extendA ? l.B : l.A;
  point ptdir = l.extendA ?
    rotate(180, l.B) * l.A : rotate(180, l.A) * l.B;
  return line(origin, false, ptdir);
}

/*<asyxml><function type="line[]" signature="complementary(explicit segment)"><code></asyxml>*/
line[] complementary(explicit segment s)
{/*<asyxml></code><documentation>Return the two half-lines of origin 's.A' and 's.B' respectively.</documentation></function></asyxml>*/
  line[] ol = new line[2];
  ol[0] = complementary(line(s.A, false, s.B));
  ol[1] = complementary(line(s.A, s.B, false));
  return ol;
}

/*<asyxml><function type="line" signature="Ox(coordsys)"><code></asyxml>*/
line Ox(coordsys R = currentcoordsys)
{/*<asyxml></code><documentation>Return the x-axis of 'R'.</documentation></function></asyxml>*/
  return line(point(R, (0, 0)), point(R, E));
}
/*<asyxml><constant type = "line" signature="Ox"><code></asyxml>*/
restricted line Ox = Ox();/*<asyxml></code><documentation>the x-axis of
                          the default coordinate system.</documentation></constant></asyxml>*/

/*<asyxml><function type="line" signature="Oy(coordsys)"><code></asyxml>*/
line Oy(coordsys R = currentcoordsys)
{/*<asyxml></code><documentation>Return the y-axis of 'R'.</documentation></function></asyxml>*/
  return line(point(R, (0, 0)), point(R, N));
}
/*<asyxml><constant type = "line" signature="Oy"><code></asyxml>*/
restricted line Oy = Oy();/*<asyxml></code><documentation>the y-axis of
                          the default coordinate system.</documentation></constant></asyxml>*/

/*<asyxml><function type="line" signature="line(real,point)"><code></asyxml>*/
line line(real a, point A = point(currentcoordsys, (0, 0)))
{/*<asyxml></code><documentation>Return the line passing through 'A' with an
   angle (in the coordinate system of A) 'a' in degrees.
   line(point, real) is also defined.</documentation></function></asyxml>*/
  return line(A, A + point(A.coordsys, A.coordsys.polar(1, radians(a))));
}
line line(point A = point(currentcoordsys, (0, 0)), real a)
{
  return line(a, A);
}
line line(int a, point A = point(currentcoordsys, (0, 0)))
{
  return line((real)a, A);
}

/*<asyxml><function type="line" signature="line(coordsys,real,real)"><code></asyxml>*/
line line(coordsys R = currentcoordsys, real slope, real origin)
{/*<asyxml></code><documentation>Return the line defined by slope and y-intercept relative to 'R'.</documentation></function></asyxml>*/
  if (slope == infinity || slope == -infinity)
    abort("The slope is infinite. Please, use the routine 'vline'.");
  return line(point(R, (0, origin)), point(R, (1, origin + slope)));
}

/*<asyxml><function type="line" signature="line(coordsys,real,real,real)"><code></asyxml>*/
line line(coordsys R = currentcoordsys, real a, real b, real c)
{/*<asyxml></code><documentation>Retrun the line defined by equation relative to 'R'.</documentation></function></asyxml>*/
  if (a == 0 && b == 0) abort("line: inconsistent equation...");
  pair M;
  M = (a == 0) ? (0, -c/b) : (-c/a, 0);
  return line(point(R, M), point(R, M + (-b, a)));
}

/*<asyxml><function type="line" signature="vline(coordsys)"><code></asyxml>*/
line vline(coordsys R = currentcoordsys)
{/*<asyxml></code><documentation>Return a vertical line in 'R' passing through the origin of 'R'.</documentation></function></asyxml>*/
  point P = point(R, (0, 0));
  point PP = point(R, (R.O + N)/R);
  return line(P, PP);
}
/*<asyxml><constant type = "line" signature="vline"><code></asyxml>*/
restricted line vline = vline();/*<asyxml></code><documentation>The vertical line in the current coordinate system passing
                                through the origin of this system.</documentation></constant></asyxml>*/

/*<asyxml><function type="line" signature="hline(coordsys)"><code></asyxml>*/
line hline(coordsys R = currentcoordsys)
{/*<asyxml></code><documentation>Return a horizontal line in 'R' passing through the origin of 'R'.</documentation></function></asyxml>*/
  point P = point(R, (0, 0));
  point PP = point(R, (R.O + E)/R);
  return line(P, PP);
}
/*<asyxml><constant type = "line" signature="hline"><code></asyxml>*/
line hline = hline();/*<asyxml></code><documentation>The horizontal line in the current coordinate system passing
                     through the origin of this system.</documentation></constant></asyxml>*/

/*<asyxml><function type="line" signature="changecoordsys(coordsys,line)"><code></asyxml>*/
line changecoordsys(coordsys R, line l)
{/*<asyxml></code><documentation>Return the line 'l' in the coordinate system 'R'.</documentation></function></asyxml>*/
  point A = changecoordsys(R, l.A);
  point B = changecoordsys(R, l.B);
  return line(A, B);
}

/*<asyxml><function type="transform" signature="scale(real,line,line,bool)"><code></asyxml>*/
transform scale(real k, line l1, line l2, bool safe = false)
{/*<asyxml></code><documentation>Return the dilatation with respect to
   'l1' in the direction of 'l2'.</documentation></function></asyxml>*/
  return scale(k, l1.A, l1.B, l2.A, l2.B, safe);
}

/*<asyxml><function type="transform" signature="reflect(line)"><code></asyxml>*/
transform reflect(line l)
{/*<asyxml></code><documentation>Return the reflect about the line 'l'.</documentation></function></asyxml>*/
  return reflect((pair)l.A, (pair)l.B);
}

/*<asyxml><function type="transform" signature="reflect(line,line)"><code></asyxml>*/
transform reflect(line l1, line l2, bool safe = false)
{/*<asyxml></code><documentation>Return the reflect about the line
   'l1' in the direction of 'l2'.</documentation></function></asyxml>*/
  return scale(-1.0, l1, l2, safe);
}


/*<asyxml><function type="point[]" signature="intersectionpoints(line,path)"><code></asyxml>*/
point[] intersectionpoints(line l, path g)
{/*<asyxml></code><documentation>Return all points of intersection of the line 'l' with the path 'g'.</documentation></function></asyxml>*/
  // TODO utiliser la version 1.44 de intersections(path g, pair p, pair q)
  // real [] t = intersections(g, l.A, l.B);
  // coordsys R = coordsys(l);
  // return sequence(new point(int n){return point(R, point(g, t[n])/R);}, t.length);
  real [] t;
  pair[] op;
  pair A = l.A;
  pair B = l.B;
  real dy = B.y - A.y,
    dx = A.x - B.x,
    lg = length(g);

  for (int i = 0; i < lg; ++i)
    {
      pair z0 = point(g, i),
        z1 = point(g, i + 1),
        c0 = postcontrol(g, i),
        c1 = precontrol(g, i + 1),
        t3 = z1 - z0 - 3 * c1 + 3 * c0,
        t2 = 3 * z0 + 3 * c1 - 6 * c0,
        t1 = 3 * c0 - 3z0;
      real a = dy * t3.x + dx * t3.y,
        b = dy * t2.x + dx * t2.y,
        c = dy * t1.x + dx * t1.y,
        d = dy * z0.x + dx * z0.y + A.y * B.x - A.x * B.y;

      t = cubicroots(a, b, c, d);
      for (int j = 0; j < t.length; ++j)
        if (
            t[j]>=0
            && (
                t[j]<1
                || (
                    t[j] == 1
                    && (i == lg - 1)
                    && !cyclic(g)
                    )
                )
            ) {
          op.push(point(g, i + t[j]));
        }
    }

  point[] opp;
  for (int i = 0; i < op.length; ++i)
    opp.push(point(coordsys(l), op[i]/coordsys(l)));
  return opp;
}

/*<asyxml><function type="point" signature="intersectionpoint(line,line)"><code></asyxml>*/
point intersectionpoint(line l1, line l2)
{/*<asyxml></code><documentation>Return the point of intersection of line 'l1' with 'l2'.
   If 'l1' and 'l2' have an infinity or none point of intersection,
   this routine return (infinity, infinity).</documentation></function></asyxml>*/
  point[] P = standardizecoordsys(l1.A, l1.B, l2.A, l2.B);
  coordsys R = P[0].coordsys;
  pair p = extension(P[0], P[1], P[2], P[3]);
  if(finite(p)){
    point p = point(R, p/R);
    if (p @ l1 && p @ l2) return p;
  }
  return point(R, (infinity, infinity));
}

/*<asyxml><function type="line" signature="parallel(point,line)"><code></asyxml>*/
line parallel(point M, line l)
{/*<asyxml></code><documentation>Return the line parallel to 'l' passing through 'M'.</documentation></function></asyxml>*/
  point A, B;
  if (M.coordsys != coordsys(l))
    {
      A = changecoordsys(M.coordsys, l.A);
      B = changecoordsys(M.coordsys, l.B);
    } else {A = l.A;B = l.B;}
  return line(M, M - A + B);
}

/*<asyxml><function type="line" signature="parallel(point,explicit vector)"><code></asyxml>*/
line parallel(point M, explicit vector dir)
{/*<asyxml></code><documentation>Return the line of direction 'dir' and passing through 'M'.</documentation></function></asyxml>*/
  return line(M, M + locate(dir));
}

/*<asyxml><function type="line" signature="parallel(point,explicit pair)"><code></asyxml>*/
line parallel(point M, explicit pair dir)
{/*<asyxml></code><documentation>Return the line of direction 'dir' and passing through 'M'.</documentation></function></asyxml>*/
  return line(M, M + vector(currentcoordsys, dir));
}

/*<asyxml><function type="bool" signature="parallel(line,line)"><code></asyxml>*/
bool parallel(line l1, line l2, bool strictly = false)
{/*<asyxml></code><documentation>Return 'true' if 'l1' and 'l2' are (strictly ?) parallel.</documentation></function></asyxml>*/
  bool coll = collinear(l1.u, l2.u);
  return strictly ? coll && (l1 != l2) : coll;
}

/*<asyxml><function type="bool" signature="concurrent(...line[])"><code></asyxml>*/
bool concurrent(... line[] l)
{/*<asyxml></code><documentation>Returns true if all the lines 'l' are concurrent.</documentation></function></asyxml>*/
  if (l.length < 3) abort("'concurrent' needs at least for three lines ...");
  pair point = intersectionpoint(l[0], l[1]);
  bool conc;
  for (int i = 2; i < l.length; ++i) {
    pair pt = intersectionpoint(l[i - 1], l[i]);
    conc = simeq(pt, point);
    if (!conc) break;
  }
  return conc;
}

/*<asyxml><function type="transform" signature="projection(line)"><code></asyxml>*/
transform projection(line l)
{/*<asyxml></code><documentation>Return the orthogonal projection on 'l'.</documentation></function></asyxml>*/
  return projection(l.A, l.B);
}

/*<asyxml><function type="transform" signature="projection(line,line,bool)"><code></asyxml>*/
transform projection(line l1, line l2, bool safe = false)
{/*<asyxml></code><documentation>Return the projection on (AB) in parallel of (CD).
   If 'safe = true' and (l1)//(l2) return the identity.
   If 'safe = false' and (l1)//(l2) return a infinity scaling.</documentation></function></asyxml>*/
  return projection(l1.A, l1.B, l2.A, l2.B, safe);
}

/*<asyxml><function type="transform" signature="vprojection(line,bool)"><code></asyxml>*/
transform vprojection(line l, bool safe = false)
{/*<asyxml></code><documentation>Return the projection on 'l' in parallel of N--S.
   If 'safe' is 'true' the projected point keeps the same place if 'l'
   is vertical.</documentation></function></asyxml>*/
  coordsys R = defaultcoordsys;
  return projection(l, line(point(R, N), point(R, S)), safe);
}

/*<asyxml><function type="transform" signature="hprojection(line,bool)"><code></asyxml>*/
transform hprojection(line l, bool safe = false)
{/*<asyxml></code><documentation>Return the projection on 'l' in parallel of E--W.
   If 'safe' is 'true' the projected point keeps the same place if 'l'
   is horizontal.</documentation></function></asyxml>*/
  coordsys R = defaultcoordsys;
  return projection(l, line(point(R, E), point(R, W)), safe);
}

/*<asyxml><function type="line" signature="perpendicular(point,line)"><code></asyxml>*/
line perpendicular(point M, line l)
{/*<asyxml></code><documentation>Return the perpendicular line of 'l' passing through 'M'.</documentation></function></asyxml>*/
  point Mp = projection(l) * M;
  point A = Mp == l.A ? l.B : l.A;
  return line(Mp, rotate(90, Mp) * A);
}

/*<asyxml><function type="line" signature="perpendicular(point,explicit vector)"><code></asyxml>*/
line perpendicular(point M, explicit vector normal)
{/*<asyxml></code><documentation>Return the line passing through 'M'
   whose normal is \param{normal}.</documentation></function></asyxml>*/
  return perpendicular(M, line(M, M + locate(normal)));
}

/*<asyxml><function type="line" signature="perpendicular(point,explicit pair)"><code></asyxml>*/
line perpendicular(point M, explicit pair normal)
{/*<asyxml></code><documentation>Return the line passing through 'M'
   whose normal is \param{normal} (given in the currentcoordsys).</documentation></function></asyxml>*/
  return perpendicular(M, line(M, M + vector(currentcoordsys, normal)));
}

/*<asyxml><function type="bool" signature="perpendicular(line,line)"><code></asyxml>*/
bool perpendicular(line l1, line l2)
{/*<asyxml></code><documentation>Return 'true' if 'l1' and 'l2' are perpendicular.</documentation></function></asyxml>*/
  return abs(dot(locate(l1.u), locate(l2.u))) < epsgeo ;
}

/*<asyxml><function type="real" signature="angle(line,coordsys)"><code></asyxml>*/
real angle(line l, coordsys R = coordsys(l))
{/*<asyxml></code><documentation>Return the angle of the oriented line 'l',
   in radian, in the interval ]-pi, pi] and relatively to 'R'.</documentation></function></asyxml>*/
  return angle(l.u, R, false);
}

/*<asyxml><function type="real" signature="degrees(line,coordsys,bool)"><code></asyxml>*/
real degrees(line l, coordsys R = coordsys(l))
{/*<asyxml></code><documentation>Returns the angle of the oriented line 'l' in degrees,
   in the interval [0, 360[ and relatively to 'R'.</documentation></function></asyxml>*/
  return degrees(angle(l, R));
}

/*<asyxml><function type="real" signature="sharpangle(line,line)"><code></asyxml>*/
real sharpangle(line l1, line l2)
{/*<asyxml></code><documentation>Return the measure in radians of the sharp angle formed by 'l1' and 'l2'.</documentation></function></asyxml>*/
  vector u1 = l1.u;
  vector u2 = (dot(l1.u, l2.u) < 0) ? -l2.u : l2.u;
  real a12 = angle(locate(u2)) - angle(locate(u1));
  a12 = a12%(sgnd(a12) * pi);
  if (a12 <= -pi/2) {
    a12 += pi;
  } else if (a12 > pi/2) {
    a12 -= pi;
  }
  return a12;
}

/*<asyxml><function type="real" signature="angle(line,line)"><code></asyxml>*/
real angle(line l1, line l2)
{/*<asyxml></code><documentation>Return the measure in radians of oriented angle (l1.u, l2.u).</documentation></function></asyxml>*/
  return angle(locate(l2.u)) - angle(locate(l1.u));
}

/*<asyxml><function type="real" signature="degrees(line,line)"><code></asyxml>*/
real degrees(line l1, line l2)
{/*<asyxml></code><documentation>Return the measure in degrees of the
   angle formed by the oriented lines 'l1' and 'l2'.</documentation></function></asyxml>*/
  return degrees(angle(l1, l2));
}

/*<asyxml><function type="real" signature="sharpdegrees(line,line)"><code></asyxml>*/
real sharpdegrees(line l1, line l2)
{/*<asyxml></code><documentation>Return the measure in degrees of the sharp angle formed by 'l1' and 'l2'.</documentation></function></asyxml>*/
  return degrees(sharpangle(l1, l2));
}

/*<asyxml><function type="line" signature="bisector(line,line,real,bool)"><code></asyxml>*/
line bisector(line l1, line l2, real angle = 0, bool sharp = true)
{/*<asyxml></code><documentation>Return the bisector of the angle formed by 'l1' and 'l2'
   rotated by the angle 'angle' (in degrees) around intersection point of 'l1' with 'l2'.
   If 'sharp' is true (the default), this routine returns the bisector of the sharp angle.
   Note that the returned line inherit of coordinate system of 'l1'.</documentation></function></asyxml>*/
  line ol;
  if (l1 == l2) return l1;
  point A = intersectionpoint(l1, l2);
  if (finite(A)) {
    if(sharp) ol = rotate(sharpdegrees(l1, l2)/2 + angle, A) * l1;
    else {
      coordsys R = coordsys(l1);
      pair a = A, b = A + l1.u, c = A + l2.u;
      pair pp = extension(a, a + dir(a--b, a--c), b, b + dir(b--a, b--c));
      return rotate(angle, A) * line(A, point(R, pp/R));
    }
  } else {
    ol = l1;
  }
  return ol;
}

/*<asyxml><function type="line" signature="sector(int,int,line,line,real,bool)"><code></asyxml>*/
line sector(int n = 2, int p = 1, line l1, line l2, real angle = 0, bool sharp = true)
{/*<asyxml></code><documentation>Return the p-th nth-sector of the angle
   formed by the oriented line 'l1' and 'l2'
   rotated by the angle 'angle' (in degrees) around the intersection point of 'l1' with 'l2'.
   If 'sharp' is true (the default), this routine returns the bisector of the sharp angle.
   Note that the returned line inherit of coordinate system of 'l1'.</documentation></function></asyxml>*/
  line ol;
  if (l1 == l2) return l1;
  point A = intersectionpoint(l1, l2);
  if (finite(A)) {
    if(sharp) ol = rotate(p * sharpdegrees(l1, l2)/n + angle, A) * l1;
    else {
      ol = rotate(p * degrees(l1, l2)/n + angle, A) * l1;
    }
  } else {
    ol = l1;
  }
  return ol;
}

/*<asyxml><function type="line" signature="bisector(point,point,point,point,real)"><code></asyxml>*/
line bisector(point A, point B, point C, point D, real angle = 0, bool sharp = true)
{/*<asyxml></code><documentation>Return the bisector of the angle formed by the lines (AB) and (CD).
   <look href = "#bisector(line, line, real, bool)"/>.</documentation></function></asyxml>*/
  point[] P = standardizecoordsys(A, B, C, D);
  return bisector(line(P[0], P[1]), line(P[2], P[3]), angle, sharp);
}

/*<asyxml><function type="line" signature="bisector(segment,real)"><code></asyxml>*/
line bisector(segment s, real angle = 0)
{/*<asyxml></code><documentation>Return the bisector of the segment line 's' rotated by 'angle' (in degrees) around the
   midpoint of 's'.</documentation></function></asyxml>*/
  coordsys R = coordsys(s);
  point m = midpoint(s);
  vector dir = rotateO(90) * unit(s.A - m);
  return rotate(angle, m) * line(m + dir, m - dir);
}

/*<asyxml><function type="line" signature="bisector(point,point,real)"><code></asyxml>*/
line bisector(point A, point B, real angle = 0)
{/*<asyxml></code><documentation>Return the bisector of the segment line [AB] rotated by 'angle' (in degrees) around the
   midpoint of [AB].</documentation></function></asyxml>*/
  point[] P = standardizecoordsys(A, B);
  return bisector(segment(P[0], P[1]), angle);
}

/*<asyxml><function type="real" signature="distance(point,line)"><code></asyxml>*/
real distance(point M, line l)
{/*<asyxml></code><documentation>Return the distance from 'M' to 'l'.
   distance(line, point) is also defined.</documentation></function></asyxml>*/
  point A = changecoordsys(defaultcoordsys, l.A);
  point B = changecoordsys(defaultcoordsys, l.B);
  line ll = line(A, B);
  pair m = locate(M);
  return abs(ll.a * m.x + ll.b * m.y + ll.c)/sqrt(ll.a^2 + ll.b^2);
}

real distance(line l, point M)
{
  return distance(M, l);
}

/*<asyxml><function type="void" signature="draw(picture,Label,line,bool,bool,align,pen,arrowbar,Label,marker)"><code></asyxml>*/
void draw(picture pic = currentpicture, Label L = "",
          line l, bool dirA = l.extendA, bool dirB = l.extendB,
          align align = NoAlign, pen p = currentpen,
          arrowbar arrow = None,
          Label legend = "", marker marker = nomarker,
          pathModifier pathModifier = NoModifier)
{/*<asyxml></code><documentation>Draw the line 'l' without altering the size of picture pic.
   The boolean parameters control the infinite section.
   The global variable 'linemargin' (default value is 0) allows to modify
   the bounding box in which the line must be drawn.</documentation></function></asyxml>*/
  if(!(dirA || dirB)) draw(l.A--l.B, invisible);// l is a segment.
  Drawline(pic, L, l.A, dirP = dirA, l.B, dirQ = dirB,
           align, p, arrow,
           legend, marker, pathModifier);
}

/*<asyxml><function type="void" signature="draw(picture,Label[], line[], align,pen[], arrowbar,Label,marker)"><code></asyxml>*/
void draw(picture pic = currentpicture, Label[] L = new Label[], line[] l,
          align align = NoAlign, pen[] p = new pen[],
          arrowbar arrow = None,
          Label[] legend = new Label[], marker marker = nomarker,
          pathModifier pathModifier = NoModifier)
{/*<asyxml></code><documentation>Draw each lines with the corresponding pen.</documentation></function></asyxml>*/
  for (int i = 0; i < l.length; ++i) {
    draw(pic, L.length>0 ? L[i] : "", l[i],
         align, p = p.length>0 ? p[i] : currentpen,
         arrow, legend.length>0 ? legend[i] : "", marker,
         pathModifier);
  }
}

/*<asyxml><function type="void" signature="draw(picture,Label[], line[], align,pen,arrowbar,Label,marker)"><code></asyxml>*/
void draw(picture pic = currentpicture, Label[] L = new Label[], line[] l,
          align align = NoAlign, pen p,
          arrowbar arrow = None,
          Label[] legend = new Label[], marker marker = nomarker,
          pathModifier pathModifier = NoModifier)
{/*<asyxml></code><documentation>Draw each lines with the same pen 'p'.</documentation></function></asyxml>*/
  pen[] tp = sequence(new pen(int i){return p;}, l.length);
  draw(pic, L, l, align, tp, arrow, legend, marker, pathModifier);
}

/*<asyxml><function type="void" signature="show(picture,line,pen)"><code></asyxml>*/
void show(picture pic = currentpicture, line l, pen p = red)
{/*<asyxml></code><documentation>Draw some informations of 'l'.</documentation></function></asyxml>*/
  dot("$A$", (pair)l.A, align = -locate(l.v), p);
  dot("$B$", (pair)l.B, align = -locate(l.v), p);
  draw(l, dotted);
  draw("$\vec{u}$", locate(l.A)--locate(l.A + l.u), p, Arrow);
  draw("$\vec{v}$", locate(l.A)--locate(l.A + l.v), p, Arrow);
}

/*<asyxml><function type="point[]" signature="sameside(point,line,line)"><code></asyxml>*/
point[] sameside(point M, line l1, line l2)
{/*<asyxml></code><documentation>Return two points on 'l1' and 'l2' respectively.
   The first point is from the same side of M relatively to 'l2',
   the second point is from the same side of M relatively to 'l1'.</documentation></function></asyxml>*/
  point[] op;
  coordsys R1 = coordsys(l1);
  coordsys R2 = coordsys(l2);
  if (parallel(l1, l2)) {
    op.push(projection(l1) * M);
    op.push(projection(l2) * M);
  } else {
    point O = intersectionpoint(l1, l2);
    if (M @ l2) op.push((sameside(M, O + l1.u, l2)) ? O + l1.u : rotate(180, O) * (O + l1.u));
    else op.push(projection(l1, l2) * M);
    if (M @ l1) op.push((sameside(M, O + l2.u, l1)) ? O + l2.u : rotate(180, O) * (O + l2.u));
    else {op.push(projection(l2, l1) * M);}
  }
  return op;
}

/*<asyxml><function type="void" signature="markangle(picture,Label,int,real,real,explicit line,explicit line,explicit pair,arrowbar,pen,filltype,margin,marker)"><code></asyxml>*/
void markangle(picture pic = currentpicture,
               Label L = "", int n = 1, real radius = 0, real space = 0,
               explicit line l1, explicit line l2, explicit pair align = dir(1),
               arrowbar arrow = None, pen p = currentpen,
               filltype filltype = NoFill,
               margin margin = NoMargin, marker marker = nomarker)
{/*<asyxml></code><documentation>Mark the angle (l1, l2) aligned in the direction 'align' relative to 'l1'.
   Commune values for 'align' are dir(real).</documentation></function></asyxml>*/
  if (parallel(l1, l2, true)) return;
  real al = degrees(l1, defaultcoordsys);
  pair O, A, B;
  if (radius == 0) radius = markangleradius(p);
  real d = degrees(locate(l1.u));
  align = rotate(d) * align;
  if (l1 == l2) {
    O = midpoint(segment(l1.A, l1.B));
    A = l1.A;B = l1.B;
    if (sameside(rotate(sgn(angle(B-A)) * 45, O) * A, O + align, l1)) {radius = -radius;}
  } else {
    O = intersectionpoint(extend(l1), extend(l2));
    pair R = O + align;
    point [] ss = sameside(point(coordsys(l1), R/coordsys(l1)), l1, l2);
    A = ss[0];
    B = ss[1];
  }
  markangle(pic = pic, L = L, n = n, radius = radius, space = space,
            O = O, A = A, B = B,
            arrow = arrow, p = p, filltype = filltype,
            margin = margin, marker = marker);
}

/*<asyxml><function type="void" signature="markangle(picture,Label,int,real,real,explicit line,explicit line,explicit vector,arrowbar,pen,filltype,margin,marker)"><code></asyxml>*/
void markangle(picture pic = currentpicture,
               Label L = "", int n = 1, real radius = 0, real space = 0,
               explicit line l1, explicit line l2, explicit vector align,
               arrowbar arrow = None, pen p = currentpen,
               filltype filltype = NoFill,
               margin margin = NoMargin, marker marker = nomarker)
{/*<asyxml></code><documentation>Mark the angle (l1, l2) in the direction 'dir' given relatively to 'l1'.</documentation></function></asyxml>*/
  markangle(pic, L, n, radius, space, l1, l2, (pair)align, arrow,
            p, filltype, margin, marker);
}

/*<asyxml><function type="void" signature="markangle(picture,Label,int,real,real,line,line,arrowbar,pen,filltype,margin,marker)"><code></asyxml>*/
// void markangle(picture pic = currentpicture,
//                Label L = "", int n = 1, real radius = 0, real space = 0,
//                explicit line l1, explicit line l2,
//                arrowbar arrow = None, pen p = currentpen,
//                filltype filltype = NoFill,
//                margin margin = NoMargin, marker marker = nomarker)
// {/*<asyxml></code><documentation>Mark the oriented angle (l1, l2).</documentation></function></asyxml>*/
//   if (parallel(l1, l2, true)) return;
//   real al = degrees(l1, defaultcoordsys);
//   pair O, A, B;
//   if (radius == 0) radius = markangleradius(p);
//   real d = degrees(locate(l1.u));
//   if (l1 == l2) {
//     O = midpoint(segment(l1.A, l1.B));
//   } else {
//     O = intersectionpoint(extend(l1), extend(l2));
//   }
//   A = O + locate(l1.u);
//   B = O + locate(l2.u);
//   markangle(pic = pic, L = L, n = n, radius = radius, space = space,
//             O = O, A = A, B = B,
//             arrow = arrow, p = p, filltype = filltype,
//             margin = margin, marker = marker);
// }

/*<asyxml><function type="void" signature="perpendicularmark(picture,line,line,real,pen,int,margin,filltype)"><code></asyxml>*/
void perpendicularmark(picture pic = currentpicture, line l1, line l2,
                       real size = 0, pen p = currentpen, int quarter = 1,
                       margin margin = NoMargin, filltype filltype = NoFill)
{/*<asyxml></code><documentation>Draw a right angle at the intersection point of lines and
   aligned in the 'quarter' nth quarter of circle formed by 'l1.u' and
   'l2.u'.</documentation></function></asyxml>*/
  point P = intersectionpoint(l1, l2);
  pair align = rotate(90 * (quarter - 1)) * dir(45);
  perpendicularmark(P, align, locate(l1.u), size, p, margin, filltype);
}
// *.........................LINES.........................*
// *=======================================================*

// *=======================================================*
// *........................CONICS.........................*
/*<asyxml><struct signature="bqe"><code></asyxml>*/
struct bqe
{/*<asyxml></code><documentation>Bivariate Quadratic Equation.</documentation></asyxml>*/
  /*<asyxml><property type = "real[]" signature="a"><code></asyxml>*/
  real[] a;/*<asyxml></code><documentation>a[0] * x^2 + a[1] * x * y + a[2] * y^2 + a[3] * x + a[4] * y + a[5] = 0</documentation></property><property type = "coordsys" signature="coordsys"><code></asyxml>*/
  coordsys coordsys;/*<asyxml></code></property></asyxml>*/
}/*<asyxml></struct></asyxml>*/

/*<asyxml><function type="bqe" signature="bqe(coordsys,real,real,real,real,real,real)"><code></asyxml>*/
bqe bqe(coordsys R = currentcoordsys,
        real a, real b, real c, real d, real e, real f)
{/*<asyxml></code><documentation>Return the bivariate quadratic equation
   a[0] * x^2 + a[1] * x * y + a[2] * y^2 + a[3] * x + a[4] * y + a[5] = 0
   relatively to the coordinate system R.</documentation></function></asyxml>*/
  bqe obqe;
  obqe.coordsys = R;
  obqe.a = new real[] {a, b, c, d, e, f};
  return obqe;
}

/*<asyxml><function type="bqe" signature="changecoordsys(coordsys,bqe)"><code></asyxml>*/
bqe changecoordsys(coordsys R, bqe bqe)
{/*<asyxml></code><documentation>Returns the bivariate quadratic equation relatively to 'R'.</documentation></function></asyxml>*/
  pair i = coordinates(changecoordsys(R, vector(defaultcoordsys,
                                             bqe.coordsys.i)));
  pair j = coordinates(changecoordsys(R, vector(defaultcoordsys,
                                             bqe.coordsys.j)));
  pair O = coordinates(changecoordsys(R, point(defaultcoordsys,
                                            bqe.coordsys.O)));
  real a = bqe.a[0], b = bqe.a[1], c = bqe.a[2], d = bqe.a[3], f = bqe.a[4], g = bqe.a[5];
  real ux = i.x, uy = i.y;
  real vx = j.x, vy = j.y;
  real ox = O.x, oy = O.y;
  real D = ux * vy - uy * vx;
  real ap = (a * vy^2 - b * uy * vy + c * uy^2)/D^2;
  real bpp = (-2 * a * vx * vy + b * ux * vy + b * uy * vx - 2 * c * ux * uy)/D^2;
  real cp = (a * vx^2 - b * ux * vx + c * ux^2)/D^2;
  real dp = (-2a * ox * vy^2 + 2a * oy * vx * vy + 2b * ox * uy * vy-
           b * oy * ux * vy - b * oy * uy * vx - 2c * ox * uy^2 + 2c * oy * uy * ux)/D^2+
    (d * vy - f * uy)/D;
  real fp = (2a * ox * vx * vy - b * ox * ux * vy - 2a * oy * vx^2-
           b * ox * uy * vx + 2 * b * oy * ux * vx + 2c * ox * ux * uy - 2c * oy * ux^2)/D^2+
    (f * ux - d * vx)/D;
  g = (a * ox^2 * vy^2 - 2a * ox * oy * vx * vy - b * ox^2 * uy * vy + b * ox * oy * ux * vy+
     a * oy^2 * vx^2 + b * ox * oy * uy * vx - b * oy^2 * ux * vx + c * ox^2 * uy^2-
     2 * c * ox * oy * ux * uy + c * oy^2 * ux^2)/D^2+
    (d * oy * vx + f * ox * uy - d * ox * vy - f * oy * ux)/D + g;
  bqe obqe;
  obqe.a = approximate(new real[] {ap, bpp, cp, dp, fp, g});
  obqe.coordsys = R;
  return obqe;
}

/*<asyxml><function type="bqe" signature="bqe(point,point,point,point,point)"><code></asyxml>*/
bqe bqe(point M1, point M2, point M3, point M4, point M5)
{/*<asyxml></code><documentation>Return the bqe of conic passing through the five points (if possible).</documentation></function></asyxml>*/
  coordsys R;
  pair[] pts;
  if (samecoordsys(M1, M2, M3, M4, M5)) {
    R = M1.coordsys;
    pts= new pair[] {M1.coordinates, M2.coordinates, M3.coordinates, M4.coordinates, M5.coordinates};
  } else {
    R = defaultcoordsys;
    pts= new pair[] {M1, M2, M3, M4, M5};
  }
  real[][] M;
  real[] x;
  bqe bqe;
  bqe.coordsys = R;
  for (int i = 0; i < 5; ++i) {// Try a = -1
    M[i] = new real[] {pts[i].x * pts[i].y, pts[i].y^2, pts[i].x, pts[i].y, 1};
    x[i] = pts[i].x^2;
  }
  if(abs(determinant(M)) < 1e-5) {// Try c = -1
    for (int i = 0; i < 5; ++i) {
      M[i] = new real[] {pts[i].x^2, pts[i].x * pts[i].y, pts[i].x, pts[i].y, 1};
      x[i] = pts[i].y^2;
    }
    real[] coef = solve(M, x);
    bqe.a = new real[] {coef[0], coef[1], -1, coef[2], coef[3], coef[4]};
  } else {
    real[] coef = solve(M, x);
    bqe.a = new real[] {-1, coef[0], coef[1], coef[2], coef[3], coef[4]};
  }
  bqe.a = approximate(bqe.a);
  return bqe;
}

/*<asyxml><function type="bool" signature="samecoordsys(bool...bqe[])"><code></asyxml>*/
bool samecoordsys(bool warn = true ... bqe[] bqes)
{/*<asyxml></code><documentation>Return true if all the bivariate quadratic equations have the same coordinate system.</documentation></function></asyxml>*/
  bool ret = true;
  coordsys t = bqes[0].coordsys;
  for (int i = 1; i < bqes.length; ++i) {
    ret = (t == bqes[i].coordsys);
    if(!ret) break;
    t = bqes[i].coordsys;
  }
  if(warn && !ret)
    warning("coodinatesystem",
            "the coordinate system of two bivariate quadratic equations are not
the same. The operation will be done relatively to the default coordinate
system.");
  return ret;
}

/*<asyxml><function type="real[]" signature="realquarticroots(real,real,real,real,real)"><code></asyxml>*/
real[] realquarticroots(real a, real b, real c, real d, real e)
{/*<asyxml></code><documentation>Return the real roots of the quartic equation ax^4 + b^x3 + cx^2 + dx = 0.</documentation></function></asyxml>*/
  static real Fuzz = sqrt(realEpsilon);
  pair[] zroots = quarticroots(a, b, c, d, e);
  real[] roots;
  real p(real x){return a * x^4 + b * x^3 + c * x^2 + d * x + e;}
  real prime(real x){return 4 * a * x^3 + 3 * b * x^2 + 2 * c * x + d;}
  real x;
  bool search = true;
  int n;
  void addroot(real x)
  {
    bool exist = false;
    for (int i = 0; i < roots.length; ++i) {
      if(abs(roots[i]-x) < 1e-5) {exist = true; break;}
    }
    if(!exist) roots.push(x);
  }
  for(int i = 0; i < zroots.length; ++i) {
    if(zroots[i].y == 0 || abs(p(zroots[i].x)) < Fuzz) addroot(zroots[i].x);
    else {
      if(abs(zroots[i].y) < 1e-3) {
        x = zroots[i].x;
        search = true;
        n = 200;
        while(search) {
          real tx = abs(p(x)) < Fuzz ? x : newton(iterations = n, p, prime, x);
          if(tx < realMax) {
            if(abs(p(tx)) < Fuzz) {
              addroot(tx);
              search = false;
            } else if(n < 200) n *=2;
            else {
              search = false;
            }
          } else search = false; //It's not a real root.
        }
      }
    }
  }
  return roots;
}

/*<asyxml><function type="point[]" signature="intersectionpoints(bqe,bqe)"><code></asyxml>*/
point[] intersectionpoints(bqe bqe1, bqe bqe2)
{/*<asyxml></code><documentation>Return the interscetion of the two conic sections whose equations are 'bqe1' and 'bqe2'.</documentation></function></asyxml>*/
  coordsys R = bqe1.coordsys;
  bqe lbqe1, lbqe2;
  real[] a, b;
  if(R != bqe2.coordsys) {
    R = currentcoordsys;
    a = changecoordsys(R, bqe1).a;
    b = changecoordsys(R, bqe2).a;
  } else {
    a = bqe1.a;
    b = bqe2.a;
  }
  static real e = 100 * sqrt(realEpsilon);
  real[] x, y, c;
  point[] P;
  if(abs(a[0]-b[0]) > e || abs(a[1]-b[1]) > e || abs(a[2]-b[2]) > e) {
    c = new real[] {-2 * a[0]*a[2]*b[0]*b[2]+a[0]*a[2]*b[1]^2 - a[0]*a[1]*b[2]*b[1]+a[1]^2 * b[0]*b[2]-
                  a[2]*a[1]*b[0]*b[1]+a[0]^2 * b[2]^2 + a[2]^2 * b[0]^2,
                  -a[2]*a[1]*b[0]*b[4]-a[2]*a[4]*b[0]*b[1]-a[1]*a[3]*b[2]*b[1]+2 * a[0]*a[2]*b[1]*b[4]-
                  a[0]*a[1]*b[2]*b[4]+a[1]^2 * b[2]*b[3]-2 * a[2]*a[3]*b[0]*b[2]-2 * a[0]*a[2]*b[2]*b[3]+
                  a[2]*a[3]*b[1]^2 - a[2]*a[1]*b[1]*b[3]+2 * a[1]*a[4]*b[0]*b[2]+2 * a[2]^2 * b[0]*b[3]-
                  a[0]*a[4]*b[2]*b[1]+2 * a[0]*a[3]*b[2]^2,
                  -a[3]*a[4]*b[2]*b[1]+a[2]*a[5]*b[1]^2 - a[1]*a[5]*b[2]*b[1]-a[1]*a[3]*b[2]*b[4]+
                  a[1]^2 * b[2]*b[5]-2 * a[2]*a[3]*b[2]*b[3]+2 * a[2]^2 * b[0]*b[5]+2 * a[0]*a[5]*b[2]^2 + a[3]^2 * b[2]^2-
                  2 * a[2]*a[5]*b[0]*b[2]+2 * a[1]*a[4]*b[2]*b[3]-a[2]*a[4]*b[1]*b[3]-2 * a[0]*a[2]*b[2]*b[5]+
                  a[2]^2 * b[3]^2 + 2 * a[2]*a[3]*b[1]*b[4]-a[2]*a[4]*b[0]*b[4]+a[4]^2 * b[0]*b[2]-a[2]*a[1]*b[3]*b[4]-
                  a[2]*a[1]*b[1]*b[5]-a[0]*a[4]*b[2]*b[4]+a[0]*a[2]*b[4]^2,
                  -a[4]*a[5]*b[2]*b[1]+a[2]*a[3]*b[4]^2 + 2 * a[3]*a[5]*b[2]^2 - a[2]*a[1]*b[4]*b[5]-
                  a[2]*a[4]*b[3]*b[4]+2 * a[2]^2 * b[3]*b[5]-2 * a[2]*a[3]*b[2]*b[5]-a[3]*a[4]*b[2]*b[4]-
                  2 * a[2]*a[5]*b[2]*b[3]-a[2]*a[4]*b[1]*b[5]+2 * a[1]*a[4]*b[2]*b[5]-a[1]*a[5]*b[2]*b[4]+
                  a[4]^2 * b[2]*b[3]+2 * a[2]*a[5]*b[1]*b[4],
                  -2 * a[2]*a[5]*b[2]*b[5]+a[4]^2 * b[2]*b[5]+a[5]^2 * b[2]^2 - a[4]*a[5]*b[2]*b[4]+a[2]*a[5]*b[4]^2+
                  a[2]^2 * b[5]^2 - a[2]*a[4]*b[4]*b[5]};
    x = realquarticroots(c[0], c[1], c[2], c[3], c[4]);
  } else {
    if(abs(b[4]-a[4]) > e){
      real D = (b[4]-a[4])^2;
      c = new real[] {(a[0]*b[4]^2 + (-a[1]*b[3]-2 * a[0]*a[4]+a[1]*a[3]) * b[4]+a[2]*b[3]^2+
                     (a[1]*a[4]-2 * a[2]*a[3]) * b[3]+a[0]*a[4]^2 - a[1]*a[3]*a[4]+a[2]*a[3]^2)/D,
                    -((a[1]*b[4]-2 * a[2]*b[3]-a[1]*a[4]+2 * a[2]*a[3]) * b[5]-a[3]*b[4]^2 + (a[4]*b[3]-a[1]*a[5]+a[3]*a[4]) * b[4]+(2 * a[2]*a[5]-a[4]^2) * b[3]+(a[1]*a[4]-2 * a[2]*a[3]) * a[5])/D,
                    a[2]*(a[5]-b[5])^2/D + a[4]*(a[5]-b[5])/(b[4]-a[4]) + a[5]};
      x = quadraticroots(c[0], c[1], c[2]);
    } else {
      if(abs(a[3]-b[3]) > e) {
        real D = b[3]-a[3];
        c = new real[] {a[2], (-a[1]*b[5] + a[4]*b[3] + a[1]*a[5] - a[3]*a[4])/D,
                      a[0]*(a[5]-b[5])^2/D^2 + a[3]*(a[5]-b[5])/D + a[5]};
        y = quadraticroots(c[0], c[1], c[2]);
        for (int i = 0; i < y.length; ++i) {
          c = new real[] {a[0], a[1]*y[i]+a[3], a[2]*y[i]^2 + a[4]*y[i]+a[5]};
          x = quadraticroots(c[0], c[1], c[2]);
          for (int j = 0; j < x.length; ++j) {
            if(abs(b[0]*x[j]^2 + b[1]*x[j]*y[i]+b[2]*y[i]^2 + b[3]*x[j]+b[4]*y[i]+b[5]) < 1e-5)
              P.push(point(R, (x[j], y[i])));
          }
        }
        return P;
      } else {
        if(abs(a[5]-b[5]) < e) abort("intersectionpoints: intersection of identical conics.");
      }
    }
  }
  for (int i = 0; i < x.length; ++i) {
    c = new real[] {a[2], a[1]*x[i]+a[4], a[0]*x[i]^2 + a[3]*x[i]+a[5]};
    y = quadraticroots(c[0], c[1], c[2]);
    for (int j = 0; j < y.length; ++j) {
      if(abs(b[0]*x[i]^2 + b[1]*x[i]*y[j]+b[2]*y[j]^2 + b[3]*x[i]+b[4]*y[j]+b[5]) < 1e-5)
        P.push(point(R, (x[i], y[j])));
    }
  }
  return P;
}

/*<asyxml><struct signature="conic"><code></asyxml>*/
struct conic
{/*<asyxml></code><documentation></documentation><property type = "real" signature="e,p,h"><code></asyxml>*/
  real e, p, h;/*<asyxml></code><documentation>BE CAREFUL: h = distance(F, D) and p = h * e (http://en.wikipedia.org/wiki/Ellipse)
                 While http://mathworld.wolfram.com/ takes p = distance(F,D).</documentation></property><property type = "point" signature="F"><code></asyxml>*/
  point F;/*<asyxml></code><documentation>Focus.</documentation></property><property type = "line" signature="D"><code></asyxml>*/
  line D;/*<asyxml></code><documentation>Directrix.</documentation></property><property type = "line" signature="l"><code></asyxml>*/
  line[] l;/*<asyxml></code><documentation>Case of degenerated conic (not yet implemented !).</documentation></property></asyxml>*/
}/*<asyxml></struct></asyxml>*/

bool degenerate(conic c)
{
  return !finite(c.p) || !finite(c.h);
}

/*ANCconic conic(point, line, real)ANC*/
conic conic(point F, line l, real e)
{/*DOC
   The conic section define by the eccentricity 'e', the focus 'F'
   and the directrix 'l'.
   Note that an eccentricity equal to 0 defines a circle centered at F,
   with a radius equal at the distance from 'F' to 'l'.
   If the coordinate system of 'F' and 'l' are not identical, the conic is
   attached to 'defaultcoordsys'.
   DOC*/
  if(e < 0) abort("conic: 'e' can't be negative.");
  conic oc;
  point[] P = standardizecoordsys(F, l.A, l.B);
  line ll;
  ll = line(P[1], P[2]);
  oc.e = e < epsgeo ? 0 : e; // Handle case of circle.
  oc.F = P[0];
  oc.D = ll;
  oc.h = distance(P[0], ll);
  oc.p = abs(e) < epsgeo ? oc.h : e * oc.h;
  return oc;
}

/*<asyxml><struct signature="circle"><code></asyxml>*/
struct circle
{/*<asyxml></code><documentation>All the calculus with this structure will be as exact as Asymptote can do.
   For a full precision, you must not cast 'circle' to 'path' excepted for drawing routines.</documentation></asyxml>*/
  /*<asyxml><property type = "point" signature="C"><code></asyxml>*/
  point C;/*<asyxml></code><documentation>Center</documentation></property><property><code></asyxml>*/
  real r;/*<asyxml></code><documentation>Radius</documentation></property><property><code></asyxml>*/
  line l;/*<asyxml></code><documentation>If the radius is infinite, this line is used instead of circle.</documentation></property></asyxml>*/
}/*<asyxml></struct></asyxml>*/

bool degenerate(circle c)
{
  return !finite(c.r);
}

line line(circle c){
  if(finite(c.r)) abort("Circle can not be casted to line here.");
  return c.l;
}

/*<asyxml><struct signature="ellipse"><code></asyxml>*/
struct ellipse
{/*<asyxml></code><documentation>Look at <html><a href = "http://mathworld.wolfram.com/Ellipse.html">http://mathworld.wolfram.com/Ellipse.html</a></html></documentation></asyxml>*/
  /*<asyxml><property type = "point" signature="F1,F2,C"><code></asyxml>*/
  restricted point F1,F2,C;/*<asyxml></code><documentation>Foci and center.</documentation></property><property type = "real" signature="a,b,c,e,p"><code></asyxml>*/
  restricted real a,b,c,e,p;/*<asyxml></code></property><property type = "real" signature="angle"><code></asyxml>*/
  restricted real angle;/*<asyxml></code><documentation>Value is degrees(F1 - F2).</documentation></property><property type = "line" signature="D1,D2"><code></asyxml>*/
  restricted line D1,D2;/*<asyxml></code><documentation>Directrices.</documentation></property><property type = "line" signature="l"><code></asyxml>*/
  line l;/*<asyxml></code><documentation>If one axis is infinite, this line is used instead of ellipse.</documentation></property></asyxml>*/
  /*<asyxml><method type = "void" signature="init(point,point,real)"><code></asyxml>*/
  void init(point f1, point f2, real a)
  {/*<asyxml></code><documentation>Ellipse given by foci and semimajor axis</documentation></method></asyxml>*/
    point[] P = standardizecoordsys(f1, f2);
    this.F1 = P[0];
    this.F2 = P[1];
    this.angle = abs(P[1]-P[0]) < 10 * epsgeo ? 0 : degrees(P[1]-P[0]);
    this.C = (P[0] + P[1])/2;
    this.a = a;
    if(!finite(a)) {
      this.l = line(P[0], P[1]);
      this.b = infinity;
      this.e = 0;
      this.c = 0;
    } else {
      this.c = abs(C - P[0]);
      this.b = this.c < epsgeo ? a : sqrt(a^2 - c^2); // Handle case of circle.
      this.e = this.c < epsgeo ? 0 : this.c/a; // Handle case of circle.
      if(this.e >= 1) abort("ellipse.init: wrong parameter: e >= 1.");
      this.p = a * (1 - this.e^2);
      if (this.c != 0) {// directrix is not set for a circle.
        point A = this.C + (a^2/this.c) * unit(P[0]-this.C);
        this.D1 = line(A, A + rotateO(90) * unit(A - this.C));
        this.D2 = reverse(rotate(180, C) * D1);
      }
    }
  }
}/*<asyxml></struct></asyxml>*/

bool degenerate(ellipse el)
{
  return (!finite(el.a) || !finite(el.b));
}

/*<asyxml><struct signature="parabola"><code></asyxml>*/
struct parabola
{/*<asyxml></code><documentation>Look at <html><a href = "http://mathworld.wolfram.com/Parabola.html">http://mathworld.wolfram.com/Parabola.html</a></html></documentation><property type = "point" signature="F,V"><code></asyxml>*/
  restricted point F,V;/*<asyxml></code><documentation>Focus and vertex</documentation></property><property type = "real" signature="a,p,e = 1"><code></asyxml>*/
  restricted real a,p,e = 1;/*<asyxml></code></property><property type = "real" signature="angle"><code></asyxml>*/
  restricted real angle;/*<asyxml></code><documentation>Angle, in degrees, of the line (FV).</documentation></property><property type = "line" signature="D"><code></asyxml>*/
  restricted line D;/*<asyxml></code><documentation>Directrix</documentation></property><property type = "pair" signature="bmin,bmax"><code></asyxml>*/
  pair bmin, bmax;/*<asyxml></code><documentation>The (left, bottom) and (right, top) coordinates of region bounding box for drawing the parabola.
                    If unset the current picture bounding box is used instead.</documentation></property></asyxml>*/

  /*<asyxml><method type = "void" signature="init(point,line)"><code></asyxml>*/
  void init(point F, line directrix)
  {/*<asyxml></code><documentation>Parabola given by focus and directrix.</documentation></method></asyxml>*/
    point[] P = standardizecoordsys(F, directrix.A, directrix.B);
    line l = line(P[1], P[2]);
    this.F = P[0];
    this.D = l;
    this.a = distance(P[0], l)/2;
    this.p = 2 * a;
    this.V = 0.5 * (F + projection(D) * P[0]);
    this.angle = degrees(F - V);
  }
}/*<asyxml></struct></asyxml>*/

/*<asyxml><struct signature="hyperbola"><code></asyxml>*/
struct hyperbola
{/*<asyxml></code><documentation><html>Look at <a href = "http://mathworld.wolfram.com/Hyperbola.html">http://mathworld.wolfram.com/Hyperbola.html</a></html></documentation><property type = "point" signature="F1,F2"><code></asyxml>*/
  restricted point F1,F2;/*<asyxml></code><documentation>Foci.</documentation></property><property type = "point" signature="C,V1,V2"><code></asyxml>*/
  restricted point C,V1,V2;/*<asyxml></code><documentation>Center and vertices.</documentation></property><property type = "real" signature="a,b,c,e,p"><code></asyxml>*/
  restricted real a,b,c,e,p;/*<asyxml></code><documentation></documentation></property><property type = "real" signature="angle"><code></asyxml>*/
  restricted real angle;/*<asyxml></code><documentation>Angle,in degrees,of the line (F1F2).</documentation></property><property type = "line" signature="D1,D2,A1,A2"><code></asyxml>*/
  restricted line D1,D2,A1,A2;/*<asyxml></code><documentation>Directrices and asymptotes.</documentation></property><property type = "pair" signature="bmin,bmax"><code></asyxml>*/
  pair bmin, bmax; /*<asyxml></code><documentation>The (left, bottom) and (right, top) coordinates of region bounding box for drawing the hyperbola.
                     If unset the current picture bounding box is used instead.</documentation></property></asyxml>*/

  /*<asyxml><method type = "void" signature="init(point,point,real)"><code></asyxml>*/
  void init(point f1, point f2, real a)
  {/*<asyxml></code><documentation>Hyperbola given by foci and semimajor axis.</documentation></method></asyxml>*/
    point[] P = standardizecoordsys(f1, f2);
    this.F1 = P[0];
    this.F2 = P[1];
    this.angle = degrees(F2 - F1);
    this.a = a;
    this.C = (P[0] + P[1])/2;
    this.c = abs(C - P[0]);
    this.e = this.c/a;
    if(this.e <= 1) abort("hyperbola.init: wrong parameter: e <= 1.");
    this.b = a * sqrt(this.e^2 - 1);
    this.p = a * (this.e^2 - 1);
    point A = this.C + (a^2/this.c) * unit(P[0]-this.C);
    this.D1 = line(A, A + rotateO(90) * unit(A - this.C));
    this.D2 = reverse(rotate(180, C) * D1);
    this.V1 = C + a * unit(F1 - C);
    this.V2 = C + a * unit(F2 - C);
    this.A1 = line(C, V1 + b * unit(rotateO(-90) * (C - V1)));
    this.A2 = line(C, V1 + b * unit(rotateO(90) * (C - V1)));
  }
}/*<asyxml></struct></asyxml>*/

/*<asyxml><variable type="int" signature="conicnodesfactor"><code></asyxml>*/
int conicnodesfactor = 1;/*<asyxml></code><documentation>Factor for the node number of all conics.</documentation></variable></asyxml>*/

/*<asyxml><variable type="int" signature="circlenodesnumberfactor"><code></asyxml>*/
int circlenodesnumberfactor = 100;/*<asyxml></code><documentation>Factor for the node number of circles.</documentation></variable></asyxml>*/
/*<asyxml><function type="int" signature="circlenodesnumber(real)"><code></asyxml>*/
int circlenodesnumber(real r)
{/*<asyxml></code><documentation>Return the number of nodes for drawing a circle of radius 'r'.</documentation></function></asyxml>*/
  if (circlenodesnumberfactor < 100)
    warning("circlenodesnumberfactor",
            "variable 'circlenodesnumberfactor' may be too small.");
  int oi = ceil(circlenodesnumberfactor * abs(r)^0.1);
  oi = 45 * floor(oi/45);
  return oi == 0 ? 4 : conicnodesfactor * oi;
}

/*<asyxml><function type="int" signature="circlenodesnumber(real,real,real)"><code></asyxml>*/
int circlenodesnumber(real r, real angle1, real angle2)
{/*<asyxml></code><documentation>Return the number of nodes to draw a circle arc.</documentation></function></asyxml>*/
  return (r > 0) ?
    ceil(circlenodesnumber(r) * abs(angle1 - angle2)/360) :
    ceil(circlenodesnumber(r) * abs((1 - abs(angle1 - angle2)/360)));
}

/*<asyxml><variable type="int" signature="ellispenodesnumberfactor"><code></asyxml>*/
int ellipsenodesnumberfactor = 250;/*<asyxml></code><documentation>Factor for the node number of ellispe (non-circle).</documentation></variable></asyxml>*/
/*<asyxml><function type="int" signature="ellipsenodesnumber(real,real)"><code></asyxml>*/
int ellipsenodesnumber(real a, real b)
{/*<asyxml></code><documentation>Return the number of nodes to draw a ellipse of axis 'a' and 'b'.</documentation></function></asyxml>*/
  if (ellipsenodesnumberfactor < 250)
    write("ellipsenodesnumberfactor",
          "variable 'ellipsenodesnumberfactor' maybe too small.");
  int tmp = circlenodesnumberfactor;
  circlenodesnumberfactor = ellipsenodesnumberfactor;
  int oi = circlenodesnumber(max(abs(a), abs(b))/min(abs(a), abs(b)));
  circlenodesnumberfactor = tmp;
  return conicnodesfactor * oi;
}

/*<asyxml><function type="int" signature="ellipsenodesnumber(real,real,real)"><code></asyxml>*/
int ellipsenodesnumber(real a, real b, real angle1, real angle2, bool dir)
{/*<asyxml></code><documentation>Return the number of nodes to draw an ellipse arc.</documentation></function></asyxml>*/
  real d;
  real da = angle2 - angle1;
  if(dir) {
    d = angle1 < angle2 ? da : 360 + da;
  } else {
    d = angle1 < angle2 ? -360 + da : da;
  }
  int n = floor(ellipsenodesnumber(a, b) * abs(d)/360);
  return n < 5 ? 5 : n;
}

/*<asyxml><variable type="int" signature="parabolanodesnumberfactor"><code></asyxml>*/
int parabolanodesnumberfactor = 100;/*<asyxml></code><documentation>Factor for the number of nodes of parabolas.</documentation></variable></asyxml>*/
/*<asyxml><function type="int" signature="parabolanodesnumber(parabola,real,real)"><code></asyxml>*/
int parabolanodesnumber(parabola p, real angle1, real angle2)
{/*<asyxml></code><documentation>Return the number of nodes for drawing a parabola.</documentation></function></asyxml>*/
  return conicnodesfactor * floor(0.01 * parabolanodesnumberfactor * abs(angle1 - angle2));
}

/*<asyxml><variable type="int" signature="hyperbolanodesnumberfactor"><code></asyxml>*/
int hyperbolanodesnumberfactor = 100;/*<asyxml></code><documentation>Factor for the number of nodes of hyperbolas.</documentation></variable></asyxml>*/
/*<asyxml><function type="int" signature="hyperbolanodesnumber(hyperbola,real,real)"><code></asyxml>*/
int hyperbolanodesnumber(hyperbola h, real angle1, real angle2)
{/*<asyxml></code><documentation>Return the number of nodes for drawing an hyperbola.</documentation></function></asyxml>*/
  return conicnodesfactor * floor(0.01 * hyperbolanodesnumberfactor * abs(angle1 - angle2)/h.e);
}

/*<asyxml><operator type = "conic" signature="+(conic,explicit point)"><code></asyxml>*/
conic operator +(conic c, explicit point M)
{/*<asyxml></code><documentation></documentation></operator></asyxml>*/
  return conic(c.F + M, c.D + M, c.e);
}
/*<asyxml><operator type = "conic" signature="-(conic,explicit point)"><code></asyxml>*/
conic operator -(conic c, explicit point M)
{/*<asyxml></code><documentation></documentation></operator></asyxml>*/
  return conic(c.F - M, c.D - M, c.e);
}
/*<asyxml><operator type = "conic" signature="+(conic,explicit pair)"><code></asyxml>*/
conic operator +(conic c, explicit pair m)
{/*<asyxml></code><documentation></documentation></operator></asyxml>*/
  point M = point(c.F.coordsys, m);
  return conic(c.F + M, c.D + M, c.e);
}
/*<asyxml><operator type = "conic" signature="-(conic,explicit pair)"><code></asyxml>*/
conic operator -(conic c, explicit pair m)
{/*<asyxml></code><documentation></documentation></operator></asyxml>*/
  point M = point(c.F.coordsys, m);
  return conic(c.F - M, c.D - M, c.e);
}
/*<asyxml><operator type = "conic" signature="+(conic,vector)"><code></asyxml>*/
conic operator +(conic c, vector v)
{/*<asyxml></code><documentation></documentation></operator></asyxml>*/
  return conic(c.F + v, c.D + v, c.e);
}
/*<asyxml><operator type = "conic" signature="-(conic,vector)"><code></asyxml>*/
conic operator -(conic c, vector v)
{/*<asyxml></code><documentation></documentation></operator></asyxml>*/
  return conic(c.F - v, c.D - v, c.e);
}

/*<asyxml><function type="coordsys" signature="coordsys(conic)"><code></asyxml>*/
coordsys coordsys(conic co)
{/*<asyxml></code><documentation>Return the coordinate system of 'co'.</documentation></function></asyxml>*/
  return co.F.coordsys;
}

/*<asyxml><function type="conic" signature="changecoordsys(coordsys,conic)"><code></asyxml>*/
conic changecoordsys(coordsys R, conic co)
{/*<asyxml></code><documentation>Change the coordinate system of 'co' to 'R'</documentation></function></asyxml>*/
  line l = changecoordsys(R, co.D);
  point F = changecoordsys(R, co.F);
  return conic(F, l, co.e);
}

/*<asyxml><typedef type = "polarconicroutine" return = "path" params = "conic, real, real, int, bool"><code></asyxml>*/
typedef path polarconicroutine(conic co, real angle1, real angle2, int n, bool direction);/*<asyxml></code><documentation>Routine type used to draw conics from 'angle1' to 'angle2'</documentation></typedef></asyxml>*/

/*<asyxml><function type="path" signature="arcfromfocus(conic,real,real,int,bool)"><code></asyxml>*/
path arcfromfocus(conic co, real angle1, real angle2, int n = 400, bool direction = CCW)
{/*<asyxml></code><documentation>Return the path of the conic section 'co' from angle1 to angle2 in degrees,
   drawing in the given direction, with n nodes.</documentation></function></asyxml>*/
  guide op;
  if (n < 1) return op;
  if (angle1 > angle2) {
    path g = arcfromfocus(co, angle2, angle1, n, !direction);
    return g == nullpath ? g : reverse(g);
  }
  point O = projection(co.D) * co.F;
  pair i = unit(locate(co.F) - locate(O));
  pair j = rotate(90) * i;
  coordsys Rp = cartesiansystem(co.F, i, j);
  real a1 = direction ? radians(angle1) : radians(angle2);
  real a2 = direction ? radians(angle2) : radians(angle1) + 2 * pi;
  real step = n == 1 ? 0 : (a2 - a1)/(n - 1);
  real a, r;
  for (int i = 0; i < n; ++i) {
    a = a1 + i * step;
    if(co.e >= 1) {
      r = 1 - co.e * cos(a);
      if(r > epsgeo) {
        r = co.p/r;
        op = op--Rp * Rp.polar(r, a);
      }
    } else {
      r = co.p/(1 - co.e * cos(a));
      op = op..Rp * Rp.polar(r, a);
    }
  }
  if(co.e < 1 && abs(abs(a2 - a1) - 2 * pi) < epsgeo) op = (path)op..cycle;

  return (direction ? op : op == nullpath ? op :reverse(op));
}

/*<asyxml><variable type="polarconicroutine" signature="currentpolarconicroutine"><code></asyxml>*/
polarconicroutine currentpolarconicroutine = arcfromfocus;/*<asyxml></code><documentation>Default routine used to cast conic section to path.</documentation></variable></asyxml>*/

/*<asyxml><function type="point" signature="angpoint(conic,real)"><code></asyxml>*/
point angpoint(conic co, real angle)
{/*<asyxml></code><documentation>Return the point of 'co' whose the angular (in degrees)
   coordinate is 'angle' (mesured from the focus of 'co', relatively
   to its 'natural coordinate system').</documentation></function></asyxml>*/
  coordsys R = coordsys(co);
  return point(R, point(arcfromfocus(co, angle, angle, 1, CCW), 0)/R);
}

/*<asyxml><operator type = "bool" signature="@(point,conic)"><code></asyxml>*/
bool operator @(point M, conic co)
{/*<asyxml></code><documentation>Return true iff 'M' on 'co'.</documentation></operator></asyxml>*/
  if(co.e == 0) return abs(abs(co.F - M) - co.p) < 10 * epsgeo;
  return abs(co.e * distance(M, co.D) - abs(co.F - M)) < 10 * epsgeo;
}

/*<asyxml><function type="coordsys" signature="coordsys(ellipse)"><code></asyxml>*/
coordsys coordsys(ellipse el)
{/*<asyxml></code><documentation>Return the coordinate system of 'el'.</documentation></function></asyxml>*/
  return el.F1.coordsys;
}

/*<asyxml><function type="coordsys" signature="canonicalcartesiansystem(ellipse)"><code></asyxml>*/
coordsys canonicalcartesiansystem(ellipse el)
{/*<asyxml></code><documentation>Return the canonical cartesian system of the ellipse 'el'.</documentation></function></asyxml>*/
  if(degenerate(el)) return cartesiansystem(el.l.A, el.l.u, el.l.v);
  pair O = locate(el.C);
  pair i = el.e == 0 ? el.C.coordsys.i : unit(locate(el.F1) - O);
  pair j = rotate(90) * i;
  return cartesiansystem(O, i, j);
}

/*<asyxml><function type="coordsys" signature="canonicalcartesiansystem(parabola)"><code></asyxml>*/
coordsys canonicalcartesiansystem(parabola p)
{/*<asyxml></code><documentation>Return the canonical cartesian system of a parabola,
   so that Origin = vertex of 'p' and directrix: x = -a.</documentation></function></asyxml>*/
  point A = projection(p.D) * p.F;
  pair O = locate((A + p.F)/2);
  pair i = unit(locate(p.F) - O);
  pair j = rotate(90) * i;
  return cartesiansystem(O, i, j);
}

/*<asyxml><function type="coordsys" signature="canonicalcartesiansystem(hyperbola)"><code></asyxml>*/
coordsys canonicalcartesiansystem(hyperbola h)
{/*<asyxml></code><documentation>Return the canonical cartesian system of an hyperbola.</documentation></function></asyxml>*/
  pair O = locate(h.C);
  pair i = unit(locate(h.F2) - O);
  pair j = rotate(90) * i;
  return cartesiansystem(O, i, j);
}

/*<asyxml><function type="ellipse" signature="ellipse(point,point,real)"><code></asyxml>*/
ellipse ellipse(point F1, point F2, real a)
{/*<asyxml></code><documentation>Return the ellipse whose the foci are 'F1' and 'F2'
   and the semimajor axis is 'a'.</documentation></function></asyxml>*/
  ellipse oe;
  oe.init(F1, F2, a);
  return oe;
}

/*<asyxml><constant type = "bool" signature="byfoci,byvertices"><code></asyxml>*/
restricted bool byfoci = true, byvertices = false;/*<asyxml></code><documentation>Constants useful for the routine 'hyperbola(point P1, point P2, real ae, bool byfoci = byfoci)'</documentation></constant></asyxml>*/

/*<asyxml><function type="hyperbola" signature="hyperbola(point,point,real,bool)"><code></asyxml>*/
hyperbola hyperbola(point P1, point P2, real ae, bool byfoci = byfoci)
{/*<asyxml></code><documentation>if 'byfoci = true':
   return the hyperbola whose the foci are 'P1' and 'P2'
   and the semimajor axis is 'ae'.
   else return the hyperbola whose vertexes are 'P1' and 'P2' with eccentricity 'ae'.</documentation></function></asyxml>*/
  hyperbola oh;
  point[] P = standardizecoordsys(P1, P2);
  if(byfoci) {
    oh.init(P[0], P[1], ae);
  } else {
    real a = abs(P[0]-P[1])/2;
    vector V = unit(P[0]-P[1]);
    point F1 = P[0] + a * (ae - 1) * V;
    point F2 = P[1]-a * (ae - 1) * V;
    oh.init(F1, F2, a);
  }
  return oh;
}

/*<asyxml><function type="ellipse" signature="ellipse(point,point,point)"><code></asyxml>*/
ellipse ellipse(point F1, point F2, point M)
{/*<asyxml></code><documentation>Return the ellipse passing through 'M' whose the foci are 'F1' and 'F2'.</documentation></function></asyxml>*/
  point P[] = standardizecoordsys(false, F1, F2, M);
  real a = abs(F1 - M) + abs(F2 - M);
  return ellipse(F1, F2, finite(a) ? a/2 : a);
}

/*<asyxml><function type="ellipse" signature="ellipse(point,real,real,real)"><code></asyxml>*/
ellipse ellipse(point C, real a, real b, real angle = 0)
{/*<asyxml></code><documentation>Return the ellipse centered at 'C' with semimajor axis 'a' along C--C + dir(angle),
   semiminor axis 'b' along the perpendicular.</documentation></function></asyxml>*/
  ellipse oe;
  coordsys R = C.coordsys;
  angle += degrees(R.i);
  if(a < b) {angle += 90; real tmp = a; a = b; b = tmp;}
  if(finite(a) && finite(b)) {
    real c = sqrt(abs(a^2 - b^2));
    point f1, f2;
    if(abs(a - b) < epsgeo) {
      f1 = C; f2 = C;
    } else {
      f1 = point(R, (locate(C) + rotate(angle) * (-c, 0))/R);
      f2 = point(R, (locate(C) + rotate(angle) * (c, 0))/R);
    }
    oe.init(f1, f2, a);
  } else {
    if(finite(b) || !finite(a)) oe.init(C, C + R.polar(1, angle), infinity);
    else oe.init(C, C + R.polar(1, 90 + angle), infinity);
  }
  return oe;
}

/*<asyxml><function type="ellipse" signature="ellipse(bqe)"><code></asyxml>*/
ellipse ellipse(bqe bqe)
{/*<asyxml></code><documentation>Return the ellipse a[0] * x^2 + a[1] * xy + a[2] * y^2 + a[3] * x + a[4] * y + a[5] = 0
   given in the coordinate system of 'bqe' with a[i] = bque.a[i].
   <url href = "http://mathworld.wolfram.com/QuadraticCurve.html"/>
   <url href = "http://mathworld.wolfram.com/Ellipse.html"/>.</documentation></function></asyxml>*/
  bqe lbqe = changecoordsys(defaultcoordsys, bqe);
  real a = lbqe.a[0], b = lbqe.a[1]/2, c = lbqe.a[2], d = lbqe.a[3]/2, f = lbqe.a[4]/2, g = lbqe.a[5];
  coordsys R = bqe.coordsys;
  string message = "ellipse: the given equation is not an equation of an ellipse.";
  real u = b^2 * g + d^2 * c + f^2 * a;
  real delta = a * c * g + b * f * d + d * b * f - u;
  if(abs(delta) < epsgeo) abort(message);
  real j = b^2 - a * c;
  real i = a + c;
  real dd = j * (sgnd(c - a) * sqrt((a - c)^2 + 4 * (b^2)) - c-a);
  real ddd = j * (-sgnd(c - a) * sqrt((a - c)^2 + 4 * (b^2)) - c-a);

  if(abs(ddd) < epsgeo || abs(dd) < epsgeo ||
     j >= -epsgeo || delta/sgnd(i) > 0) abort(message);

  real x = (c * d - b * f)/j, y = (a * f - b * d)/j;
  // real dir = abs(b) < epsgeo ? 0 : pi/2-0.5 * acot(0.5 * (c-a)/b);
  real dir = abs(b) < epsgeo ? 0 : 0.5 * acot(0.5 * (c - a)/b);
  if(dir * (c - a) * b < 0) dir = dir < 0 ? dir + pi/2 : dir - pi/2;
  real cd = cos(dir), sd = sin(dir);
  real t = a * cd^2 - 2 * b * cd * sd + c * sd^2;
  real tt = a * sd^2 + 2 * b * cd * sd + c * cd^2;
  real gg = -g + ((d * cd - f * sd)^2)/t + ((d * sd + f * cd)^2)/tt;
  t = t/gg; tt = tt/gg;
  // The equation of the ellipse is t * (x - center.x)^2 + tt * (y - center.y)^2 = 1;
  real aa, bb;
  aa = sqrt(2 * (u - 2 * b * d * f - a * c * g)/dd);
  bb = sqrt(2 * (u - 2 * b * d * f - a * c * g)/ddd);
  a = t > tt ? max(aa, bb) : min(aa, bb);
  b = t > tt ? min(aa, bb) : max(aa, bb);
  return ellipse(point(R, (x, y)/R),
                 a, b, degrees(pi/2 - dir - angle(R.i)));
}

/*<asyxml><function type="ellipse" signature="ellipse(point,point,point,point,point)"><code></asyxml>*/
ellipse ellipse(point M1, point M2, point M3, point M4, point M5)
{/*<asyxml></code><documentation>Return the ellipse passing through the five points (if possible)</documentation></function></asyxml>*/
  return ellipse(bqe(M1, M2, M3, M4, M5));
}

/*<asyxml><function type="bool" signature="inside(ellipse,point)"><code></asyxml>*/
bool inside(ellipse el, point M)
{/*<asyxml></code><documentation>Return 'true' iff 'M' is inside 'el'.</documentation></function></asyxml>*/
  return abs(el.F1 - M) + abs(el.F2 - M) - 2 * el.a < -epsgeo;
}

/*<asyxml><function type="bool" signature="inside(parabola,point)"><code></asyxml>*/
bool inside(parabola p, point M)
{/*<asyxml></code><documentation>Return 'true' if 'M' is inside 'p'.</documentation></function></asyxml>*/
  return distance(p.D, M) - abs(p.F - M) > epsgeo;
}

/*<asyxml><function type="parabola" signature="parabola(point,line)"><code></asyxml>*/
parabola parabola(point F, line l)
{/*<asyxml></code><documentation>Return the parabola whose focus is 'F' and directrix is 'l'.</documentation></function></asyxml>*/
  parabola op;
  op.init(F, l);
  return op;
}

/*<asyxml><function type="parabola" signature="parabola(point,point)"><code></asyxml>*/
parabola parabola(point F, point vertex)
{/*<asyxml></code><documentation>Return the parabola whose focus is 'F' and vertex is 'vertex'.</documentation></function></asyxml>*/
  parabola op;
  point[] P = standardizecoordsys(F, vertex);
  point A = rotate(180, P[1]) * P[0];
  point B = A + rotateO(90) * unit(P[1]-A);
  op.init(P[0], line(A, B));
  return op;
}

/*<asyxml><function type="parabola" signature="parabola(point,real,real)"><code></asyxml>*/
parabola parabola(point F, real a, real angle)
{/*<asyxml></code><documentation>Return the parabola whose focus is F, latus rectum is 4a and
   the angle of the axis of symmetry (in the coordinate system of F) is 'angle'.</documentation></function></asyxml>*/
  parabola op;
  coordsys R = F.coordsys;
  point A = F - point(R, R.polar(2a, radians(angle)));
  point B = A + point(R, R.polar(1, radians(90 + angle)));
  op.init(F, line(A, B));
  return op;
}

/*<asyxml><function type="bool" signature="isparabola(bqe)"><code></asyxml>*/
bool isparabola(bqe bqe)
{/*<asyxml></code><documentation>Return true iff 'bqe' is the equation of a parabola.</documentation></function></asyxml>*/
  bqe lbqe = changecoordsys(defaultcoordsys, bqe);
  real a = lbqe.a[0], b = lbqe.a[1]/2, c = lbqe.a[2], d = lbqe.a[3]/2, f = lbqe.a[4]/2, g = lbqe.a[5];
  real delta = a * c * g + b * f * d + d * b * f - (b^2 * g + d^2 * c + f^2 * a);
  return (abs(delta) > epsgeo && abs(b^2 - a * c) < epsgeo);
}

/*<asyxml><function type="parabola" signature="parabola(bqe)"><code></asyxml>*/
parabola parabola(bqe bqe)
{/*<asyxml></code><documentation>Return the parabola a[0]x^2 + a[1]xy + a[2]y^2 + a[3]x + a[4]y + a[5]] = 0 (a[n] means bqe.a[n]).
   <url href = "http://mathworld.wolfram.com/QuadraticCurve.html"/>
   <url href = "http://mathworld.wolfram.com/Parabola.html"/></documentation></function></asyxml>*/
  bqe lbqe = changecoordsys(defaultcoordsys, bqe);
  real a = lbqe.a[0], b = lbqe.a[1]/2, c = lbqe.a[2], d = lbqe.a[3]/2, f = lbqe.a[4]/2, g = lbqe.a[5];
  string message = "parabola: the given equation is not an equation of a parabola.";
  real delta = a * c * g + b * f * d + d * b * f - (b^2 * g + d^2 * c + f^2 * a);
  if(abs(delta) < 10 * epsgeo || abs(b^2 - a * c) > 10 * epsgeo) abort(message);
  real dir = abs(b) < epsgeo ? 0 : 0.5 * acot(0.5 * (c - a)/b);
  if(dir * (c - a) * b < 0) dir = dir < 0 ? dir + pi/2 : dir - pi/2;
  real cd = cos(dir), sd = sin(dir);
  real ap = a * cd^2 - 2 * b * cd * sd + c * sd^2;
  real cp = a * sd^2 + 2 * b * cd * sd + c * cd^2;
  real dp = d * cd - f * sd;
  real fp = d * sd + f * cd;
  real gp = g;
  parabola op;
  coordsys R = bqe.coordsys;
  // The equation of the parabola is ap * x'^2 + cp * y'^2 + 2dp * x'+2fp * y'+gp = 0
  if (abs(ap) < epsgeo) {/* directrix parallel to the rotated(dir) y-axis
                            equation: (y-vertex.y)^2 = 4 * a * (x-vertex)
                         */
    pair pvertex = rotate(degrees(-dir)) * (0.5(-gp + fp^2/cp)/dp, -fp/cp);
    real a = -0.5 * dp/cp;
    point vertex = point(R, pvertex/R);
    point focus = point(R, (pvertex + a * expi(-dir))/R);
    op = parabola(focus, vertex);

  } else {/* directrix parallel to the rotated(dir) x-axis
             equation: (x-vertex)^2 = 4 * a * (y-vertex.y)
          */
    pair pvertex = rotate(degrees(-dir)) * (-dp/ap, 0.5 * (-gp + dp^2/ap)/fp);
    real a = -0.5 * fp/ap;
    point vertex = point(R, pvertex/R);
    point focus = point(R, (pvertex + a * expi(pi/2 - dir))/R);
    op = parabola(focus, vertex);
  }
  return op;
}

/*<asyxml><function type="parabola" signature="parabola(point,point,point,line)"><code></asyxml>*/
parabola parabola(point M1, point M2, point M3, line l)
{/*<asyxml></code><documentation>Return the parabola passing through the three points with its directix
   parallel to the line 'l'.</documentation></function></asyxml>*/
  coordsys R;
  pair[] pts;
  if (samecoordsys(M1, M2, M3)) {
    R = M1.coordsys;
  } else {
    R = defaultcoordsys;
  }
  real gle = degrees(l);
  coordsys Rp = cartesiansystem(R.O, rotate(gle) * R.i, rotate(gle) * R.j);
  pts = new pair[] {coordinates(changecoordsys(Rp, M1)),
                  coordinates(changecoordsys(Rp, M2)),
                  coordinates(changecoordsys(Rp, M3))};
  real[][] M;
  real[] x;
  for (int i = 0; i < 3; ++i) {
    M[i] = new real[] {pts[i].x, pts[i].y, 1};
    x[i] = -pts[i].x^2;
  }
  real[] coef = solve(M, x);
  return parabola(changecoordsys(R, bqe(Rp, 1, 0, 0, coef[0], coef[1], coef[2])));
}

/*<asyxml><function type="parabola" signature="parabola(point,point,point,point,point)"><code></asyxml>*/
parabola parabola(point M1, point M2, point M3, point M4, point M5)
{/*<asyxml></code><documentation>Return the parabola passing through the five points.</documentation></function></asyxml>*/
  return parabola(bqe(M1, M2, M3, M4, M5));
}

/*<asyxml><function type="hyperbola" signature="hyperbola(point,real,real,real)"><code></asyxml>*/
hyperbola hyperbola(point C, real a, real b, real angle = 0)
{/*<asyxml></code><documentation>Return the hyperbola centered at 'C' with semimajor axis 'a' along C--C + dir(angle),
   semiminor axis 'b' along the perpendicular.</documentation></function></asyxml>*/
  hyperbola oh;
  coordsys R = C.coordsys;
  angle += degrees(R.i);
  real c = sqrt(a^2 + b^2);
  point f1 = point(R, (locate(C) + rotate(angle) * (-c, 0))/R);
  point f2 = point(R, (locate(C) + rotate(angle) * (c, 0))/R);
  oh.init(f1, f2, a);
  return oh;
}

/*<asyxml><function type="hyperbola" signature="hyperbola(bqe)"><code></asyxml>*/
hyperbola hyperbola(bqe bqe)
{/*<asyxml></code><documentation>Return the hyperbola a[0]x^2 + a[1]xy + a[2]y^2 + a[3]x + a[4]y + a[5]] = 0 (a[n] means bqe.a[n]).
   <url href = "http://mathworld.wolfram.com/QuadraticCurve.html"/>
   <url href = "http://mathworld.wolfram.com/Hyperbola.html"/></documentation></function></asyxml>*/
  bqe lbqe = changecoordsys(defaultcoordsys, bqe);
  real a = lbqe.a[0], b = lbqe.a[1]/2, c = lbqe.a[2], d = lbqe.a[3]/2, f = lbqe.a[4]/2, g = lbqe.a[5];
  string message = "hyperbola: the given equation is not an equation of a hyperbola.";
  real delta = a * c * g + b * f * d + d * b * f - (b^2 * g + d^2 * c + f^2 * a);
  if(abs(delta) < 10 * epsgeo || abs(b^2 - a * c) < 0) abort(message);
  real dir = abs(b) < epsgeo ? 0 : 0.5 * acot(0.5 * (c - a)/b);
  real cd = cos(dir), sd = sin(dir);
  real ap = a * cd^2 - 2 * b * cd * sd + c * sd^2;
  real cp = a * sd^2 + 2 * b * cd * sd + c * cd^2;
  real dp = d * cd - f * sd;
  real fp = d * sd + f * cd;
  real gp = -g + dp^2/ap + fp^2/cp;
  hyperbola op;
  coordsys R = bqe.coordsys;
  real j = b^2 - a * c;
  point C = point(R, ((c * d - b * f)/j, (a * f - b * d)/j)/R);
  real aa = gp/ap, bb = gp/cp;
  real a = sqrt(abs(aa)), b = sqrt(abs(bb));
  if(aa < 0) {dir -= pi/2; aa = a; a = b; b = aa;}
  return hyperbola(C, a, b, degrees(-dir - angle(R.i)));
}

/*<asyxml><function type="hyperbola" signature="hyperbola(point,point,point,point,point)"><code></asyxml>*/
hyperbola hyperbola(point M1, point M2, point M3, point M4, point M5)
{/*<asyxml></code><documentation>Return the hyperbola passing through the five points (if possible).</documentation></function></asyxml>*/
  return hyperbola(bqe(M1, M2, M3, M4, M5));
}

/*<asyxml><function type="hyperbola" signature="conj(hyperbola)"><code></asyxml>*/
hyperbola conj(hyperbola h)
{/*<asyxml></code><documentation>Conjugate.</documentation></function></asyxml>*/
  return hyperbola(h.C, h.b, h.a, 90 + h.angle);
}

/*<asyxml><function type="circle" signature="circle(explicit point,real)"><code></asyxml>*/
circle circle(explicit point C, real r)
{/*<asyxml></code><documentation>Circle given by center and radius.</documentation></function></asyxml>*/
  circle oc = new circle;
  oc.C = C;
  oc.r = r;
  if(!finite(r)) oc.l = line(C, C + vector(C.coordsys, (1, 0)));
  return oc;
}

/*<asyxml><function type="circle" signature="circle(point,point)"><code></asyxml>*/
circle circle(point A, point B)
{/*<asyxml></code><documentation>Return the circle of diameter AB.</documentation></function></asyxml>*/
  real r;
  circle oc;
  real a = abs(A), b = abs(B);
  if(finite(a) && finite(b)) {
    oc = circle((A + B)/2, abs(A - B)/2);
  } else {
    oc.r = infinity;
    if(finite(abs(A))) oc.l = line(A, A + unit(B));
    else {
      if(finite(abs(B))) oc.l = line(B, B + unit(A));
      else if(finite(abs(A - B)/2)) oc = circle((A + B)/2, abs(A - B)/2); else
        oc.l = line(A, B);
    }
  }
  return oc;
}

/*<asyxml><function type="circle" signature="circle(segment)"><code></asyxml>*/
circle circle(segment s)
{/*<asyxml></code><documentation>Return the circle of diameter 's'.</documentation></function></asyxml>*/
  return circle(s.A, s.B);
}

/*<asyxml><function type="point" signature="circumcenter(point,point,point)"><code></asyxml>*/
point circumcenter(point A, point B, point C)
{/*<asyxml></code><documentation>Return the circumcenter of triangle ABC.</documentation></function></asyxml>*/
  point[] P = standardizecoordsys(A, B, C);
  coordsys R = P[0].coordsys;
  pair a = A, b = B, c = C;
  pair mAB = (a + b)/2;
  pair mAC = (a + c)/2;
  pair pp = extension(mAB, rotate(90, mAB) * a, mAC, rotate(90, mAC) * c);
  return point(R, pp/R);
}

/*<asyxml><function type="circle" signature="circle(point,point,point)"><code></asyxml>*/
circle circle(point A, point B, point C)
{/*<asyxml></code><documentation>Return the circumcircle of the triangle ABC.</documentation></function></asyxml>*/
  if(collinear(A - B, A - C)) {
    circle oc;
    oc.r = infinity;
    oc.C = (A + B + C)/3;
    oc.l = line(oc.C, oc.C == A ? B : A);
    return oc;
  }
  point c = circumcenter(A, B, C);
  return circle(c, abs(c - A));
}

/*<asyxml><function type="circle" signature="circumcircle(point,point,point)"><code></asyxml>*/
circle circumcircle(point A, point B, point C)
{/*<asyxml></code><documentation>Return the circumcircle of the triangle ABC.</documentation></function></asyxml>*/
  return circle(A, B, C);
}

/*<asyxml><operator type = "circle" signature="*(real,explicit circle)"><code></asyxml>*/
circle operator *(real x, explicit circle c)
{/*<asyxml></code><documentation>Multiply the radius of 'c'.</documentation></operator></asyxml>*/
  return finite(c.r) ? circle(c.C, x * c.r) : c;
}
circle operator *(int x, explicit circle c)
{
  return finite(c.r) ? circle(c.C, x * c.r) : c;
}
/*<asyxml><operator type = "circle" signature="/(explicit circle,real)"><code></asyxml>*/
circle operator /(explicit circle c, real x)
{/*<asyxml></code><documentation>Divide the radius of 'c'</documentation></operator></asyxml>*/
  return finite(c.r) ? circle(c.C, c.r/x) : c;
}
circle operator /(explicit circle c, int x)
{
  return finite(c.r) ? circle(c.C, c.r/x) : c;
}
/*<asyxml><operator type = "circle" signature="+(explicit circle,explicit point)"><code></asyxml>*/
circle operator +(explicit circle c, explicit point M)
{/*<asyxml></code><documentation>Translation of 'c'.</documentation></operator></asyxml>*/
  return circle(c.C + M, c.r);
}
/*<asyxml><operator type = "circle" signature="-(explicit circle,explicit point)"><code></asyxml>*/
circle operator -(explicit circle c, explicit point M)
{/*<asyxml></code><documentation>Translation of 'c'.</documentation></operator></asyxml>*/
  return circle(c.C - M, c.r);
}
/*<asyxml><operator type = "circle" signature="+(explicit circle,pair)"><code></asyxml>*/
circle operator +(explicit circle c, pair m)
{/*<asyxml></code><documentation>Translation of 'c'.
   'm' represent coordinates in the coordinate system where 'c' is defined.</documentation></operator></asyxml>*/
  return circle(c.C + m, c.r);
}
/*<asyxml><operator type = "circle" signature="-(explicit circle,pair)"><code></asyxml>*/
circle operator -(explicit circle c, pair m)
{/*<asyxml></code><documentation>Translation of 'c'.
   'm' represent coordinates in the coordinate system where 'c' is defined.</documentation></operator></asyxml>*/
  return circle(c.C - m, c.r);
}
/*<asyxml><operator type = "circle" signature="+(explicit circle,vector)"><code></asyxml>*/
circle operator +(explicit circle c, vector m)
{/*<asyxml></code><documentation>Translation of 'c'.</documentation></operator></asyxml>*/
  return circle(c.C + m, c.r);
}
/*<asyxml><operator type = "circle" signature="-(explicit circle,vector)"><code></asyxml>*/
circle operator -(explicit circle c, vector m)
{/*<asyxml></code><documentation>Translation of 'c'.</documentation></operator></asyxml>*/
  return circle(c.C - m, c.r);
}
/*<asyxml><operator type = "real" signature="^(point,explicit circle)"><code></asyxml>*/
real operator ^(point M, explicit circle c)
{/*<asyxml></code><documentation>The power of 'M' with respect to the circle 'c'</documentation></operator></asyxml>*/
  return xpart((abs(locate(M) - locate(c.C)), c.r)^2);
}
/*<asyxml><operator type = "bool" signature="@(point,explicit circle)"><code></asyxml>*/
bool operator @(point M, explicit circle c)
{/*<asyxml></code><documentation>Return true iff 'M' is on the circle 'c'.</documentation></operator></asyxml>*/
  return finite(c.r) ?
    abs(abs(locate(M) - locate(c.C)) - abs(c.r)) <= 10 * epsgeo :
    M @ c.l;
}

/*<asyxml><operator type = "ellipse" signature="cast(circle)"><code></asyxml>*/
ellipse operator cast(circle c)
{/*<asyxml></code><documentation></documentation></operator></asyxml>*/
  return finite(c.r) ? ellipse(c.C, c.r, c.r, 0) : ellipse(c.l.A, c.l.B, infinity);
}

/*<asyxml><operator type = "circle" signature="cast(ellipse)"><code></asyxml>*/
circle operator cast(ellipse el)
{/*<asyxml></code><documentation></documentation></operator></asyxml>*/
  circle oc;
  bool infb = (!finite(el.a) || !finite(el.b));
  if(!infb && abs(el.a - el.b) > epsgeo)
    abort("Can not cast ellipse with different axis values to circle");
  oc = circle(el.C, infb ? infinity : el.a);
  oc.l = el.l.copy();
  return oc;
}

/*<asyxml><operator type = "ellipse" signature="cast(conic)"><code></asyxml>*/
ellipse operator cast(conic co)
{/*<asyxml></code><documentation>Cast a conic to an ellipse (can be a circle).</documentation></operator></asyxml>*/
  if(degenerate(co) && co.e < 1) return ellipse(co.l[0].A, co.l[0].B, infinity);
  ellipse oe;
  if(co.e < 1) {
    real a = co.p/(1 - co.e^2);
    real c = co.e * a;
    vector v = co.D.v;
    if(!sameside(co.D.A + v, co.F, co.D)) v = -v;
    point f2 = co.F + 2 * c * v;
    f2 = changecoordsys(co.F.coordsys, f2);
    oe = a == 0 ? ellipse(co.F, co.p, co.p, 0) : ellipse(co.F, f2, a);
  } else
    abort("casting: The conic section is not an ellipse.");
  return oe;
}

/*<asyxml><operator type = "parabola" signature="cast(conic)"><code></asyxml>*/
parabola operator cast(conic co)
{/*<asyxml></code><documentation>Cast a conic to a parabola.</documentation></operator></asyxml>*/
  parabola op;
  if(abs(co.e - 1) > epsgeo) abort("casting: The conic section is not a parabola.");
  op.init(co.F, co.D);
  return op;
}

/*<asyxml><operator type = "conic" signature="cast(parabola)"><code></asyxml>*/
conic operator cast(parabola p)
{/*<asyxml></code><documentation>Cast a parabola to a conic section.</documentation></operator></asyxml>*/
  return conic(p.F, p.D, 1);
}

/*<asyxml><operator type = "hyperbola" signature="cast(conic)"><code></asyxml>*/
hyperbola operator cast(conic co)
{/*<asyxml></code><documentation>Cast a conic section to an hyperbola.</documentation></operator></asyxml>*/
  hyperbola oh;
  if(co.e > 1) {
    real a = co.p/(co.e^2 - 1);
    real c = co.e * a;
    vector v = co.D.v;
    if(sameside(co.D.A + v, co.F, co.D)) v = -v;
    point f2 = co.F + 2 * c * v;
    f2 = changecoordsys(co.F.coordsys, f2);
    oh = hyperbola(co.F, f2, a);
  } else
    abort("casting: The conic section is not an hyperbola.");
  return oh;
}

/*<asyxml><operator type = "conic" signature="cast(hyperbola)"><code></asyxml>*/
conic operator cast(hyperbola h)
{/*<asyxml></code><documentation>Hyperbola to conic section.</documentation></operator></asyxml>*/
  return conic(h.F1, h.D1, h.e);
}

/*<asyxml><operator type = "conic" signature="cast(ellipse)"><code></asyxml>*/
conic operator cast(ellipse el)
{/*<asyxml></code><documentation>Ellipse to conic section.</documentation></operator></asyxml>*/
  conic oc;
  if(abs(el.c) > epsgeo) {
    real x = el.a^2/el.c;
    point O = (el.F1 + el.F2)/2;
    point A = O + x * unit(el.F1 - el.F2);
    oc = conic(el.F1, perpendicular(A, line(el.F1, el.F2)), el.e);
  } else {//The ellipse is a circle
    coordsys R = coordsys(el);
    point M = el.F1 + point(R, R.polar(el.a, 0));
    line l = line(rotate(90, M) * el.F1, M);
    oc = conic(el.F1, l, 0);
  }
  if(degenerate(el)) {
    oc.p = infinity;
    oc.h = infinity;
    oc.l = new line[]{el.l};
  }
  return oc;
}

/*<asyxml><operator type = "conic" signature="cast(circle)"><code></asyxml>*/
conic operator cast(circle c)
{/*<asyxml></code><documentation>Circle to conic section.</documentation></operator></asyxml>*/
  return (conic)((ellipse)c);
}

/*<asyxml><operator type = "circle" signature="cast(conic)"><code></asyxml>*/
circle operator cast(conic c)
{/*<asyxml></code><documentation>Conic section to circle.</documentation></operator></asyxml>*/
  ellipse el = (ellipse)c;
  circle oc;
  if(abs(el.a - el.b) < epsgeo) {
    oc = circle(el.C, el.a);
    if(degenerate(c)) oc.l = c.l[0];
  }
  else abort("Can not cast this conic to a circle");
  return oc;
}

/*<asyxml><operator type = "ellipse" signature="*(transform,ellipse)"><code></asyxml>*/
ellipse operator *(transform t, ellipse el)
{/*<asyxml></code><documentation>Provide transform * ellipse.</documentation></operator></asyxml>*/
  if(!degenerate(el)) {
    point[] ep;
    for (int i = 0; i < 360; i += 72) {
      ep.push(t * angpoint(el, i));
    }
    ellipse oe = ellipse(ep[0], ep[1], ep[2], ep[3], ep[4]);
    if(angpoint(oe, 0) != ep[0]) return ellipse(oe.F2, oe.F1, oe.a);
    return oe;
  }
  return ellipse(t * el.l.A, t * el.l.B, infinity);
}

/*<asyxml><operator type = "parabola" signature="*(transform,parabola)"><code></asyxml>*/
parabola operator *(transform t, parabola p)
{/*<asyxml></code><documentation>Provide transform * parabola.</documentation></operator></asyxml>*/
  point[] P;
  P.push(t * angpoint(p, 45));
  P.push(t * angpoint(p, -45));
  P.push(t * angpoint(p, 180));
  parabola op = parabola(P[0], P[1], P[2], t * p.D);
  op.bmin = p.bmin;
  op.bmax = p.bmax;

  return op;
}

/*<asyxml><operator type = "ellipse" signature="*(transform,circle)"><code></asyxml>*/
ellipse operator *(transform t, circle c)
{/*<asyxml></code><documentation>Provide transform * circle.
   For example, 'circle C = scale(2) * circle' and 'ellipse E = xscale(2) * circle' are valid
   but 'circle C = xscale(2) * circle' is invalid.</documentation></operator></asyxml>*/
  return t * ((ellipse)c);
}

/*<asyxml><operator type = "hyperbola" signature="*(transform,hyperbola)"><code></asyxml>*/
hyperbola operator *(transform t, hyperbola h)
{/*<asyxml></code><documentation>Provide transform * hyperbola.</documentation></operator></asyxml>*/
  if (t == identity()) {
    return h;
  }

  point[] ep;
  for (int i = 90; i <= 270; i += 45) {
    ep.push(t * angpoint(h, i));
  }

  hyperbola oe = hyperbola(ep[0], ep[1], ep[2], ep[3], ep[4]);
  if(angpoint(oe, 90) != ep[0]) {
    oe = hyperbola(oe.F2, oe.F1, oe.a);
  }

  oe.bmin = h.bmin;
  oe.bmax = h.bmax;

  return oe;
}

/*<asyxml><operator type = "conic" signature="*(transform,conic)"><code></asyxml>*/
conic operator *(transform t, conic co)
{/*<asyxml></code><documentation>Provide transform * conic.</documentation></operator></asyxml>*/
  if(co.e < 1) return (t * ((ellipse)co));
  if(co.e == 1) return (t * ((parabola)co));
  return (t * ((hyperbola)co));
}

/*<asyxml><operator type = "ellipse" signature="*(real,ellipse)"><code></asyxml>*/
ellipse operator *(real x, ellipse el)
{/*<asyxml></code><documentation>Identical but more efficient (rapid) than 'scale(x, el.C) * el'.</documentation></operator></asyxml>*/
  return degenerate(el) ? el : ellipse(el.C, x * el.a, x * el.b, el.angle);
}

/*<asyxml><operator type = "ellipse" signature="/(ellipse,real)"><code></asyxml>*/
ellipse operator /(ellipse el, real x)
{/*<asyxml></code><documentation>Identical but more efficient (rapid) than 'scale(1/x, el.C) * el'.</documentation></operator></asyxml>*/
  return degenerate(el) ? el : ellipse(el.C, el.a/x, el.b/x, el.angle);
}

/*<asyxml><function type="path" signature="arcfromcenter(ellipse,real,real,int,bool)"><code></asyxml>*/
path arcfromcenter(ellipse el, real angle1, real angle2,
                   bool direction=CCW,
                   int n=ellipsenodesnumber(el.a,el.b,angle1,angle2,direction))
{/*<asyxml></code><documentation>Return the path of the ellipse 'el' from angle1 to angle2 in degrees,
   drawing in the given direction, with n nodes.
   The angles are mesured relatively to the  axis (C,x-axis) where C is
   the center of the ellipse.</documentation></function></asyxml>*/
  if(degenerate(el)) abort("arcfromcenter: can not convert degenerated ellipse to path.");
  if (angle1 > angle2)
    return reverse(arcfromcenter(el, angle2, angle1, !direction, n));

  guide op;
  coordsys Rp=coordsys(el);
  if (n < 1) return op;

  interpolate join = operator ..;
  real stretch = max(el.a/el.b, el.b/el.a);

  if (stretch > 10) {
    n *= floor(stretch/5);
    join = operator --;
  }

  real a1 = direction ? radians(angle1) : radians(angle2);
  real a2 = direction ? radians(angle2) : radians(angle1) + 2 * pi;
  real step=(a2 - a1)/(n != 1 ? n-1 : 1);
  real a, r;
  real da = radians(el.angle);

  for (int i=0; i < n; ++i) {
    a = a1 + i * step;
    r = el.b/sqrt(1 - (el.e * cos(a))^2);
    op = join(op, Rp*Rp.polar(r, da + a));
  }

  return shift(el.C.x*Rp.i + el.C.y*Rp.j) * (direction ? op : reverse(op));
}

/*<asyxml><function type="path" signature="arcfromcenter(hyperbola,real,real,int,bool)"><code></asyxml>*/
path arcfromcenter(hyperbola h, real angle1, real angle2,
                   int n = hyperbolanodesnumber(h, angle1, angle2),
                   bool direction = CCW)
{/*<asyxml></code><documentation>Return the path of the hyperbola 'h' from angle1 to angle2 in degrees,
   drawing in the given direction, with n nodes.
   The angles are mesured relatively to the axis (C, x-axis) where C is
   the center of the hyperbola.</documentation></function></asyxml>*/
  guide op;
  coordsys Rp = coordsys(h);
  if (n < 1) return op;
  if (angle1 > angle2) {
    path g = reverse(arcfromcenter(h, angle2, angle1, n, !direction));
    return g == nullpath ? g : reverse(g);
  }
  real a1 = direction ? radians(angle1) : radians(angle2);
  real a2 = direction ? radians(angle2) : radians(angle1) + 2 * pi;
  real step = (a2 - a1)/(n != 1 ? n - 1 : 1);
  real a, r;
  typedef guide interpolate(... guide[]);
  interpolate join = operator ..;
  real da = radians(h.angle);
  for (int i = 0; i < n; ++i) {
    a = a1 + i * step;
    r = (h.b * cos(a))^2 - (h.a * sin(a))^2;
    if(r > epsgeo) {
      r = sqrt(h.a^2 * h.b^2/r);
      op = join(op, Rp * Rp.polar(r, a + da));
      join = operator ..;
    } else join = operator --;
  }
  return shift(h.C.x * Rp.i + h.C.y * Rp.j)*
    (direction ? op : op == nullpath ? op : reverse(op));
}

/*<asyxml><function type="path" signature="arcfromcenter(explicit conic,real,real,int,bool)"><code></asyxml>*/
path arcfromcenter(explicit conic co, real angle1, real angle2,
                   int n, bool direction = CCW)
{/*<asyxml></code><documentation>Use arcfromcenter(ellipse, ...) or arcfromcenter(hyperbola, ...) depending of
   the eccentricity of 'co'.</documentation></function></asyxml>*/
  path g;
  if(co.e < 1)
    g = arcfromcenter((ellipse)co, angle1,
                    angle2, direction, n);
  else if(co.e > 1)
    g = arcfromcenter((hyperbola)co, angle1,
                    angle2, n, direction);
  else abort("arcfromcenter: does not exist for a parabola.");
  return g;
}

/*<asyxml><constant type = "polarconicroutine" signature="fromCenter"><code></asyxml>*/
restricted polarconicroutine fromCenter = arcfromcenter;/*<asyxml></code><documentation></documentation></constant></asyxml>*/
/*<asyxml><constant type = "polarconicroutine" signature="fromFocus"><code></asyxml>*/
restricted polarconicroutine fromFocus = arcfromfocus;/*<asyxml></code><documentation></documentation></constant></asyxml>*/

/*<asyxml><function type="bqe" signature="equation(ellipse)"><code></asyxml>*/
bqe equation(ellipse el)
{/*<asyxml></code><documentation>Return the coefficients of the equation of the ellipse in its coordinate system:
   bqe.a[0] * x^2 + bqe.a[1] * x * y + bqe.a[2] * y^2 + bqe.a[3] * x + bqe.a[4] * y + bqe.a[5] = 0.
   One can change the coordinate system of 'bqe' using the routine 'changecoordsys'.</documentation></function></asyxml>*/
  pair[] pts;
  for (int i = 0; i < 360; i += 72)
    pts.push(locate(angpoint(el, i)));

  real[][] M;
  real[] x;
  for (int i = 0; i < 5; ++i) {
    M[i] = new real[] {pts[i].x * pts[i].y, pts[i].y^2, pts[i].x, pts[i].y, 1};
    x[i] = -pts[i].x^2;
  }
  real[] coef = solve(M, x);
  bqe bqe = changecoordsys(coordsys(el),
                         bqe(defaultcoordsys,
                             1, coef[0], coef[1], coef[2], coef[3], coef[4]));
  bqe.a = approximate(bqe.a);
  return bqe;
}

/*<asyxml><function type="bqe" signature="equation(parabola)"><code></asyxml>*/
bqe equation(parabola p)
{/*<asyxml></code><documentation>Return the coefficients of the equation of the parabola in its coordinate system.
   bqe.a[0] * x^2 + bqe.a[1] * x * y + bqe.a[2] * y^2 + bqe.a[3] * x + bqe.a[4] * y + bqe.a[5] = 0
   One can change the coordinate system of 'bqe' using the routine 'changecoordsys'.</documentation></function></asyxml>*/
  coordsys R = canonicalcartesiansystem(p);
  parabola tp = changecoordsys(R, p);
  point A = projection(tp.D) * point(R, (0, 0));
  real a = abs(A);
  return changecoordsys(coordsys(p),
                        bqe(R, 0, 0, 1, -4 * a, 0, 0));
}

/*<asyxml><function type="bqe" signature="equation(hyperbola)"><code></asyxml>*/
bqe equation(hyperbola h)
{/*<asyxml></code><documentation>Return the coefficients of the equation of the hyperbola in its coordinate system.
   bqe.a[0] * x^2 + bqe.a[1] * x * y + bqe.a[2] * y^2 + bqe.a[3] * x + bqe.a[4] * y + bqe.a[5] = 0
   One can change the coordinate system of 'bqe' using the routine 'changecoordsys'.</documentation></function></asyxml>*/
  coordsys R = canonicalcartesiansystem(h);
  return changecoordsys(coordsys(h),
                        bqe(R, 1/h.a^2, 0, -1/h.b^2, 0, 0, -1));
}

/*<asyxml><operator type = "path" signature="cast(ellipse)"><code></asyxml>*/
path operator cast(ellipse el)
{/*<asyxml></code><documentation>Cast ellipse to path.</documentation></operator></asyxml>*/
  if(degenerate(el))
    abort("Casting degenerated ellipse to path is not possible.");
  int n = el.e == 0 ? circlenodesnumber(el.a) : ellipsenodesnumber(el.a, el.b);
  return arcfromcenter(el, 0.0, 360, CCW, n)&cycle;
}

/*<asyxml><operator type = "path" signature="cast(circle)"><code></asyxml>*/
path operator cast(circle c)
{/*<asyxml></code><documentation>Cast circle to path.</documentation></operator></asyxml>*/
  return (path)((ellipse)c);
}

/*<asyxml><function type="real[]" signature="bangles(picture,parabola)"><code></asyxml>*/
real[] bangles(picture pic = currentpicture, parabola p)
{/*<asyxml></code><documentation>Return the array {ma, Ma} where 'ma' and 'Ma' are respectively
   the smaller and the larger angles for which the parabola 'p' is included
   in the bounding box of the picture 'pic'.</documentation></function></asyxml>*/
  pair bmin, bmax;
  pair[] b;
  if (p.bmin == p.bmax) {
    bmin = pic.userMin();
    bmax = pic.userMax();
  } else {
    bmin = p.bmin;bmax = p.bmax;
  }
  if(bmin.x == bmax.x || bmin.y == bmax.y || !finite(abs(bmin)) || !finite(abs(bmax)))
    return new real[] {0, 0};
  b[0] = bmin;
  b[1] = (bmax.x, bmin.y);
  b[2] = bmax;
  b[3] = (bmin.x, bmax.y);
  real[] eq = changecoordsys(defaultcoordsys, equation(p)).a;
  pair[] inter;
  for (int i = 0; i < 4; ++i) {
    pair[] tmp = intersectionpoints(b[i], b[(i + 1)%4], eq);
    for (int j = 0; j < tmp.length; ++j) {
      if(dot(b[i]-tmp[j], b[(i + 1)%4]-tmp[j]) <= epsgeo)
        inter.push(tmp[j]);
    }
  }
  pair F = p.F, V = p.V;
  real d = degrees(F - V);
  real[] a = sequence(new real(int n){
      return (360 - d + degrees(inter[n]-F))%360;
    }, inter.length);
  real ma = a.length != 0 ? min(a) : 0, Ma= a.length != 0 ? max(a) : 0;
  return new real[] {ma, Ma};
}

/*<asyxml><function type="real[][]" signature="bangles(picture,hyperbola)"><code></asyxml>*/
real[][] bangles(picture pic = currentpicture, hyperbola h)
{/*<asyxml></code><documentation>Return the array {{ma1, Ma1}, {ma2, Ma2}} where 'maX' and 'MaX' are respectively
   the smaller and the bigger angles (from h.FX) for which the hyperbola 'h' is included
   in the bounding box of the picture 'pic'.</documentation></function></asyxml>*/
  pair bmin, bmax;
  pair[] b;
  if (h.bmin == h.bmax) {
    bmin = pic.userMin();
    bmax = pic.userMax();
  } else {
    bmin = h.bmin;bmax = h.bmax;
  }
  if(bmin.x == bmax.x || bmin.y == bmax.y || !finite(abs(bmin)) || !finite(abs(bmax)))
    return new real[][] {{0, 0}, {0, 0}};
  b[0] = bmin;
  b[1] = (bmax.x, bmin.y);
  b[2] = bmax;
  b[3] = (bmin.x, bmax.y);
  real[] eq = changecoordsys(defaultcoordsys, equation(h)).a;
  pair[] inter0, inter1;
  pair C = locate(h.C);
  pair F1 = h.F1;
  for (int i = 0; i < 4; ++i) {
    pair[] tmp = intersectionpoints(b[i], b[(i + 1)%4], eq);
    for (int j = 0; j < tmp.length; ++j) {
      if(dot(b[i]-tmp[j], b[(i + 1)%4]-tmp[j]) <= epsgeo) {
        if(dot(F1 - C, tmp[j]-C) > 0) inter0.push(tmp[j]);
        else inter1.push(tmp[j]);
      }
    }
  }
  real d = degrees(F1 - C);
  real[] ma, Ma;
  pair[][] inter = new pair[][] {inter0, inter1};
  for (int i = 0; i < 2; ++i) {
    real[] a = sequence(new real(int n){
        return (360 - d + degrees(inter[i][n]-F1))%360;
      }, inter[i].length);
    ma[i] = a.length != 0 ? min(a) : 0;
    Ma[i] = a.length != 0 ? max(a) : 0;
  }
  return new real[][] {{ma[0], Ma[0]}, {ma[1], Ma[1]}};
}

/*<asyxml><operator type = "path" signature="cast(parabola)"><code></asyxml>*/
path operator cast(parabola p)
{/*<asyxml></code><documentation>Cast parabola to path.
   If possible, the returned path is restricted to the actual bounding box
   of the current picture if the variables 'p.bmin' and 'p.bmax' are not set else
   the bounding box of box(p.bmin, p.bmax) is used instead.</documentation></operator></asyxml>*/
  real[] bangles = bangles(p);
  int n = parabolanodesnumber(p, bangles[0], bangles[1]);
  return arcfromfocus(p, bangles[0], bangles[1], n, CCW);
}


/*<asyxml><function type="void" signature="draw(picture,Label,circle,align,pen,arrowbar,arrowbar,margin,Label,marker)"><code></asyxml>*/
void draw(picture pic = currentpicture, Label L = "", circle c,
          align align = NoAlign, pen p = currentpen,
          arrowbar arrow = None, arrowbar bar = None,
          margin margin = NoMargin, Label legend = "", marker marker = nomarker)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  if(degenerate(c)) draw(pic, L, c.l, align, p, arrow, legend, marker);
  else draw(pic, L, (path)c, align, p, arrow, bar, margin, legend, marker);
}

/*<asyxml><function type="void" signature="draw(picture,Label,ellipse,align,pen,arrowbar,arrowbar,margin,Label,marker)"><code></asyxml>*/
void draw(picture pic = currentpicture, Label L = "", ellipse el,
          align align = NoAlign, pen p = currentpen,
          arrowbar arrow = None, arrowbar bar = None,
          margin margin = NoMargin, Label legend = "", marker marker = nomarker)
{/*<asyxml></code><documentation></documentation>Draw the ellipse 'el' if it is not degenerated else draw 'el.l'.</function></asyxml>*/
  if(degenerate(el)) draw(pic, L, el.l, align, p, arrow, legend, marker);
  else draw(pic, L, (path)el, align, p, arrow, bar, margin, legend, marker);
}

/*<asyxml><function type="void" signature="draw(picture,Label,parabola,align,pen,arrowbar,arrowbar,margin,Label,marker)"><code></asyxml>*/
void draw(picture pic = currentpicture, Label L = "", parabola parabola,
          align align = NoAlign, pen p = currentpen,
          arrowbar arrow = None, arrowbar bar = None,
          margin margin = NoMargin, Label legend = "", marker marker = nomarker)
{/*<asyxml></code><documentation>Draw the parabola 'p' on 'pic' without (if possible) altering the
   size of picture pic.</documentation></function></asyxml>*/
  pic.add(new void (frame f, transform t, transform T, pair m, pair M) {
      // Reduce the bounds by the size of the pen and the margins.
      m -= min(p); M -= max(p);
      parabola.bmin = inverse(t) * m;
      parabola.bmax = inverse(t) * M;
      picture tmp;
      path pp = t * ((path) (T * parabola));

      if (pp != nullpath) {
        draw(tmp, L, pp, align, p, arrow, bar, NoMargin, legend, marker);
        add(f, tmp.fit());
      }
    }, true);

  pair m = pic.userMin(), M = pic.userMax();
  if(m != M) {
    pic.addBox(truepoint(SW), truepoint(NE));
  }
}

/*<asyxml><operator type = "path" signature="cast(hyperbola)"><code></asyxml>*/
path operator cast(hyperbola h)
{/*<asyxml></code><documentation>Cast hyperbola to path.
   If possible, the returned path is restricted to the actual bounding box
   of the current picture unless the variables 'h.bmin' and 'h.bmax'
   are set; in this case the bounding box of box(h.bmin, h.bmax) is used instead.
   Only the branch on the side of 'h.F1' is considered.</documentation></operator></asyxml>*/
  real[][] bangles = bangles(h);
  int n = hyperbolanodesnumber(h, bangles[0][0], bangles[0][1]);
  return arcfromfocus(h, bangles[0][0], bangles[0][1], n, CCW);
}

/*<asyxml><function type="void" signature="draw(picture,Label,hyperbola,align,pen,arrowbar,arrowbar,margin,Label,marker)"><code></asyxml>*/
void draw(picture pic = currentpicture, Label L = "", hyperbola h,
          align align = NoAlign, pen p = currentpen,
          arrowbar arrow = None, arrowbar bar = None,
          margin margin = NoMargin, Label legend = "", marker marker = nomarker)
{/*<asyxml></code><documentation>Draw the hyperbola 'h' on 'pic' without (if possible) altering the
   size of the picture pic.</documentation></function></asyxml>*/
  pic.add(new void (frame f, transform t, transform T, pair m, pair M) {
      // Reduce the bounds by the size of the pen and the margins.
      m -= min(p); M -= max(p);
      h.bmin = inverse(t) * m;
      h.bmax = inverse(t) * M;
      path hp;

      picture tmp;
      hp = t * ((path) (T * h));
      if (hp != nullpath) {
        draw(tmp, L, hp, align, p, arrow, bar, NoMargin, legend, marker);
      }

      hyperbola ht = hyperbola(h.F2, h.F1, h.a);
      ht.bmin = h.bmin;
      ht.bmax = h.bmax;

      hp = t * ((path) (T * ht));
      if (hp != nullpath) {
        draw(tmp, "", hp, align, p, arrow, bar, NoMargin, marker);
      }

      add(f, tmp.fit());
    }, true);

  pair m = pic.userMin(), M = pic.userMax();
  if(m != M)
    pic.addBox(truepoint(SW), truepoint(NE));
}

/*<asyxml><function type="void" signature="draw(picture,Label,explicit conic,align,pen,arrowbar,arrowbar,margin,Label,marker)"><code></asyxml>*/
void draw(picture pic = currentpicture, Label L = "", explicit conic co,
          align align = NoAlign, pen p = currentpen,
          arrowbar arrow = None, arrowbar bar = None,
          margin margin = NoMargin, Label legend = "", marker marker = nomarker)
{/*<asyxml></code><documentation>Use one of the routine 'draw(ellipse, ...)',
   'draw(parabola, ...)' or 'draw(hyperbola, ...)' depending of the value of eccentricity of 'co'.</documentation></function></asyxml>*/
  if(co.e == 0)
    draw(pic, L, (circle)co, align, p, arrow, bar, margin, legend, marker);
  else
    if(co.e < 1) draw(pic, L, (ellipse)co, align, p, arrow, bar, margin, legend, marker);
    else
      if(co.e == 1) draw(pic, L, (parabola)co, align, p, arrow, bar, margin, legend, marker);
      else
        if(co.e > 1) draw(pic, L, (hyperbola)co, align, p, arrow, bar, margin, legend, marker);
        else abort("draw: unknown conic.");
}

/*<asyxml><function type="int" signature="conicnodesnumber(conic,real,real)"><code></asyxml>*/
int conicnodesnumber(conic co, real angle1, real angle2, bool dir = CCW)
{/*<asyxml></code><documentation>Return the number of node to draw a conic arc.</documentation></function></asyxml>*/
  int oi;
  if(co.e == 0) {
    circle c = (circle)co;
    oi = circlenodesnumber(c.r, angle1, angle2);
  } else if(co.e < 1) {
    ellipse el = (ellipse)co;
    oi = ellipsenodesnumber(el.a, el.b, angle1, angle2, dir);
  } else if(co.e == 1) {
    parabola p = (parabola)co;
    oi = parabolanodesnumber(p, angle1, angle2);
  } else {
    hyperbola h = (hyperbola)co;
    oi = hyperbolanodesnumber(h, angle1, angle2);
  }
  return oi;
}

/*<asyxml><operator type = "path" signature="cast(conic)"><code></asyxml>*/
path operator cast(conic co)
{/*<asyxml></code><documentation>Cast conic section to path.</documentation></operator></asyxml>*/
  if(co.e < 1) return (path)((ellipse)co);
  if(co.e == 1) return (path)((parabola)co);
  return (path)((hyperbola)co);
}

/*<asyxml><function type="bqe" signature="equation(explicit conic)"><code></asyxml>*/
bqe equation(explicit conic co)
{/*<asyxml></code><documentation>Return the coefficients of the equation of conic section in its coordinate system:
   bqe.a[0] * x^2 + bqe.a[1] * x * y + bqe.a[2] * y^2 + bqe.a[3] * x + bqe.a[4] * y + bqe.a[5] = 0.
   One can change the coordinate system of 'bqe' using the routine 'changecoordsys'.</documentation></function></asyxml>*/
  bqe obqe;
  if(co.e == 0)
    obqe = equation((circle)co);
  else
    if(co.e < 1) obqe = equation((ellipse)co);
    else
      if(co.e == 1) obqe = equation((parabola)co);
      else
        if(co.e > 1) obqe = equation((hyperbola)co);
        else abort("draw: unknown conic.");
  return obqe;
}

/*<asyxml><function type="string" signature="conictype(bqe)"><code></asyxml>*/
string conictype(bqe bqe)
{/*<asyxml></code><documentation>Returned values are "ellipse" or "parabola" or "hyperbola"
   depending of the conic section represented by 'bqe'.</documentation></function></asyxml>*/
  bqe lbqe = changecoordsys(defaultcoordsys, bqe);
  string os = "degenerated";
  real a = lbqe.a[0], b = lbqe.a[1]/2, c = lbqe.a[2], d = lbqe.a[3]/2, f = lbqe.a[4]/2, g = lbqe.a[5];
  real delta = a * c * g + b * f * d + d * b * f - (b^2 * g + d^2 * c + f^2 * a);
  if(abs(delta) < 10 * epsgeo) return os;
  real J = a * c - b^2;
  real I = a + c;
  if(J > epsgeo) {
    if(delta/I < -epsgeo);
    os = "ellipse";
  } else {
    if(abs(J) < epsgeo) os = "parabola"; else os = "hyperbola";
  }
  return os;
}

/*<asyxml><function type="conic" signature="conic(point,point,point,point,point)"><code></asyxml>*/
conic conic(point M1, point M2, point M3, point M4, point M5)
{/*<asyxml></code><documentation>Return the conic passing through 'M1', 'M2', 'M3', 'M4' and 'M5' if the conic is not degenerated.</documentation></function></asyxml>*/
  bqe bqe = bqe(M1, M2, M3, M4, M5);
  string ct = conictype(bqe);
  if(ct == "degenerated") abort("conic: degenerated conic passing through five points.");
  if(ct == "ellipse") return ellipse(bqe);
  if(ct == "parabola") return parabola(bqe);
  return hyperbola(bqe);
}

/*<asyxml><function type="coordsys" signature="canonicalcartesiansystem(hyperbola)"><code></asyxml>*/
coordsys canonicalcartesiansystem(explicit conic co)
{/*<asyxml></code><documentation>Return the canonical cartesian system of the conic 'co'.</documentation></function></asyxml>*/
  if(co.e < 1) return canonicalcartesiansystem((ellipse)co);
  else if(co.e == 1) return canonicalcartesiansystem((parabola)co);
  return canonicalcartesiansystem((hyperbola)co);
}

/*<asyxml><function type="bqe" signature="canonical(bqe)"><code></asyxml>*/
bqe canonical(bqe bqe)
{/*<asyxml></code><documentation>Return the bivariate quadratic equation relative to the
   canonical coordinate system of the conic section represented by 'bqe'.</documentation></function></asyxml>*/
  string type = conictype(bqe);
  if(type == "") abort("canonical: the equation can not be performed.");
  bqe obqe;
  if(type == "ellipse") {
    ellipse el = ellipse(bqe);
    obqe = changecoordsys(canonicalcartesiansystem(el), equation(el));
  } else {
    if(type == "parabola") {
      parabola p = parabola(bqe);
      obqe = changecoordsys(canonicalcartesiansystem(p), equation(p));
    } else {
      hyperbola h = hyperbola(bqe);
      obqe = changecoordsys(canonicalcartesiansystem(h), equation(h));
    }
  }
  return obqe;
}

/*<asyxml><function type="conic" signature="conic(bqe)"><code></asyxml>*/
conic conic(bqe bqe)
{/*<asyxml></code><documentation>Return the conic section represented by the bivariate quartic equation 'bqe'.</documentation></function></asyxml>*/
  string type = conictype(bqe);
  if(type == "") abort("canonical: the equation can not be performed.");
  conic oc;
  if(type == "ellipse") {
    oc = ellipse(bqe);
  } else {
    if(type == "parabola") oc = parabola(bqe); else oc = hyperbola(bqe);
  }
  return oc;
}

/*<asyxml><function type="real" signature="arclength(circle)"><code></asyxml>*/
real arclength(circle c)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  return c.r * 2 * pi;
}

/*<asyxml><function type="real" signature="focusToCenter(ellipse,real)"><code></asyxml>*/
real focusToCenter(ellipse el, real a)
{/*<asyxml></code><documentation>Return the angle relatively to the center of 'el' for the angle 'a'
   given relatively to the focus of 'el'.</documentation></function></asyxml>*/
  pair p = point(fromFocus(el, a, a, 1, CCW), 0);
  pair c = locate(el.C);
  real d = degrees(p - c) - el.angle;
  d = abs(d) < epsgeo ? 0 : d; // Avoid -1e-15
  return d%(sgnd(a) * 360);
}

/*<asyxml><function type="real" signature="centerToFocus(ellipse,real)"><code></asyxml>*/
real centerToFocus(ellipse el, real a)
{/*<asyxml></code><documentation>Return the angle relatively to the focus of 'el' for the angle 'a'
   given relatively to the center of 'el'.</documentation></function></asyxml>*/
  pair P = point(fromCenter(el, a, a, 1, CCW), 0);
  pair F1 = locate(el.F1);
  pair F2 = locate(el.F2);
  real d = degrees(P - F1) - degrees(F2 - F1);
  d = abs(d) < epsgeo ? 0 : d; // Avoid -1e-15
  return d%(sgnd(a) * 360);
}

/*<asyxml><function type="real" signature="arclength(ellipse)"><code></asyxml>*/
real arclength(ellipse el)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  return degenerate(el) ? infinity : 4 * el.a * elle(pi/2, el.e);
}

/*<asyxml><function type="real" signature="arclength(ellipse,real,real,bool,polarconicroutine)"><code></asyxml>*/
real arclength(ellipse el, real angle1, real angle2,
               bool direction = CCW,
               polarconicroutine polarconicroutine = currentpolarconicroutine)
{/*<asyxml></code><documentation>Return the length of the arc of the ellipse between 'angle1'
   and 'angle2'.
   'angle1' and 'angle2' must be in the interval ]-360;+oo[ if polarconicroutine = fromFocus,
   ]-oo;+oo[ if polarconicroutine = fromCenter.</documentation></function></asyxml>*/
  if(degenerate(el)) return infinity;
  if(angle1 > angle2) return arclength(el, angle2, angle1, !direction, polarconicroutine);
  //   path g;int n = 1000;
  //   if(el.e == 0) g = arcfromcenter(el, angle1, angle2, n, direction);
  //   if(el.e != 1) g = polarconicroutine(el, angle1, angle2, n, direction);
  //   write("with path = ", arclength(g));
  if(polarconicroutine == fromFocus) {
    //   dot(point(fromFocus(el, angle1, angle1, 1, CCW), 0), 2mm + blue);
    //   dot(point(fromFocus(el, angle2, angle2, 1, CCW), 0), 2mm + blue);
    //   write("fromfocus1 = ", angle1);
    //   write("fromfocus2 = ", angle2);
    real gle1 = focusToCenter(el, angle1);
    real gle2 = focusToCenter(el, angle2);
    if((gle1 - gle2) * (angle1 - angle2) > 0) {
      angle1 = gle1; angle2 = gle2;
    } else {
      angle1 = gle2; angle2 = gle1;
    }
    //   dot(point(fromCenter(el, angle1, angle1, 1, CCW), 0), 1mm + red);
    //   dot(point(fromCenter(el, angle2, angle2, 1, CCW), 0), 1mm + red);
    //   write("fromcenter1 = ", angle1);
    //   write("fromcenter2 = ", angle2);
  }
  if(angle1 < 0 || angle2 < 0) return arclength(el, 180 + angle1, 180 + angle2, direction, fromCenter);
  real a1 = direction ? angle1 : angle2;
  real a2 = direction ? angle2 : angle1 + 360;
  real elleq = el.a * elle(pi/2, el.e);
  real S(real a)
  {//Return the arclength from 0 to the angle 'a' (in degrees)
    // given form the center of the ellipse.
    real gle = atan(el.a * Tan(a)/el.b)+
      pi * (((a%90 == 0 && a != 0) ? floor(a/90) - 1 : floor(a/90)) -
          ((a%180 == 0) ? 0 : floor(a/180)) -
          (a%360 == 0 ? floor(a/(360)) : 0));
    /* // Uncomment to visualize the used branches
       unitsize(2cm, 1cm);
       import graph;

       real xmin = 0, xmax = 3pi;

       xlimits( xmin, xmax);
       ylimits( 0, 10);
       yaxis( "y" , LeftRight(), RightTicks(pTick=.8red, ptick = lightgrey, extend = true));
       xaxis( "x - value", BottomTop(), Ticks(Label("$%.2f$", red), Step = pi/2, step = pi/4, pTick=.8red, ptick = lightgrey, extend = true));

       real p2 = pi/2;
       real f(real t)
       {
       return atan(0.6 * tan(t))+
       pi * ((t%p2 == 0 && t != 0) ? floor(t/p2) - 1 : floor(t/p2)) -
       ((t%pi == 0) ? 0 : pi * floor(t/pi)) - (t%(2pi) == 0 ? pi * floor(t/(2 * pi)) : 0);
       }

       draw(graph(f, xmin, xmax, 100));
       write(degrees(f(pi/2)));
       write(degrees(f(pi)));
       write(degrees(f(3pi/2)));
       write(degrees(f(2pi)));
       draw(graph(new real(real t){return t;}, xmin, xmax, 3));
    */
    return elleq - el.a * elle(pi/2 - gle, el.e);
  }
  return S(a2) - S(a1);
}

/*<asyxml><function type="real" signature="arclength(parabola,real)"><code></asyxml>*/
real arclength(parabola p, real angle)
{/*<asyxml></code><documentation>Return the arclength from 180 to 'angle' given from focus in the
   canonical coordinate system of 'p'.</documentation></function></asyxml>*/
  real a = p.a; /* In canonicalcartesiansystem(p) the equation of p
                 is x = y^2/(4a) */
  // integrate(sqrt(1 + (x/(2 * a))^2), x);
  real S(real t){return 0.5 * t * sqrt(1 + t^2/(4 * a^2)) + a * asinh(t/(2 * a));}
  real R(real gle){return 2 * a/(1 - Cos(gle));}
  real t = Sin(angle) * R(angle);
  return S(t);
}

/*<asyxml><function type="real" signature="arclength(parabola,real,real)"><code></asyxml>*/
real arclength(parabola p, real angle1, real angle2)
{/*<asyxml></code><documentation>Return the arclength from 'angle1' to 'angle2' given from
   focus in the canonical coordinate system of 'p'</documentation></function></asyxml>*/
  return arclength(p, angle1) - arclength(p, angle2);
}

/*<asyxml><function type="real" signature="arclength(parabola p)"><code></asyxml>*/
real arclength(parabola p)
{/*<asyxml></code><documentation>Return the length of the arc of the parabola bounded to the bounding
   box of the current picture.</documentation></function></asyxml>*/
  real[] b = bangles(p);
  return arclength(p, b[0], b[1]);
}
// *........................CONICS.........................*
// *=======================================================*

// *=======================================================*
// *.......................ABSCISSA........................*
/*<asyxml><struct signature="abscissa"><code></asyxml>*/
struct abscissa
{/*<asyxml></code><documentation>Provide abscissa structure on a curve used in the routine-like 'point(object, abscissa)'
   where object can be 'line','segment','ellipse','circle','conic'...</documentation><property type = "real" signature="x"><code></asyxml>*/
  real x;/*<asyxml></code><documentation>The abscissa value.</documentation></property><property type = "int" signature="system"><code></asyxml>*/
  int system;/*<asyxml></code><documentation>0 = relativesystem; 1 = curvilinearsystem; 2 = angularsystem; 3 = nodesystem</documentation></property><property type = "polarconicroutine" signature="polarconicroutine"><code></asyxml>*/
  polarconicroutine polarconicroutine = fromCenter;/*<asyxml></code><documentation>The routine used with angular system and two foci conic section.
                                                   Possible values are 'formCenter' and 'formFocus'.</documentation></property></asyxml>*/
  /*<asyxml><method type = "abscissa" signature="copy()"><code></asyxml>*/
  abscissa copy()
  {/*<asyxml></code><documentation>Return a copy of this abscissa.</documentation></method></asyxml>*/
    abscissa oa = new abscissa;
    oa.x = this.x;
    oa.system = this.system;
    oa.polarconicroutine = this.polarconicroutine;
    return oa;
  }
}/*<asyxml></struct></asyxml>*/

/*<asyxml><constant type = "int" signature="relativesystem,curvilinearsystem,angularsystem,nodesystem"><code></asyxml>*/
restricted int relativesystem = 0, curvilinearsystem = 1, angularsystem = 2, nodesystem = 3;/*<asyxml></code><documentation>Constant used to set the abscissa system.</documentation></constant></asyxml>*/

/*<asyxml><operator type = "abscissa" signature="cast(explicit position)"><code></asyxml>*/
abscissa operator cast(explicit position position)
{/*<asyxml></code><documentation>Cast position to abscissa.
   If 'position' is relative, the abscissa is relative else it's a curvilinear abscissa.</documentation></operator></asyxml>*/
  abscissa oarcc;
  oarcc.x = position.position.x;
  oarcc.system = position.relative ? relativesystem : curvilinearsystem;
  return oarcc;
}

/*<asyxml><operator type = "abscissa" signature="+(real,explicit abscissa)"><code></asyxml>*/
abscissa operator +(real x, explicit abscissa a)
{/*<asyxml></code><documentation>Provide 'real + abscissa'.
   Return abscissa b so that b.x = a.x + x.
   +(explicit abscissa, real), -(real, explicit abscissa) and -(explicit abscissa, real) are also defined.</documentation></operator></asyxml>*/
  abscissa oa = a.copy();
  oa.x = a.x + x;
  return oa;
}

abscissa operator +(explicit abscissa a, real x)
{
  return x + a;
}
abscissa operator +(int x, explicit abscissa a)
{
  return ((real)x) + a;
}

/*<asyxml><operator type = "abscissa" signature="-(explicit abscissa a)"><code></asyxml>*/
abscissa operator -(explicit abscissa a)
{/*<asyxml></code><documentation>Return the abscissa b so that b.x = -a.x.</documentation></operator></asyxml>*/
  abscissa oa;
  oa.system = a.system;
  oa.x = -a.x;
  return oa;
}

abscissa operator -(real x, explicit abscissa a)
{
  abscissa oa;
  oa.system = a.system;
  oa.x = x - a.x;
  return oa;
}
abscissa operator -(explicit abscissa a, real x)
{
  abscissa oa;
  oa.system = a.system;
  oa.x = a.x - x;
  return oa;
}
abscissa operator -(int x, explicit abscissa a)
{
  return ((real)x) - a;
}

/*<asyxml><operator type = "abscissa" signature="*(real,abscissa)"><code></asyxml>*/
abscissa operator *(real x, explicit abscissa a)
{/*<asyxml></code><documentation>Provide 'real * abscissa'.
   Return abscissa b so that b.x = x * a.x.
   *(explicit abscissa, real), /(real, explicit abscissa) and /(explicit abscissa, real) are also defined.</documentation></operator></asyxml>*/
  abscissa oa;
  oa.system = a.system;
  oa.x = a.x * x;
  return oa;
}
abscissa operator *(explicit abscissa a, real x)
{
  return x * a;
}

abscissa operator /(real x, explicit abscissa a)
{
  abscissa oa;
  oa.system = a.system;
  oa.x = x/a.x;
  return oa;
}
abscissa operator /(explicit abscissa a, real x)
{
  abscissa oa;
  oa.system = a.system;
  oa.x = a.x/x;
  return oa;
}

abscissa operator /(int x, explicit abscissa a)
{
  return ((real)x)/a;
}

/*<asyxml><function type="abscissa" signature="relabscissa(real)"><code></asyxml>*/
abscissa relabscissa(real x)
{/*<asyxml></code><documentation>Return a relative abscissa.</documentation></function></asyxml>*/
  return (abscissa)(Relative(x));
}
abscissa relabscissa(int x)
{
  return (abscissa)(Relative(x));
}

/*<asyxml><function type="abscissa" signature="curabscissa(real)"><code></asyxml>*/
abscissa curabscissa(real x)
{/*<asyxml></code><documentation>Return a curvilinear abscissa.</documentation></function></asyxml>*/
  return (abscissa)((position)x);
}
abscissa curabscissa(int x)
{
  return (abscissa)((position)x);
}

/*<asyxml><function type="abscissa" signature="angabscissa(real,polarconicroutine)"><code></asyxml>*/
abscissa angabscissa(real x, polarconicroutine polarconicroutine = currentpolarconicroutine)
{/*<asyxml></code><documentation>Return a angular abscissa.</documentation></function></asyxml>*/
  abscissa oarcc;
  oarcc.x = x;
  oarcc.polarconicroutine = polarconicroutine;
  oarcc.system = angularsystem;
  return oarcc;
}
abscissa angabscissa(int x, polarconicroutine polarconicroutine = currentpolarconicroutine)
{
  return angabscissa((real)x, polarconicroutine);
}

/*<asyxml><function type="abscissa" signature="nodabscissa(real)"><code></asyxml>*/
abscissa nodabscissa(real x)
{/*<asyxml></code><documentation>Return an abscissa as time on the path.</documentation></function></asyxml>*/
  abscissa oarcc;
  oarcc.x = x;
  oarcc.system = nodesystem;
  return oarcc;
}
abscissa nodabscissa(int x)
{
  return nodabscissa((real)x);
}

/*<asyxml><operator type = "abscissa" signature="cast(real)"><code></asyxml>*/
abscissa operator cast(real x)
{/*<asyxml></code><documentation>Cast real to abscissa, precisely 'nodabscissa'.</documentation></operator></asyxml>*/
  return nodabscissa(x);
}
abscissa operator cast(int x)
{
  return nodabscissa((real)x);
}

/*<asyxml><function type="point" signature="point(circle,abscissa)"><code></asyxml>*/
point point(circle c, abscissa l)
{/*<asyxml></code><documentation>Return the point of 'c' which has the abscissa 'l.x'
   according to the abscissa system 'l.system'.</documentation></function></asyxml>*/
  coordsys R = c.C.coordsys;
  if (l.system == nodesystem)
    return point(R, point((path)c, l.x)/R);
  if (l.system == relativesystem)
    return c.C + point(R, R.polar(c.r, 2 * pi * l.x));
  if (l.system == curvilinearsystem)
    return c.C + point(R, R.polar(c.r, l.x/c.r));
  if (l.system == angularsystem)
    return c.C + point(R, R.polar(c.r, radians(l.x)));
  abort("point: bad abscissa system.");
  return (0, 0);
}

/*<asyxml><function type="point" signature="point(ellipse,abscissa)"><code></asyxml>*/
point point(ellipse el, abscissa l)
{/*<asyxml></code><documentation>Return the point of 'el' which has the abscissa 'l.x'
   according to the abscissa system 'l.system'.</documentation></function></asyxml>*/
  if(el.e == 0) return point((circle)el, l);
  coordsys R = coordsys(el);
  if (l.system == nodesystem)
    return point(R, point((path)el, l.x)/R);
  if (l.system == relativesystem) {
    return point(el, curabscissa((l.x%1) * arclength(el)));
  }
  if (l.system == curvilinearsystem) {
    real a1 = 0, a2 = 360, cx = 0;
    real aout = a1;
    real x = abs(l.x)%arclength(el);
    while (abs(cx - x) > epsgeo) {
      aout = (a1 + a2)/2;
      cx = arclength(el, 0, aout, CCW, fromCenter); //fromCenter is speeder
      if(cx > x) a2 = (a1 + a2)/2; else a1 = (a1 + a2)/2;
    }
    path pel = fromCenter(el, sgn(l.x) * aout, sgn(l.x) * aout, 1, CCW);
    return point(R, point(pel, 0)/R);
  }
  if (l.system == angularsystem) {
    return point(R, point(l.polarconicroutine(el, l.x, l.x, 1, CCW), 0)/R);
  }
  abort("point: bad abscissa system.");
  return (0, 0);
}

/*<asyxml><function type="point" signature="point(parabola,abscissa)"><code></asyxml>*/
point point(parabola p, abscissa l)
{/*<asyxml></code><documentation>Return the point of 'p' which has the abscissa 'l.x'
   according to the abscissa system 'l.system'.</documentation></function></asyxml>*/
  coordsys R = coordsys(p);
  if (l.system == nodesystem)
    return point(R, point((path)p, l.x)/R);
  if (l.system == relativesystem) {
    real[] b = bangles(p);
    real al = sgn(l.x) > 0 ? arclength(p, 180, b[1]) : arclength(p, 180, b[0]);
    return point(p, curabscissa(abs(l.x) * al));
  }
  if (l.system == curvilinearsystem) {
    real a1 = 1e-3, a2 = 360 - 1e-3, cx = infinity;
    while (abs(cx - l.x) > epsgeo) {
      cx = arclength(p, 180, (a1 + a2)/2);
      if(cx > l.x) a2 = (a1 + a2)/2; else a1 = (a1 + a2)/2;
    }
    path pp = fromFocus(p, a1, a1, 1, CCW);
    return point(R, point(pp, 0)/R);
  }
  if (l.system == angularsystem) {
    return point(R, point(fromFocus(p, l.x, l.x, 1, CCW), 0)/R);
  }
  abort("point: bad abscissa system.");
  return (0, 0);
}

/*<asyxml><function type="point" signature="point(hyperbola,abscissa)"><code></asyxml>*/
point point(hyperbola h, abscissa l)
{/*<asyxml></code><documentation>Return the point of 'h' which has the abscissa 'l.x'
   according to the abscissa system 'l.system'.</documentation></function></asyxml>*/
  coordsys R = coordsys(h);
  if (l.system == nodesystem)
    return point(R, point((path)h, l.x)/R);
  if (l.system == relativesystem) {
    abort("point(hyperbola, relativeSystem) is not implemented...
Try relpoint((path)your_hyperbola, x);");
  }
  if (l.system == curvilinearsystem) {
    abort("point(hyperbola, curvilinearSystem) is not implemented...");
  }
  if (l.system == angularsystem) {
    return point(R, point(l.polarconicroutine(h, l.x, l.x, 1, CCW), 0)/R);
  }
  abort("point: bad abscissa system.");
  return (0, 0);
}

/*<asyxml><function type="abscissa" signature="point(conic,point)"><code></asyxml>*/
point point(explicit conic co, abscissa l)
{/*<asyxml></code><documentation>Return the curvilinear abscissa of 'M' on the conic 'co'.</documentation></function></asyxml>*/
  if(co.e == 0) return point((circle)co, l);
  if(co.e < 1) return point((ellipse)co, l);
  if(co.e == 1) return point((parabola)co, l);
  return point((hyperbola)co, l);
}


/*<asyxml><function type="point" signature="point(line,abscissa)"><code></asyxml>*/
point point(line l, abscissa x)
{/*<asyxml></code><documentation>Return the point of 'l' which has the abscissa 'l.x' according to the abscissa system 'l.system'.
   Note that the origin is l.A, and point(l, relabscissa(x)) returns l.A + x.x * vector(l.B - l.A).</documentation></function></asyxml>*/
  coordsys R = l.A.coordsys;
  if (x.system == nodesystem)
    return l.A + (x.x < 0 ? 0 : x.x > 1 ? 1 : x.x) * vector(l.B - l.A);
  if (x.system == relativesystem)
    return l.A + x.x * vector(l.B - l.A);
  if (x.system == curvilinearsystem)
    return l.A + x.x * l.u;
  if (x.system == angularsystem)
    abort("point: what the meaning of angular abscissa on line ?.");
  abort("point: bad abscissa system.");
  return (0, 0);
}

/*<asyxml><function type="point" signature="point(line,real)"><code></asyxml>*/
point point(line l, explicit real x)
{/*<asyxml></code><documentation>Return the point between node l.A and l.B (x <= 0 means l.A, x >=1 means l.B).</documentation></function></asyxml>*/
  return point(l, nodabscissa(x));
}
point point(line l, explicit int x)
{
  return point(l, nodabscissa(x));
}

/*<asyxml><function type="circle" signature="point(explicit circle,explicit real)"><code></asyxml>*/
point point(explicit circle c, explicit real x)
{/*<asyxml></code><documentation>Return the point between node floor(x) and floor(x) + 1.</documentation></function></asyxml>*/
  return point(c, nodabscissa(x));
}
point point(explicit circle c, explicit int x)
{
  return point(c, nodabscissa(x));
}

/*<asyxml><function type="point" signature="point(explicit ellipse,explicit real)"><code></asyxml>*/
point point(explicit ellipse el, explicit real x)
{/*<asyxml></code><documentation>Return the point between node floor(x) and floor(x) + 1.</documentation></function></asyxml>*/
  return point(el, nodabscissa(x));
}
point point(explicit ellipse el, explicit int x)
{
  return point(el, nodabscissa(x));
}

/*<asyxml><function type="point" signature="point(explicit parabola,explicit real)"><code></asyxml>*/
point point(explicit parabola p, explicit real x)
{/*<asyxml></code><documentation>Return the point between node floor(x) and floor(x) + 1.</documentation></function></asyxml>*/
  return point(p, nodabscissa(x));
}
point point(explicit parabola p, explicit int x)
{
  return point(p, nodabscissa(x));
}

/*<asyxml><function type="point" signature="point(explicit hyperbola,explicit real)"><code></asyxml>*/
point point(explicit hyperbola h, explicit real x)
{/*<asyxml></code><documentation>Return the point between node floor(x) and floor(x) + 1.</documentation></function></asyxml>*/
  return point(h, nodabscissa(x));
}
point point(explicit hyperbola h, explicit int x)
{
  return point(h, nodabscissa(x));
}

/*<asyxml><function type="point" signature="point(explicit conic,explicit real)"><code></asyxml>*/
point point(explicit conic co, explicit real x)
{/*<asyxml></code><documentation>Return the point between node floor(x) and floor(x) + 1.</documentation></function></asyxml>*/
  point op;
  if(co.e == 0) op = point((circle)co, nodabscissa(x));
  else if(co.e < 1) op = point((ellipse)co, nodabscissa(x));
  else if(co.e == 1) op = point((parabola)co, nodabscissa(x));
  else op = point((hyperbola)co, nodabscissa(x));
  return op;
}
point point(explicit conic co, explicit int x)
{
  return point(co, (real)x);
}

/*<asyxml><function type="point" signature="relpoint(line,real)"><code></asyxml>*/
point relpoint(line l, real x)
{/*<asyxml></code><documentation>Return the relative point of 'l' (0 means l.A,
   1 means l.B, x means l.A + x * vector(l.B - l.A) ).</documentation></function></asyxml>*/
  return point(l, Relative(x));
}

/*<asyxml><function type="point" signature="relpoint(explicit circle,real)"><code></asyxml>*/
point relpoint(explicit circle c, real x)
{/*<asyxml></code><documentation>Return the relative point of 'c' (0 means origin, 1 means end).
   Origin is c.center + c.r * (1, 0).</documentation></function></asyxml>*/
  return point(c, Relative(x));
}

/*<asyxml><function type="point" signature="relpoint(explicit ellipse,real)"><code></asyxml>*/
point relpoint(explicit ellipse el, real x)
{/*<asyxml></code><documentation>Return the relative point of 'el' (0 means origin, 1 means end).</documentation></function></asyxml>*/
  return point(el, Relative(x));
}

/*<asyxml><function type="point" signature="relpoint(explicit parabola,real)"><code></asyxml>*/
point relpoint(explicit parabola p, real x)
{/*<asyxml></code><documentation>Return the relative point of the path of the parabola
   bounded by the bounding box of the current picture.
   0 means origin, 1 means end, where the origin is the vertex of 'p'.</documentation></function></asyxml>*/
  return point(p, Relative(x));
}

/*<asyxml><function type="point" signature="relpoint(explicit hyperbola,real)"><code></asyxml>*/
point relpoint(explicit hyperbola h, real x)
{/*<asyxml></code><documentation>Not yet implemented... <look href = "point(hyperbola, abscissa)"/></documentation></function></asyxml>*/
  return point(h, Relative(x));
}

/*<asyxml><function type="point" signature="relpoint(explicit conic,explicit real)"><code></asyxml>*/
point relpoint(explicit conic co, explicit real x)
{/*<asyxml></code><documentation>Return the relative point of 'co' (0 means origin, 1 means end).</documentation></function></asyxml>*/
  point op;
  if(co.e == 0) op = point((circle)co, Relative(x));
  else if(co.e < 1) op = point((ellipse)co, Relative(x));
  else if(co.e == 1) op = point((parabola)co, Relative(x));
  else op = point((hyperbola)co, Relative(x));
  return op;
}
point relpoint(explicit conic co, explicit int x)
{
  return relpoint(co, (real)x);
}

/*<asyxml><function type="point" signature="angpoint(explicit circle,real)"><code></asyxml>*/
point angpoint(explicit circle c, real x)
{/*<asyxml></code><documentation>Return the point of 'c' in the direction 'x' measured in degrees.</documentation></function></asyxml>*/
  return point(c, angabscissa(x));
}

/*<asyxml><function type="point" signature="angpoint(explicit ellipse,real,polarconicroutine)"><code></asyxml>*/
point angpoint(explicit ellipse el, real x,
               polarconicroutine polarconicroutine = currentpolarconicroutine)
{/*<asyxml></code><documentation>Return the point of 'el' in the direction 'x'
   measured in degrees according to 'polarconicroutine'.</documentation></function></asyxml>*/
  return el.e == 0 ? angpoint((circle) el, x) : point(el, angabscissa(x, polarconicroutine));
}

/*<asyxml><function type="point" signature="angpoint(explicit parabola,real)"><code></asyxml>*/
point angpoint(explicit parabola p, real x)
{/*<asyxml></code><documentation>Return the point of 'p' in the direction 'x' measured in degrees.</documentation></function></asyxml>*/
  return point(p, angabscissa(x));
}

/*<asyxml><function type="point" signature="angpoint(explicit hyperbola,real,polarconicroutine)"><code></asyxml>*/
point angpoint(explicit hyperbola h, real x,
               polarconicroutine polarconicroutine = currentpolarconicroutine)
{/*<asyxml></code><documentation>Return the point of 'h' in the direction 'x'
   measured in degrees according to 'polarconicroutine'.</documentation></function></asyxml>*/
  return point(h, angabscissa(x, polarconicroutine));
}

/*<asyxml><function type="point" signature="curpoint(line,real)"><code></asyxml>*/
point curpoint(line l, real x)
{/*<asyxml></code><documentation>Return the point of 'l' which has the curvilinear abscissa 'x'.
   Origin is l.A.</documentation></function></asyxml>*/
  return point(l, curabscissa(x));
}

/*<asyxml><function type="point" signature="curpoint(explicit circle,real)"><code></asyxml>*/
point curpoint(explicit circle c, real x)
{/*<asyxml></code><documentation>Return the point of 'c' which has the curvilinear abscissa 'x'.
   Origin is c.center + c.r * (1, 0).</documentation></function></asyxml>*/
  return point(c, curabscissa(x));
}

/*<asyxml><function type="point" signature="curpoint(explicit ellipse,real)"><code></asyxml>*/
point curpoint(explicit ellipse el, real x)
{/*<asyxml></code><documentation>Return the point of 'el' which has the curvilinear abscissa 'el'.</documentation></function></asyxml>*/
  return point(el, curabscissa(x));
}

/*<asyxml><function type="point" signature="curpoint(explicit parabola,real)"><code></asyxml>*/
point curpoint(explicit parabola p, real x)
{/*<asyxml></code><documentation>Return the point of 'p' which has the curvilinear abscissa 'x'.
   Origin is the vertex of 'p'.</documentation></function></asyxml>*/
  return point(p, curabscissa(x));
}

/*<asyxml><function type="point" signature="curpoint(conic,real)"><code></asyxml>*/
point curpoint(conic co, real x)
{/*<asyxml></code><documentation>Return the point of 'co' which has the curvilinear abscissa 'x'.</documentation></function></asyxml>*/
  point op;
  if(co.e == 0) op = point((circle)co, curabscissa(x));
  else if(co.e < 1) op = point((ellipse)co, curabscissa(x));
  else if(co.e == 1) op = point((parabola)co, curabscissa(x));
  else op = point((hyperbola)co, curabscissa(x));
  return op;
}

/*<asyxml><function type="abscissa" signature="angabscissa(circle,point)"><code></asyxml>*/
abscissa angabscissa(circle c, point M)
{/*<asyxml></code><documentation>Return the angular abscissa of 'M' on the circle 'c'.</documentation></function></asyxml>*/
  if(!(M @ c)) abort("angabscissa: the point is not on the circle.");
  abscissa oa;
  oa.system = angularsystem;
  oa.x = degrees(M - c.C);
  if(oa.x < 0) oa.x+=360;
  return oa;
}

/*<asyxml><function type="abscissa" signature="angabscissa(ellipse,point,polarconicroutine)"><code></asyxml>*/
abscissa angabscissa(ellipse el, point M,
                     polarconicroutine polarconicroutine = currentpolarconicroutine)
{/*<asyxml></code><documentation>Return the angular abscissa of 'M' on the ellipse 'el' according to 'polarconicroutine'.</documentation></function></asyxml>*/
  if(!(M @ el)) abort("angabscissa: the point is not on the ellipse.");
  abscissa oa;
  oa.system = angularsystem;
  oa.polarconicroutine = polarconicroutine;
  oa.x = polarconicroutine == fromCenter ? degrees(M - el.C) : degrees(M - el.F1);
  oa.x -= el.angle;
  if(oa.x < 0) oa.x += 360;
  return oa;
}

/*<asyxml><function type="abscissa" signature="angabscissa(hyperbola,point,polarconicroutine)"><code></asyxml>*/
abscissa angabscissa(hyperbola h, point M,
                     polarconicroutine polarconicroutine = currentpolarconicroutine)
{/*<asyxml></code><documentation>Return the angular abscissa of 'M' on the hyperbola 'h' according to 'polarconicroutine'.</documentation></function></asyxml>*/
  if(!(M @ h)) abort("angabscissa: the point is not on the hyperbola.");
  abscissa oa;
  oa.system = angularsystem;
  oa.polarconicroutine = polarconicroutine;
  oa.x = polarconicroutine == fromCenter ? degrees(M - h.C) : degrees(M - h.F1) + 180;
  oa.x -= h.angle;
  if(oa.x < 0) oa.x += 360;
  return oa;
}

/*<asyxml><function type="abscissa" signature="angabscissa(parabola,point)"><code></asyxml>*/
abscissa angabscissa(parabola p, point M)
{/*<asyxml></code><documentation>Return the angular abscissa of 'M' on the parabola 'p'.</documentation></function></asyxml>*/
  if(!(M @ p)) abort("angabscissa: the point is not on the parabola.");
  abscissa oa;
  oa.system = angularsystem;
  oa.polarconicroutine = fromFocus;// Not used
  oa.x = degrees(M - p.F);
  oa.x -= p.angle;
  if(oa.x < 0) oa.x += 360;
  return oa;
}

/*<asyxml><function type="abscissa" signature="angabscissa(conic,point)"><code></asyxml>*/
abscissa angabscissa(explicit conic co, point M)
{/*<asyxml></code><documentation>Return the angular abscissa of 'M' on the conic 'co'.</documentation></function></asyxml>*/
  if(co.e == 0) return angabscissa((circle)co, M);
  if(co.e < 1) return angabscissa((ellipse)co, M);
  if(co.e == 1) return angabscissa((parabola)co, M);
  return angabscissa((hyperbola)co, M);
}

/*<asyxml><function type="abscissa" signature="curabscissa(line,point)"><code></asyxml>*/
abscissa curabscissa(line l, point M)
{/*<asyxml></code><documentation>Return the curvilinear abscissa of 'M' on the line 'l'.</documentation></function></asyxml>*/
  if(!(M @ extend(l))) abort("curabscissa: the point is not on the line.");
  abscissa oa;
  oa.system = curvilinearsystem;
  oa.x = sgn(dot(M - l.A, l.B - l.A)) * abs(M - l.A);
  return oa;
}

/*<asyxml><function type="abscissa" signature="curabscissa(circle,point)"><code></asyxml>*/
abscissa curabscissa(circle c, point M)
{/*<asyxml></code><documentation>Return the curvilinear abscissa of 'M' on the circle 'c'.</documentation></function></asyxml>*/
  if(!(M @ c)) abort("curabscissa: the point is not on the circle.");
  abscissa oa;
  oa.system = curvilinearsystem;
  oa.x = pi * angabscissa(c, M).x * c.r/180;
  return oa;
}

/*<asyxml><function type="abscissa" signature="curabscissa(ellipse,point)"><code></asyxml>*/
abscissa curabscissa(ellipse el, point M)
{/*<asyxml></code><documentation>Return the curvilinear abscissa of 'M' on the ellipse 'el'.</documentation></function></asyxml>*/
  if(!(M @ el)) abort("curabscissa: the point is not on the ellipse.");
  abscissa oa;
  oa.system = curvilinearsystem;
  real a = angabscissa(el, M, fromCenter).x;
  oa.x = arclength(el, 0, a, fromCenter);
  oa.polarconicroutine = fromCenter;
  return oa;
}

/*<asyxml><function type="abscissa" signature="curabscissa(parabola,point)"><code></asyxml>*/
abscissa curabscissa(parabola p, point M)
{/*<asyxml></code><documentation>Return the curvilinear abscissa of 'M' on the parabola 'p'.</documentation></function></asyxml>*/
  if(!(M @ p)) abort("curabscissa: the point is not on the parabola.");
  abscissa oa;
  oa.system = curvilinearsystem;
  real a = angabscissa(p, M).x;
  oa.x = arclength(p, 180, a);
  oa.polarconicroutine = fromFocus; // Not used.
  return oa;
}

/*<asyxml><function type="abscissa" signature="curabscissa(conic,point)"><code></asyxml>*/
abscissa curabscissa(conic co, point M)
{/*<asyxml></code><documentation>Return the curvilinear abscissa of 'M' on the conic 'co'.</documentation></function></asyxml>*/
  if(co.e > 1) abort("curabscissa: not implemented for this hyperbola.");
  if(co.e == 0) return curabscissa((circle)co, M);
  if(co.e < 1) return curabscissa((ellipse)co, M);
  return curabscissa((parabola)co, M);
}

/*<asyxml><function type="abscissa" signature="nodabscissa(line,point)"><code></asyxml>*/
abscissa nodabscissa(line l, point M)
{/*<asyxml></code><documentation>Return the node abscissa of 'M' on the line 'l'.</documentation></function></asyxml>*/
  if(!(M @ (segment)l)) abort("nodabscissa: the point is not on the segment.");
  abscissa oa;
  oa.system = nodesystem;
  oa.x = abs(M - l.A)/abs(l.A - l.B);
  return oa;
}

/*<asyxml><function type="abscissa" signature="nodabscissa(circle,point)"><code></asyxml>*/
abscissa nodabscissa(circle c, point M)
{/*<asyxml></code><documentation>Return the node abscissa of 'M' on the circle 'c'.</documentation></function></asyxml>*/
  if(!(M @ c)) abort("nodabscissa: the point is not on the circle.");
  abscissa oa;
  oa.system = nodesystem;
  oa.x = intersect((path)c, locate(M))[0];
  return oa;
}

/*<asyxml><function type="abscissa" signature="nodabscissa(ellipse,point)"><code></asyxml>*/
abscissa nodabscissa(ellipse el, point M)
{/*<asyxml></code><documentation>Return the node abscissa of 'M' on the ellipse 'el'.</documentation></function></asyxml>*/
  if(!(M @ el)) abort("nodabscissa: the point is not on the ellipse.");
  abscissa oa;
  oa.system = nodesystem;
  oa.x = intersect((path)el, M)[0];
  return oa;
}

/*<asyxml><function type="abscissa" signature="nodabscissa(parabola,point)"><code></asyxml>*/
abscissa nodabscissa(parabola p, point M)
{/*<asyxml></code><documentation>Return the node abscissa OF 'M' on the parabola 'p'.</documentation></function></asyxml>*/
  if(!(M @ p)) abort("nodabscissa: the point is not on the parabola.");
  abscissa oa;
  oa.system = nodesystem;
  path pg = p;
  real[] t = intersect(pg, M, 1e-5);
  if(t.length == 0) abort("nodabscissa: the point is not on the path of the parabola.");
  oa.x = t[0];
  return oa;
}

/*<asyxml><function type="abscissa" signature="nodabscissa(conic,point)"><code></asyxml>*/
abscissa nodabscissa(conic co, point M)
{/*<asyxml></code><documentation>Return the node abscissa of 'M' on the conic 'co'.</documentation></function></asyxml>*/
  if(co.e > 1) abort("nodabscissa: not implemented for hyperbola.");
  if(co.e == 0) return nodabscissa((circle)co, M);
  if(co.e < 1) return nodabscissa((ellipse)co, M);
  return nodabscissa((parabola)co, M);
}


/*<asyxml><function type="abscissa" signature="relabscissa(line,point)"><code></asyxml>*/
abscissa relabscissa(line l, point M)
{/*<asyxml></code><documentation>Return the relative abscissa of 'M' on the line 'l'.</documentation></function></asyxml>*/
  if(!(M @ extend(l))) abort("relabscissa: the point is not on the line.");
  abscissa oa;
  oa.system = relativesystem;
  oa.x = sgn(dot(M - l.A, l.B - l.A)) * abs(M - l.A)/abs(l.A - l.B);
  return oa;
}

/*<asyxml><function type="abscissa" signature="relabscissa(circle,point)"><code></asyxml>*/
abscissa relabscissa(circle c, point M)
{/*<asyxml></code><documentation>Return the relative abscissa of 'M' on the circle 'c'.</documentation></function></asyxml>*/
  if(!(M @ c)) abort("relabscissa: the point is not on the circle.");
  abscissa oa;
  oa.system = relativesystem;
  oa.x = angabscissa(c, M).x/360;
  return oa;
}

/*<asyxml><function type="abscissa" signature="relabscissa(ellipse,point)"><code></asyxml>*/
abscissa relabscissa(ellipse el, point M)
{/*<asyxml></code><documentation>Return the relative abscissa of 'M' on the ellipse 'el'.</documentation></function></asyxml>*/
  if(!(M @ el)) abort("relabscissa: the point is not on the ellipse.");
  abscissa oa;
  oa.system = relativesystem;
  oa.x = curabscissa(el, M).x/arclength(el);
  oa.polarconicroutine = fromFocus;
  return oa;
}

/*<asyxml><function type="abscissa" signature="relabscissa(conic,point)"><code></asyxml>*/
abscissa relabscissa(conic co, point M)
{/*<asyxml></code><documentation>Return the relative abscissa of 'M'
   on the conic 'co'.</documentation></function></asyxml>*/
  if(co.e > 1) abort("relabscissa: not implemented for hyperbola and parabola.");
  if(co.e == 1) return relabscissa((parabola)co, M);
  if(co.e == 0) return relabscissa((circle)co, M);
  return relabscissa((ellipse)co, M);
}
// *.......................ABSCISSA........................*
// *=======================================================*

// *=======================================================*
// *.........................ARCS..........................*
/*<asyxml><struct signature="arc"><code></asyxml>*/
struct arc {
  /*<asyxml></code><documentation>Implement oriented ellipse (included circle) arcs.
    All the calculus with this structure will be as exact as Asymptote can do.
    For a full precision, you must not cast 'arc' to 'path' excepted for drawing routines.
    </documentation><property type = "ellipse" signature="el"><code></asyxml>*/
  ellipse el;/*<asyxml></code><documentation>The support of the arc.</documentation></property><property type = "real" signature="angle0"><code></asyxml>*/
  restricted real angle0 = 0;/*<asyxml></code><documentation>Internal use: rotating a circle does not modify the origin point,this variable stocks the eventual angle rotation. This value is not used for ellipses which are not circles.</documentation></property><property type = "real" signature="angle1,angle2"><code></asyxml>*/
  restricted  real angle1, angle2;/*<asyxml></code><documentation>Values (in degrees) in ]-360, 360[.</documentation></property><property type = "bool" signature="direction"><code></asyxml>*/
  bool direction = CCW;/*<asyxml></code><documentation>The arc will be drawn from 'angle1' to 'angle2' rotating in the direction 'direction'.</documentation></property><property type = "polarconicroutine" signature="polarconicroutine"><code></asyxml>*/
  polarconicroutine polarconicroutine = currentpolarconicroutine;/*<asyxml></code><documentation>The routine to which the angles refer.
                                                                 If 'el' is a circle 'fromCenter' is always used.</documentation></property></asyxml>*/

  /*<asyxml><method type = "void" signature="setangles(real,real,real)"><code></asyxml>*/
  void setangles(real a0, real a1, real a2)
  {/*<asyxml></code><documentation>Set the angles.</documentation></method></asyxml>*/
    if (a1 < 0 && a2 < 0) {
      a1 += 360;
      a2 += 360;
    }
    this.angle0 = a0%(sgnd(a0) * 360);
    this.angle1 = a1%(sgnd(a1) * 360);
    this.angle2 = a2%(sgnd(2) * 360);
  }

  /*<asyxml><method type = "void" signature="init(ellipse,real,real,real,polarconicroutine,bool)"><code></asyxml>*/
  void init(ellipse el, real angle0 = 0, real angle1, real angle2,
            polarconicroutine polarconicroutine,
            bool direction = CCW)
  {/*<asyxml></code><documentation>Constructor.</documentation></method></asyxml>*/
    if(abs(angle1 - angle2) > 360) abort("arc: |angle1 - angle2| > 360.");
    this.el = el;
    this.setangles(angle0, angle1, angle2);
    this.polarconicroutine = polarconicroutine;
    this.direction = direction;
  }

  /*<asyxml><method type = "arc" signature="copy()"><code></asyxml>*/
  arc copy()
  {/*<asyxml></code><documentation>Copy the arc.</documentation></method></asyxml>*/
    arc oa = new arc;
    oa.el = this.el;
    oa.direction = this.direction;
    oa.polarconicroutine = this.polarconicroutine;
    oa.angle1 = this.angle1;
    oa.angle2 = this.angle2;
    oa.angle0 = this.angle0;
    return oa;
  }
}/*<asyxml></struct></asyxml>*/

/*<asyxml><function type="polarconicroutine" signature="polarconicroutine(ellipse)"><code></asyxml>*/
polarconicroutine polarconicroutine(conic co)
{/*<asyxml></code><documentation>Return the default routine used to draw a conic.</documentation></function></asyxml>*/
  if(co.e == 0) return fromCenter;
  if(co.e == 1) return fromFocus;
  return currentpolarconicroutine;
}

/*<asyxml><function type="arc" signature="arc(ellipse,real,real,polarconicroutine,bool)"><code></asyxml>*/
arc arc(ellipse el, real angle1, real angle2,
        polarconicroutine polarconicroutine = polarconicroutine(el),
        bool direction = CCW)
{/*<asyxml></code><documentation>Return the ellipse arc from 'angle1' to 'angle2' with respect to 'polarconicroutine' and rotating in the direction 'direction'.</documentation></function></asyxml>*/
  arc oa;
  oa.init(el, 0, angle1, angle2, polarconicroutine, direction);
  return oa;
}

/*<asyxml><function type="arc" signature="complementary(arc)"><code></asyxml>*/
arc complementary(arc a)
{/*<asyxml></code><documentation>Return the complementary of 'a'.</documentation></function></asyxml>*/
  arc oa;
  oa.init(a.el, a.angle0, a.angle2, a.angle1, a.polarconicroutine, a.direction);
  return oa;
}

/*<asyxml><function type="arc" signature="reverse(arc)"><code></asyxml>*/
arc reverse(arc a)
{/*<asyxml></code><documentation>Return arc 'a' oriented in reverse direction.</documentation></function></asyxml>*/
  arc oa;
  oa.init(a.el, a.angle0, a.angle2, a.angle1, a.polarconicroutine, !a.direction);
  return oa;
}

/*<asyxml><function type="real" signature="degrees(arc)"><code></asyxml>*/
real degrees(arc a)
{/*<asyxml></code><documentation>Return the measure in degrees of the oriented arc 'a'.</documentation></function></asyxml>*/
  real or;
  real da = a.angle2 - a.angle1;
  if(a.direction) {
    or = a.angle1 < a.angle2 ? da : 360 + da;
  } else {
    or = a.angle1 < a.angle2 ? -360 + da : da;
  }
  return or;
}

/*<asyxml><function type="real" signature="angle(a)"><code></asyxml>*/
real angle(arc a)
{/*<asyxml></code><documentation>Return the measure in radians of the oriented arc 'a'.</documentation></function></asyxml>*/
  return radians(degrees(a));
}

/*<asyxml><function type="int" signature="arcnodesnumber(explicit arc)"><code></asyxml>*/
int arcnodesnumber(explicit arc a)
{/*<asyxml></code><documentation>Return the number of nodes to draw the arc 'a'.</documentation></function></asyxml>*/
  return ellipsenodesnumber(a.el.a, a.el.b, a.angle1, a.angle2, a.direction);
}

private path arctopath(arc a, int n)
{
  if(a.el.e == 0) return arcfromcenter(a.el, a.angle0 + a.angle1, a.angle0 + a.angle2, a.direction, n);
  if(a.el.e != 1) return a.polarconicroutine(a.el, a.angle1, a.angle2, n, a.direction);
  return arcfromfocus(a.el, a.angle1, a.angle2, n, a.direction);
}

/*<asyxml><function type="point" signature="angpoint(arc,real)"><code></asyxml>*/
point angpoint(arc a, real angle)
{/*<asyxml></code><documentation>Return the point given by its angular position (in degrees) relative to the arc 'a'.
   If 'angle > degrees(a)' or 'angle < 0' the returned point is on the extended arc.</documentation></function></asyxml>*/
  pair p;
  if(a.el.e == 0) {
    real gle = a.angle0 + a.angle1 + (a.direction ? angle : -angle);
    p = point(arcfromcenter(a.el, gle, gle, CCW, 1), 0);
  }
  else {
    real gle = a.angle1 + (a.direction ? angle : -angle);
    p = point(a.polarconicroutine(a.el, gle, gle, 1, CCW), 0);
  }
  return point(coordsys(a.el), p/coordsys(a.el));
}

/*<asyxml><operator type = "path" signature="cast(explicit arc)"><code></asyxml>*/
path operator cast(explicit arc a)
{/*<asyxml></code><documentation>Cast arc to path.</documentation></operator></asyxml>*/
  return arctopath(a, arcnodesnumber(a));
}

/*<asyxml><operator type = "guide" signature="cast(explicit arc)"><code></asyxml>*/
guide operator cast(explicit arc a)
{/*<asyxml></code><documentation>Cast arc to guide.</documentation></operator></asyxml>*/
  return arctopath(a, arcnodesnumber(a));
}

/*<asyxml><operator type = "arc" signature="*(transform,explicit arc)"><code></asyxml>*/
arc operator *(transform t, explicit arc a)
{/*<asyxml></code><documentation>Provide transform * arc.</documentation></operator></asyxml>*/
  pair[] P, PP;
  path g = arctopath(a, 3);
  real a0, a1 = a.angle1, a2 = a.angle2, ap1, ap2;
  bool dir = a.direction;
  P[0] = t * point(g, 0);
  P[1] = t * point(g, 2);
  ellipse el = t * a.el;
  arc oa;
  a0 = (a.angle0 + angle(shiftless(t)))%360;
  pair C;
  if(a.polarconicroutine == fromCenter) C = el.C; else C = el.F1;
  real d = abs(locate(el.F2 - el.F1)) > epsgeo ?
    degrees(locate(el.F2 - el.F1)) : a0 + degrees(el.C.coordsys.i);
  ap1 = (degrees(P[0]-C, false) - d)%360;
  ap2 = (degrees(P[1]-C, false) - d)%360;
  oa.init(el, a0, ap1, ap2, a.polarconicroutine, dir);
  g = arctopath(oa, 3);
  PP[0] = point(g, 0);
  PP[1] = point(g, 2);
  if((a1 - a2) * (ap1 - ap2) < 0) {// Handle reflection.
    dir=!a.direction;
    oa.init(el, a0, ap1, ap2, a.polarconicroutine, dir);
  }
  return oa;
}

/*<asyxml><operator type = "arc" signature="*(real,explicit arc)"><code></asyxml>*/
arc operator *(real x, explicit arc a)
{/*<asyxml></code><documentation>Provide real * arc.
   Return the arc subtracting and adding '(x - 1) * degrees(a)/2' to 'a.angle1' and 'a.angle2' respectively.</documentation></operator></asyxml>*/
  real a1, a2, gle;
  gle = (x - 1) * degrees(a)/2;
  a1 = a.angle1 - gle;
  a2 = a.angle2 + gle;
  arc oa;
  oa.init(a.el, a.angle0, a1, a2, a.polarconicroutine, a.direction);
  return oa;
}
arc operator *(int x, explicit arc a){return (real)x * a;}
/*<asyxml><operator type = "arc" signature="/(real,explicit arc)"><code></asyxml>*/
arc operator /(explicit arc a, real x)
{/*<asyxml></code><documentation>Provide arc/real.
   Return the arc subtracting and adding '(1/x - 1) * degrees(a)/2' to 'a.angle1' and 'a.angle2' respectively.</documentation></operator></asyxml>*/
  return (1/x) * a;
}
/*<asyxml><operator type = "arc" signature="+(explicit arc,point)"><code></asyxml>*/
arc operator +(explicit arc a, point M)
{/*<asyxml></code><documentation>Provide arc + point.
   Return shifted arc.
   'operator +(explicit arc, point)', 'operator +(explicit arc, vector)' and 'operator -(explicit arc, vector)' are also defined.</documentation></operator></asyxml>*/
  return shift(M) * a;
}
arc operator -(explicit arc a, point M){return a + (-M);}
arc operator +(explicit arc a, vector v){return shift(locate(v)) * a;}
arc operator -(explicit arc a, vector v){return a + (-v);}


/*<asyxml><operator type = "bool" signature="@(point,arc)"><code></asyxml>*/
bool operator @(point M, arc a)
{/*<asyxml></code><documentation>Return true iff 'M' is on the arc 'a'.</documentation></operator></asyxml>*/
  if (!(M @ a.el)) return false;
  coordsys R = defaultcoordsys;
  path ap = arctopath(a, 3);
  line l = line(point(R, point(ap, 0)), point(R, point(ap, 2)));
  return sameside(M, point(R, point(ap, 1)), l);
}

/*<asyxml><function type="void" signature="draw(picture,Label,arc,align,pen,arrowbar,arrowbar,margin,Label,marker)"><code></asyxml>*/
void draw(picture pic = currentpicture, Label L = "", arc a,
          align align = NoAlign, pen p = currentpen,
          arrowbar arrow = None, arrowbar bar = None, margin margin = NoMargin,
          Label legend = "", marker marker = nomarker)
{/*<asyxml></code><documentation>Draw 'arc' adding the pen returned by 'addpenarc(p)' to the pen 'p'.
   <look href = "#addpenarc"/></documentation></function></asyxml>*/
  draw(pic, L, (path)a, align, addpenarc(p), arrow, bar, margin, legend, marker);
}

/*<asyxml><function type="real" signature="arclength(arc)"><code></asyxml>*/
real arclength(arc a)
{/*<asyxml></code><documentation>The arc length of 'a'.</documentation></function></asyxml>*/
  return arclength(a.el, a.angle1, a.angle2, a.direction, a.polarconicroutine);
}

private point ppoint(arc a, real x)
{// Return the point of the arc proportionally to its length.
  point oP;
  if(a.el.e == 0) { // Case of circle.
    oP = angpoint(a, x * abs(degrees(a)));
  } else { // Ellipse and not circle.
    if(!a.direction) {
      transform t = reflect(line(a.el.F1, a.el.F2));
      return t * ppoint(t * a, x);
    }

    real angle1 = a.angle1, angle2 = a.angle2;
    if(a.polarconicroutine == fromFocus) {
      //       dot(point(fromFocus(a.el, angle1, angle1, 1, CCW), 0), 2mm + blue);
      //       dot(point(fromFocus(a.el, angle2, angle2, 1, CCW), 0), 2mm + blue);
      //       write("fromfocus1 = ", angle1);
      //       write("fromfocus2 = ", angle2);
      real gle1 = focusToCenter(a.el, angle1);
      real gle2 = focusToCenter(a.el, angle2);
      if((gle1 - gle2) * (angle1 - angle2) > 0) {
        angle1 = gle1; angle2 = gle2;
      } else {
        angle1 = gle2; angle2 = gle1;
      }
      //       write("fromcenter1 = ", angle1);
      //       write("fromcenter2 = ", angle2);
      //       dot(point(fromCenter(a.el, angle1, angle1, 1, CCW), 0), 1mm + red);
      //       dot(point(fromCenter(a.el, angle2, angle2, 1, CCW), 0), 1mm + red);
    }

    if(angle1 > angle2) {
      arc ta = a.copy();
      ta.polarconicroutine = fromCenter;
      ta.setangles(a0 = a.angle0, a1 = angle1 - 360, a2 = angle2);
      return ppoint(ta, x);
    }
    ellipse co = a.el;
    real gle, a1, a2, cx = 0;
    bool direction;
    if(x >= 0) {
      a1 = angle1;
      a2 = a1 + 360;
      direction = CCW;
    } else {
      a1 = angle1 - 360;
      a2 = a1 - 360;
      direction = CW;
    }
    gle = a1;
    real L = arclength(co, angle1, angle2, a.direction, fromCenter);
    real tx = L * abs(x)%arclength(co);
    real aout = a1;
    while(abs(cx - tx) > epsgeo) {
      aout = (a1 + a2)/2;
      cx = abs(arclength(co, gle, aout, direction, fromCenter));
      if(cx > tx) a2 = (a1 + a2)/2 ; else a1 = (a1 + a2)/2;
    }
    pair p = point(arcfromcenter(co, aout, aout, CCW, 1), 0);
    oP = point(coordsys(co), p/coordsys(co));
  }
  return oP;
}

/*<asyxml><function type="point" signature="point(arc,abscissa)"><code></asyxml>*/
point point(arc a, abscissa l)
{/*<asyxml></code><documentation>Return the point of 'a' which has the abscissa 'l.x'
   according to the abscissa system 'l.system'.
   Note that 'a.polarconicroutine' is used instead of 'l.polarconicroutine'.
   <look href = "#struct abscissa"/></documentation></function></asyxml>*/
  real posx;
  arc ta = a.copy();
  ellipse co = a.el;
  if (l.system == relativesystem) {
    posx = l.x;
  } else
    if (l.system == curvilinearsystem) {
      real tl;
      if(co.e == 0) {
        tl = curabscissa(a.el, angpoint(a.el, a.angle0 + a.angle1)).x;
        return curpoint(a.el, tl + (a.direction ? l.x : -l.x));
      } else {
        tl = curabscissa(a.el, angpoint(a.el, a.angle1, a.polarconicroutine)).x;
        return curpoint(a.el, tl + (a.direction ? l.x : -l.x));
      }
    } else
      if (l.system == nodesystem) {
        coordsys R = coordsys(co);
        return point(R, point((path)a, l.x)/R);
      } else
        if (l.system == angularsystem) {
          return angpoint(a, l.x);
        } else abort("point: bad abscissa system.");
  return ppoint(ta, posx);
}


/*<asyxml><function type="point" signature="point(arc,real)"><code></asyxml>*/
point point(arc a, real x)
{/*<asyxml></code><documentation>Return the point between node floor(t) and floor(t) + 1.</documentation></function></asyxml>*/
  return point(a, nodabscissa(x));
}
pair point(explicit arc a, int x)
{
  return point(a, nodabscissa(x));
}

/*<asyxml><function type="point" signature="relpoint(arc,real)"><code></asyxml>*/
point relpoint(arc a, real x)
{/*<asyxml></code><documentation>Return the relative point of 'a'.
   If x > 1 or x < 0, the returned point is on the extended arc.</documentation></function></asyxml>*/
  return point(a, relabscissa(x));
}

/*<asyxml><function type="point" signature="curpoint(arc,real)"><code></asyxml>*/
point curpoint(arc a, real x)
{/*<asyxml></code><documentation>Return the point of 'a' which has the curvilinear abscissa 'x'.
   If x < 0 or x > arclength(a), the returned point is on the extended arc.</documentation></function></asyxml>*/
  return point(a, curabscissa(x));
}

/*<asyxml><function type="abscissa" signature="angabscissa(arc,point)"><code></asyxml>*/
abscissa angabscissa(arc a, point M)
{/*<asyxml></code><documentation>Return the angular abscissa of 'M' according to the arc 'a'.</documentation></function></asyxml>*/
  if(!(M @ a.el))
    abort("angabscissa: the point is not on the extended arc.");
  abscissa oa;
  oa.system = angularsystem;
  oa.polarconicroutine = a.polarconicroutine;
  real am = angabscissa(a.el, M, a.polarconicroutine).x;
  oa.x = (am - a.angle1 - (a.el.e == 0 ? a.angle0 : 0))%360;
  oa.x = a.direction ? oa.x : 360 - oa.x;
  return oa;
}

/*<asyxml><function type="abscissa" signature="curabscissa(arc,point)"><code></asyxml>*/
abscissa curabscissa(arc a, point M)
{/*<asyxml></code><documentation>Return the curvilinear abscissa according to the arc 'a'.</documentation></function></asyxml>*/
  ellipse el = a.el;
  if(!(M @ el))
    abort("angabscissa: the point is not on the extended arc.");
  abscissa oa;
  oa.system = curvilinearsystem;
  real xm = curabscissa(el, M).x;
  real a0 = el.e == 0 ? a.angle0 : 0;
  real am = curabscissa(el, angpoint(el, a.angle1 + a0, a.polarconicroutine)).x;
  real l = arclength(el);
  oa.x = (xm - am)%l;
  oa.x = a.direction ? oa.x : l - oa.x;
  return oa;
}

/*<asyxml><function type="abscissa" signature="nodabscissa(arc,point)"><code></asyxml>*/
abscissa nodabscissa(arc a, point M)
{/*<asyxml></code><documentation>Return the node abscissa according to the arc 'a'.</documentation></function></asyxml>*/
  if(!(M @ a))
    abort("nodabscissa: the point is not on the arc.");
  abscissa oa;
  oa.system = nodesystem;
  oa.x = intersect((path)a, M)[0];
  return oa;
}

/*<asyxml><function type="abscissa" signature="relabscissa(arc,point)"><code></asyxml>*/
abscissa relabscissa(arc a, point M)
{/*<asyxml></code><documentation>Return the relative abscissa according to the arc 'a'.</documentation></function></asyxml>*/
  ellipse el = a.el;
  if(!( M @ el))
    abort("relabscissa: the point is not on the prolonged arc.");
  abscissa oa;
  oa.system = relativesystem;
  oa.x = curabscissa(a, M).x/arclength(a);
  return oa;
}

/*<asyxml><function type="void" signature="markarc(picture,Label,int,real,real,arc,arrowbar,pen,pen,margin,marker)"><code></asyxml>*/
void markarc(picture pic = currentpicture,
             Label L = "",
             int n = 1, real radius = 0, real space = 0,
             arc a,
             pen sectorpen = currentpen,
             pen markpen = sectorpen,
             margin margin = NoMargin,
             arrowbar arrow = None,
             marker marker = nomarker)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  real Da = degrees(a);
  pair p1 = point(a, 0);
  pair p2 = relpoint(a, 1);
  pair c = a.polarconicroutine == fromCenter ? locate(a.el.C) : locate(a.el.F1);
  if(radius == 0) radius = markangleradius(markpen);
  if(abs(Da) > 180) radius = -radius;
  radius = (a.direction ? 1 : -1) * sgnd(Da) * radius;
  draw(c--p1^^c--p2, sectorpen);
  markangle(pic = pic, L = L, n = n, radius = radius, space = space,
            A = p1, O = c, B = p2,
            arrow = arrow, p = markpen, margin = margin,
            marker = marker);
}
// *.........................ARCS..........................*
// *=======================================================*

// *=======================================================*
// *........................MASSES.........................*
/*<asyxml><struct signature="mass"><code></asyxml>*/
struct mass {/*<asyxml></code><documentation></documentation><property type = "point" signature="M"><code></asyxml>*/
  point M;/*<asyxml></code><documentation></documentation></property><property type = "real" signature="m"><code></asyxml>*/
  real m;/*<asyxml></code><documentation></documentation></property></asyxml>*/
}/*<asyxml></struct></asyxml>*/

/*<asyxml><function type="mass" signature="mass(point,real)"><code></asyxml>*/
mass mass(point M, real m)
{/*<asyxml></code><documentation>Constructor of mass point.</documentation></function></asyxml>*/
  mass om;
  om.M = M;
  om.m = m;
  return om;
}

/*<asyxml><operator type = "point" signature="cast(mass)"><code></asyxml>*/
point operator cast(mass m)
{/*<asyxml></code><documentation>Cast mass point to point.</documentation></operator></asyxml>*/
  point op;
  op = m.M;
  op.m = m.m;
  return op;
}
/*<asyxml><function type="point" signature="point(explicit mass)"><code></asyxml>*/
point point(explicit mass m){return m;}/*<asyxml></code><documentation>Cast
                                         'm' to point</documentation></function></asyxml>*/

/*<asyxml><operator type = "mass" signature="cast(point)"><code></asyxml>*/
mass operator cast(point M)
{/*<asyxml></code><documentation>Cast point to mass point.</documentation></operator></asyxml>*/
  mass om;
  om.M = M;
  om.m = M.m;
  return om;
}
/*<asyxml><function type="mass" signature="mass(explicit point)"><code></asyxml>*/
mass mass(explicit point P)
{/*<asyxml></code><documentation>Cast 'P' to mass.</documentation></function></asyxml>*/
  return mass(P, P.m);
}

/*<asyxml><operator type = "point[]" signature="cast(mass[])"><code></asyxml>*/
point[] operator cast(mass[] m)
{/*<asyxml></code><documentation>Cast mass[] to point[].</documentation></operator></asyxml>*/
  point[] op;
  for(mass am : m) op.push(point(am));
  return op;
}

/*<asyxml><operator type = "mass[]" signature="cast(point[])"><code></asyxml>*/
mass[] operator cast(point[] P)
{/*<asyxml></code><documentation>Cast point[] to mass[].</documentation></operator></asyxml>*/
  mass[] om;
  for(point op : P) om.push(mass(op));
  return om;
}

/*<asyxml><function type="mass" signature="mass(coordsys,explicit pair,real)"><code></asyxml>*/
mass mass(coordsys R, explicit pair p, real m)
{/*<asyxml></code><documentation>Return the mass which has coordinates
   'p' with respect to 'R' and weight 'm'.</documentation></function></asyxml>*/
  return point(R, p, m);// Using casting.
}

/*<asyxml><operator type = "mass" signature="cast(pair)"><code></asyxml>*/
mass operator cast(pair m){return mass((point)m, 1);}/*<asyxml></code><documentation>Cast pair to mass point.</documentation></operator></asyxml>*/
/*<asyxml><operator type = "path" signature="cast(mass)"><code></asyxml>*/
path operator cast(mass M){return M.M;}/*<asyxml></code><documentation>Cast mass point to path.</documentation></operator></asyxml>*/
/*<asyxml><operator type = "guide" signature="cast(mass)"><code></asyxml>*/
guide operator cast(mass M){return M.M;}/*<asyxml></code><documentation>Cast mass to guide.</documentation></operator></asyxml>*/

/*<asyxml><operator type = "mass" signature="+(mass,mass)"><code></asyxml>*/
mass operator +(mass M1, mass M2)
{/*<asyxml></code><documentation>Provide mass + mass.
   mass - mass is also defined.</documentation></operator></asyxml>*/
  return mass(M1.M + M2.M, M1.m + M2.m);
}
mass operator -(mass M1, mass M2)
{
  return mass(M1.M - M2.M, M1.m - M2.m);
}

/*<asyxml><operator type = "mass" signature="*(real,mass)"><code></asyxml>*/
mass operator *(real x, explicit mass M)
{/*<asyxml></code><documentation>Provide real * mass.
   The resulted mass is the mass of 'M' multiplied by 'x' .
   mass/real, mass + real and mass - real are also defined.</documentation></operator></asyxml>*/
  return mass(M.M, x * M.m);
}
mass operator *(int x, explicit mass M){return mass(M.M, x * M.m);}
mass operator /(explicit mass M, real x){return mass(M.M, M.m/x);}
mass operator /(explicit mass M, int x){return mass(M.M, M.m/x);}
mass operator +(explicit mass M, real x){return mass(M.M, M.m + x);}
mass operator +(explicit mass M, int x){return mass(M.M, M.m + x);}
mass operator -(explicit mass M, real x){return mass(M.M, M.m - x);}
mass operator -(explicit mass M, int x){return mass(M.M, M.m - x);}
/*<asyxml><operator type = "mass" signature="*(transform,mass)"><code></asyxml>*/
mass operator *(transform t, mass M)
{/*<asyxml></code><documentation>Provide transform * mass.</documentation></operator></asyxml>*/
  return mass(t * M.M, M.m);
}

/*<asyxml><function type="mass" signature="masscenter(... mass[])"><code></asyxml>*/
mass masscenter(... mass[] M)
{/*<asyxml></code><documentation>Return the center of the masses 'M'.</documentation></function></asyxml>*/
  point[] P;
  for (int i = 0; i < M.length; ++i)
    P.push(M[i].M);
  P = standardizecoordsys(currentcoordsys, true ... P);
  real m = M[0].m;
  point oM = M[0].m * P[0];
  for (int i = 1; i < M.length; ++i) {
    oM += M[i].m * P[i];
    m += M[i].m;
  }
  if (m == 0) abort("masscenter: the sum of masses is null.");
  return mass(oM/m, m);
}

/*<asyxml><function type="string" signature="massformat(string,string,mass)"><code></asyxml>*/
string massformat(string format = defaultmassformat,
                  string s, mass M)
{/*<asyxml></code><documentation>Return the string formated by 'format' with the mass value.
   In the parameter 'format', %L will be replaced by 's'.
   <look href = "#defaultmassformat"/>.</documentation></function></asyxml>*/
  return format == "" ? s :
    format(replace(format, "%L", replace(s, "$", "")), M.m);
}

/*<asyxml><function type="void" signature="label(picture,Label,explicit mass,align,string,pen,filltype)"><code></asyxml>*/
void label(picture pic = currentpicture, Label L, explicit mass M,
           align align = NoAlign, string format = defaultmassformat,
           pen p = nullpen, filltype filltype = NoFill)
{/*<asyxml></code><documentation>Draw label returned by massformat(format, L, M) at coordinates of M.
   <look href = "#massformat(string, string, mass)"/>.</documentation></function></asyxml>*/
  Label lL = L.copy();
  lL.s = massformat(format, lL.s, M);
  Label L = Label(lL, M.M, align, p, filltype);
  add(pic, L);
}

/*<asyxml><function type="void" signature="dot(picture,Label,explicit mass,align,string,pen)"><code></asyxml>*/
void dot(picture pic = currentpicture, Label L, explicit mass M, align align = NoAlign,
         string format = defaultmassformat, pen p = currentpen)
{/*<asyxml></code><documentation>Draw a dot with label 'L' as
   label(picture, Label, explicit mass, align, string, pen, filltype) does.
   <look href = "#label(picture, Label, mass, align, string, pen, filltype)"/>.</documentation></function></asyxml>*/
  Label lL = L.copy();
  lL.s = massformat(format, lL.s, M);
  lL.position(locate(M.M));
  lL.align(align, E);
  lL.p(p);
  dot(pic, M.M, p);
  add(pic, lL);
}
// *........................MASSES.........................*
// *=======================================================*

// *=======================================================*
// *.......................TRIANGLES.......................*
/*<asyxml><function type="point" signature="orthocentercenter(point,point,point)"><code></asyxml>*/
point orthocentercenter(point A, point B, point C)
{/*<asyxml></code><documentation>Return the orthocenter of the triangle ABC.</documentation></function></asyxml>*/
  point[] P = standardizecoordsys(A, B, C);
  coordsys R = P[0].coordsys;
  pair pp = extension(A, projection(P[1], P[2]) * P[0], B, projection(P[0], P[2]) * P[1]);
  return point(R, pp/R);
}

/*<asyxml><function type="point" signature="centroid(point,point,point)"><code></asyxml>*/
point centroid(point A, point B, point C)
{/*<asyxml></code><documentation>Return the centroid of the triangle ABC.</documentation></function></asyxml>*/
  return (A + B + C)/3;
}

/*<asyxml><function type="point" signature="incenter(point,point,point)"><code></asyxml>*/
point incenter(point A, point B, point C)
{/*<asyxml></code><documentation>Return the center of the incircle of the triangle ABC.</documentation></function></asyxml>*/
  point[] P = standardizecoordsys(A, B, C);
  coordsys R = P[0].coordsys;
  pair a = A, b = B, c = C;
  pair pp = extension(a, a + dir(a--b, a--c), b, b + dir(b--a, b--c));
  return point(R, pp/R);
}

/*<asyxml><function type="real" signature="inradius(point,point,point)"><code></asyxml>*/
real inradius(point A, point B, point C)
{/*<asyxml></code><documentation>Return the radius of the incircle of the triangle ABC.</documentation></function></asyxml>*/
  point IC = incenter(A, B, C);
  return abs(IC - projection(A, B) * IC);
}

/*<asyxml><function type="circle" signature="incircle(point,point,point)"><code></asyxml>*/
circle incircle(point A, point B, point C)
{/*<asyxml></code><documentation>Return the incircle of the triangle ABC.</documentation></function></asyxml>*/
  point IC = incenter(A, B, C);
  return circle(IC, abs(IC - projection(A, B) * IC));
}

/*<asyxml><function type="point" signature="excenter(point,point,point)"><code></asyxml>*/
point excenter(point A, point B, point C)
{/*<asyxml></code><documentation>Return the center of the excircle of the triangle tangent with (AB).</documentation></function></asyxml>*/
  point[] P = standardizecoordsys(A, B, C);
  coordsys R = P[0].coordsys;
  pair a = A, b = B, c = C;
  pair pp = extension(a, a + rotate(90) * dir(a--b, a--c), b, b + rotate(90) * dir(b--a, b--c));
  return point(R, pp/R);
}

/*<asyxml><function type="real" signature="exradius(point,point,point)"><code></asyxml>*/
real exradius(point A, point B, point C)
{/*<asyxml></code><documentation>Return the radius of the excircle of the triangle ABC with (AB).</documentation></function></asyxml>*/
  point EC = excenter(A, B, C);
  return abs(EC - projection(A, B) * EC);
}

/*<asyxml><function type="circle" signature="excircle(point,point,point)"><code></asyxml>*/
circle excircle(point A, point B, point C)
{/*<asyxml></code><documentation>Return the excircle of the triangle ABC tangent with (AB).</documentation></function></asyxml>*/
  point center = excenter(A, B, C);
  real radius = abs(center - projection(B, C) * center);
  return circle(center, radius);
}

private int[] numarray = {1, 2, 3};
numarray.cyclic = true;

/*<asyxml><struct signature="triangle"><code></asyxml>*/
struct triangle {/*<asyxml></code><documentation></documentation></asyxml>*/

  /*<asyxml><struct signature="vertex"><code></asyxml>*/
  struct vertex {/*<asyxml></code><documentation>Structure used to communicate the vertex of a triangle.</documentation><property type = "int" signature="n"><code></asyxml>*/
    int n;/*<asyxml></code><documentation>1 means VA,2 means VB,3 means VC,4 means VA etc...</documentation></property><property type = "triangle" signature="triangle"><code></asyxml>*/
    triangle t;/*<asyxml></code><documentation>The triangle to which the vertex refers.</documentation></property></asyxml>*/
  }/*<asyxml></struct></asyxml>*/

  /*<asyxml><property type = "point" signature="A,B,C"><code></asyxml>*/
  restricted point A, B, C;/*<asyxml></code><documentation>The vertices of the triangle (as point).</documentation></property><property type = "vertex" signature="VA, VB, VC"><code></asyxml>*/
  restricted vertex VA, VB, VC;/*<asyxml></code><documentation>The vertices of the triangle (as vertex).
                                 Note that the vertex structure contains the triangle to wish it refers.</documentation></property></asyxml>*/
  VA.n = 1;VB.n = 2;VC.n = 3;

  /*<asyxml><method type = "vertex" signature="vertex(int)"><code></asyxml>*/
  vertex vertex(int n)
  {/*<asyxml></code><documentation>Return numbered vertex.
     'n' is 1 means VA, 2 means VB, 3 means VC, 4 means VA etc...</documentation></method></asyxml>*/
    n = numarray[n - 1];
    if(n == 1) return VA;
    else if(n == 2) return VB;
    return VC;
  }

  /*<asyxml><method type = "point" signature="point(int)"><code></asyxml>*/
  point point(int n)
  {/*<asyxml></code><documentation>Return numbered point.
     n is 1 means A, 2 means B, 3 means C, 4 means A etc...</documentation></method></asyxml>*/
    n = numarray[n - 1];
    if(n == 1) return A;
    else if(n == 2) return B;
    return C;
  }

  /*<asyxml><method type = "void" signature="init(point,point,point)"><code></asyxml>*/
  void init(point A, point B, point C)
  {/*<asyxml></code><documentation>Constructor.</documentation></method></asyxml>*/
    point[] P = standardizecoordsys(A, B, C);
    this.A = P[0];
    this.B = P[1];
    this.C = P[2];
    VA.t = this; VB.t = this; VC.t = this;
  }

  /*<asyxml><method type = "void" signature="operator init(point,point,point)"><code></asyxml>*/
  void operator init(point A, point B, point C)
  {/*<asyxml></code><documentation>For backward compatibility.
     Provide the routine 'triangle(point A, point B, point C)'.</documentation></method></asyxml>*/
    this.init(A, B, C);
  }

  /*<asyxml><method type = "void" signature="init(real,real,real,real,point)"><code></asyxml>*/
  void operator init(real b, real alpha, real c, real angle = 0, point A = (0, 0))
  {/*<asyxml></code><documentation>For backward compatibility.
     Provide the routine 'triangle(real b, real alpha, real c, real angle = 0, point A = (0, 0))
     which returns the triangle ABC rotated by 'angle' (in degrees) and where b = AC, degrees(A) = alpha, AB = c.</documentation></method></asyxml>*/
    coordsys R = A.coordsys;
    this.init(A, A + R.polar(c, radians(angle)), A + R.polar(b, radians(angle + alpha)));
  }

  /*<asyxml><method type = "real" signature="a(),b(),c()"><code></asyxml>*/
  real a()
  {/*<asyxml></code><documentation>Return the length BC.
     b() and c() are also defined and return the length AC and AB respectively.</documentation></method></asyxml>*/
    return length(C - B);
  }
  real b() {return length(A - C);}
  real c() {return length(B - A);}

  private real det(pair a, pair b) {return a.x * b.y - a.y * b.x;}

  /*<asyxml><method type = "real" signature="area()"><code></asyxml>*/
  real area()
  {/*<asyxml></code><documentation></documentation></method></asyxml>*/
    pair a = locate(A), b = locate(B), c = locate(C);
    return 0.5 * abs(det(a, b) + det(b, c) + det(c, a));
  }

  /*<asyxml><method type = "real" signature="alpha(),beta(),gamma()"><code></asyxml>*/
  real alpha()
  {/*<asyxml></code><documentation>Return the measure (in degrees) of the angle A.
     beta() and gamma() are also defined and return the measure of the angles B and C respectively.</documentation></method></asyxml>*/
    return degrees(acos((b()^2 + c()^2 - a()^2)/(2b() * c())));
  }
  real beta()  {return degrees(acos((c()^2 + a()^2 - b()^2)/(2c() * a())));}
  real gamma() {return degrees(acos((a()^2 + b()^2 - c()^2)/(2a() * b())));}

  /*<asyxml><method type = "path" signature="Path()"><code></asyxml>*/
  path Path()
  {/*<asyxml></code><documentation>The path of the triangle.</documentation></method></asyxml>*/
    return A--C--B--cycle;
  }

  /*<asyxml><struct signature="side"><code></asyxml>*/
  struct side
  {/*<asyxml></code><documentation>Structure used to communicate the side of a triangle.</documentation><property type = "int" signature="n"><code></asyxml>*/
    int n;/*<asyxml></code><documentation>1 or 0 means [AB],-1 means [BA],2 means [BC],-2 means [CB] etc.</documentation></property><property type = "triangle" signature="triangle"><code></asyxml>*/
    triangle t;/*<asyxml></code><documentation>The triangle to which the side refers.</documentation></property></asyxml>*/
  }/*<asyxml></struct></asyxml>*/

  /*<asyxml><property type = "side" signature="AB"><code></asyxml>*/
  side AB;/*<asyxml></code><documentation>For the routines using the structure 'side', triangle.AB means 'side AB'.
            BA, AC, CA etc are also defined.</documentation></property></asyxml>*/
  AB.n = 1; AB.t = this;
  side BA; BA.n = -1; BA.t = this;
  side BC; BC.n = 2; BC.t = this;
  side CB; CB.n = -2; CB.t = this;
  side CA; CA.n = 3; CA.t = this;
  side AC; AC.n = -3; AC.t = this;

  /*<asyxml><method type = "side" signature="side(int)"><code></asyxml>*/
  side side(int n)
  {/*<asyxml></code><documentation>Return numbered side.
     n is 1 means AB, -1 means BA, 2 means BC, -2 means CB, etc.</documentation></method></asyxml>*/
    if(n == 0) abort('Invalid side number.');
    int an = numarray[abs(n)-1];
    if(an == 1) return n > 0 ? AB : BA;
    else if(an == 2) return n > 0 ? BC : CB;
    return n > 0 ? CA : AC;
  }

  /*<asyxml><method type = "line" signature="line(int)"><code></asyxml>*/
  line line(int n)
  {/*<asyxml></code><documentation>Return the numbered line.</documentation></method></asyxml>*/
    if(n == 0) abort('Invalid line number.');
    int an = numarray[abs(n)-1];
    if(an == 1) return n > 0 ? line(A, B) : line(B, A);
    else if(an == 2) return n > 0 ? line(B, C) : line(C, B);
    return n > 0 ? line(C, A) : line(A, C);
  }

}/*<asyxml></struct></asyxml>*/

from triangle unravel side; // The structure 'side' is now available outside the triangle structure.
from triangle unravel vertex; // The structure 'vertex' is now available outside the triangle structure.

triangle[] operator ^^(triangle[] t1, triangle t2)
{
  triangle[] T;
  for (int i = 0; i < t1.length; ++i) T.push(t1[i]);
  T.push(t2);
  return T;
}

triangle[] operator ^^(... triangle[] t)
{
  triangle[] T;
  for (int i = 0; i < t.length; ++i) {
    T.push(t[i]);
  }
  return T;
}

/*<asyxml><operator type = "line" signature="cast(side)"><code></asyxml>*/
line operator cast(side side)
{/*<asyxml></code><documentation>Cast side to (infinite) line.
   Most routine with line parameters works with side parameters.
   One can use the code 'segment(a_side)' to obtain a line segment.</documentation></operator></asyxml>*/
  triangle t = side.t;
  return t.line(side.n);
}

/*<asyxml><function type="line" signature="line(explicit side)"><code></asyxml>*/
line line(explicit side side)
{/*<asyxml></code><documentation>Return 'side' as line.</documentation></function></asyxml>*/
  return (line)side;
}

/*<asyxml><function type="segment" signature="segment(explicit side)"><code></asyxml>*/
segment segment(explicit side side)
{/*<asyxml></code><documentation>Return 'side' as segment.</documentation></function></asyxml>*/
  return (segment)(line)side;
}

/*<asyxml><operator type = "point" signature="cast(vertex)"><code></asyxml>*/
point operator cast(vertex V)
{/*<asyxml></code><documentation>Cast vertex to point.
   Most routine with point parameters works with vertex parameters.</documentation></operator></asyxml>*/
  return V.t.point(V.n);
}

/*<asyxml><function type="point" signature="point(explicit vertex)"><code></asyxml>*/
point point(explicit vertex V)
{/*<asyxml></code><documentation>Return the point corresponding to the vertex 'V'.</documentation></function></asyxml>*/
  return (point)V;
}

/*<asyxml><function type="side" signature="opposite(vertex)"><code></asyxml>*/
side opposite(vertex V)
{/*<asyxml></code><documentation>Return the opposite side of vertex 'V'.</documentation></function></asyxml>*/
  return V.t.side(numarray[abs(V.n)]);
}

/*<asyxml><function type="vertex" signature="opposite(side)"><code></asyxml>*/
vertex opposite(side side)
{/*<asyxml></code><documentation>Return the opposite vertex of side 'side'.</documentation></function></asyxml>*/
  return side.t.vertex(numarray[abs(side.n) + 1]);
}

/*<asyxml><function type="point" signature="midpoint(side)"><code></asyxml>*/
point midpoint(side side)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  return midpoint(segment(side));
}

/*<asyxml><operator type = "triangle" signature="*(transform,triangle)"><code></asyxml>*/
triangle operator *(transform T, triangle t)
{/*<asyxml></code><documentation>Provide transform * triangle.</documentation></operator></asyxml>*/
  return triangle(T * t.A, T * t.B, T * t.C);
}

/*<asyxml><function type="triangle" signature="triangleAbc(real,real,real,real,point)"><code></asyxml>*/
triangle triangleAbc(real alpha, real b, real c, real angle = 0, point A = (0, 0))
{/*<asyxml></code><documentation>Return the triangle ABC rotated by 'angle' with BAC = alpha, AC = b and AB = c.</documentation></function></asyxml>*/
  triangle T;
  coordsys R = A.coordsys;
  T.init(A, A + R.polar(c, radians(angle)), A + R.polar(b, radians(angle + alpha)));
  return T;
}

/*<asyxml><function type="triangle" signature="triangleabc(real,real,real,real,point)"><code></asyxml>*/
triangle triangleabc(real a, real b, real c, real angle = 0, point A = (0, 0))
{/*<asyxml></code><documentation>Return the triangle ABC rotated by 'angle' with BC = a, AC = b and AB = c.</documentation></function></asyxml>*/
  triangle T;
  coordsys R = A.coordsys;
  T.init(A, A + R.polar(c, radians(angle)), A + R.polar(b, radians(angle) + acos((b^2 + c^2 - a^2)/(2 * b * c))));
  return T;
}

/*<asyxml><function type="triangle" signature="triangle(line,line,line)"><code></asyxml>*/
triangle triangle(line l1, line l2, line l3)
{/*<asyxml></code><documentation>Return the triangle defined by three line.</documentation></function></asyxml>*/
  point P1, P2, P3;
  P1 = intersectionpoint(l1, l2);
  P2 = intersectionpoint(l1, l3);
  P3 = intersectionpoint(l2, l3);
  if(!(defined(P1) && defined(P2) && defined(P3))) abort("triangle: two lines are parallel.");
  return triangle(P1, P2, P3);
}

/*<asyxml><function type="point" signature="foot(vertex)"><code></asyxml>*/
point foot(vertex V)
{/*<asyxml></code><documentation>Return the endpoint of the altitude from V.</documentation></function></asyxml>*/
  return projection((line)opposite(V)) * ((point)V);
}

/*<asyxml><function type="point" signature="foot(side)"><code></asyxml>*/
point foot(side side)
{/*<asyxml></code><documentation>Return the endpoint of the altitude on 'side'.</documentation></function></asyxml>*/
  return projection((line)side) * point(opposite(side));
}

/*<asyxml><function type="line" signature="altitude(vertex)"><code></asyxml>*/
line altitude(vertex V)
{/*<asyxml></code><documentation>Return the altitude passing through 'V'.</documentation></function></asyxml>*/
  return line(point(V), foot(V));
}

/*<asyxml><function type="line" signature="altitude(vertex)"><code></asyxml>*/
line altitude(side side)
{/*<asyxml></code><documentation>Return the altitude cutting 'side'.</documentation></function></asyxml>*/
  return altitude(opposite(side));
}

/*<asyxml><function type="point" signature="orthocentercenter(triangle)"><code></asyxml>*/
point orthocentercenter(triangle t)
{/*<asyxml></code><documentation>Return the orthocenter of the triangle t.</documentation></function></asyxml>*/
  return orthocentercenter(t.A, t.B, t.C);
}

/*<asyxml><function type="point" signature="centroid(triangle)"><code></asyxml>*/
point centroid(triangle t)
{/*<asyxml></code><documentation>Return the centroid of the triangle 't'.</documentation></function></asyxml>*/
  return (t.A + t.B + t.C)/3;
}

/*<asyxml><function type="point" signature="circumcenter(triangle)"><code></asyxml>*/
point circumcenter(triangle t)
{/*<asyxml></code><documentation>Return the circumcenter of the triangle 't'.</documentation></function></asyxml>*/
  return circumcenter(t.A, t.B, t.C);
}

/*<asyxml><function type="circle" signature="circle(triangle)"><code></asyxml>*/
circle circle(triangle t)
{/*<asyxml></code><documentation>Return the circumcircle of the triangle 't'.</documentation></function></asyxml>*/
  return circle(t.A, t.B, t.C);
}

/*<asyxml><function type="circle" signature="circumcircle(triangle)"><code></asyxml>*/
circle circumcircle(triangle t)
{/*<asyxml></code><documentation>Return the circumcircle of the triangle 't'.</documentation></function></asyxml>*/
  return circle(t.A, t.B, t.C);
}

/*<asyxml><function type="point" signature="incenter(triangle)"><code></asyxml>*/
point incenter(triangle t)
{/*<asyxml></code><documentation>Return the center of the incircle of the triangle 't'.</documentation></function></asyxml>*/
  return incenter(t.A, t.B, t.C);
}

/*<asyxml><function type="real" signature="inradius(triangle)"><code></asyxml>*/
real inradius(triangle t)
{/*<asyxml></code><documentation>Return the radius of the incircle of the triangle 't'.</documentation></function></asyxml>*/
  return inradius(t.A, t.B, t.C);
}

/*<asyxml><function type="circle" signature="incircle(triangle)"><code></asyxml>*/
circle incircle(triangle t)
{/*<asyxml></code><documentation>Return the the incircle of the triangle 't'.</documentation></function></asyxml>*/
  return incircle(t.A, t.B, t.C);
}

/*<asyxml><function type="point" signature="excenter(side,triangle)"><code></asyxml>*/
point excenter(side side)
{/*<asyxml></code><documentation>Return the center of the excircle tangent with the side 'side' of its triangle.
   side = 0 means AB, 1 means AC, other means BC.
   One must use the predefined sides t.AB, t.AC where 't' is a triangle....</documentation></function></asyxml>*/
  point op;
  triangle t = side.t;
  int n = numarray[abs(side.n) - 1];
  if(n == 1) op = excenter(t.A, t.B, t.C);
  else  if(n == 2) op = excenter(t.B, t.C, t.A);
  else op = excenter(t.C, t.A, t.B);
  return op;
}

/*<asyxml><function type="real" signature="exradius(side,triangle)"><code></asyxml>*/
real exradius(side side)
{/*<asyxml></code><documentation>Return radius of the excircle tangent with the side 'side' of its triangle.
   side = 0 means AB, 1 means BC, other means CA.
   One must use the predefined sides t.AB, t.AC where 't' is a triangle....</documentation></function></asyxml>*/
  real or;
  triangle t = side.t;
  int n = numarray[abs(side.n) - 1];
  if(n == 1) or = exradius(t.A, t.B, t.C);
  else  if(n == 2) or = exradius(t.B, t.C, t.A);
  else or = exradius(t.A, t.C, t.B);
  return or;
}

/*<asyxml><function type="circle" signature="excircle(side,triangle)"><code></asyxml>*/
circle excircle(side side)
{/*<asyxml></code><documentation>Return the excircle tangent with the side 'side' of its triangle.
   side = 0 means AB, 1 means AC, other means BC.
   One must use the predefined sides t.AB, t.AC where 't' is a triangle....</documentation></function></asyxml>*/
  circle oc;
  int n = numarray[abs(side.n) - 1];
  triangle t = side.t;
  if(n == 1) oc = excircle(t.A, t.B, t.C);
  else  if(n == 2) oc = excircle(t.B, t.C, t.A);
  else oc = excircle(t.A, t.C, t.B);
  return oc;
}

/*<asyxml><struct signature="trilinear"><code></asyxml>*/
struct trilinear
{/*<asyxml></code><documentation>Trilinear coordinates 'a:b:c' relative to triangle 't'.
   <url href = "http://mathworld.wolfram.com/TrilinearCoordinates.html"/></documentation><property type = "real" signature="a,b,c"><code></asyxml>*/
  real a,b,c;/*<asyxml></code><documentation>The trilinear coordinates.</documentation></property><property type = "triangle" signature="t"><code></asyxml>*/
  triangle t;/*<asyxml></code><documentation>The reference triangle.</documentation></property></asyxml>*/
}/*<asyxml></struct></asyxml>*/

/*<asyxml><function type="trilinear" signature="trilinear(triangle,real,real,real)"><code></asyxml>*/
trilinear trilinear(triangle t, real a, real b, real c)
{/*<asyxml></code><documentation>Return the trilinear coordinates relative to 't'.
   <url href = "http://mathworld.wolfram.com/TrilinearCoordinates.html"/></documentation></function></asyxml>*/
  trilinear ot;
  ot.a = a; ot.b = b; ot.c = c;
  ot.t = t;
  return ot;
}

/*<asyxml><function type="trilinear" signature="trilinear(triangle,point)"><code></asyxml>*/
trilinear trilinear(triangle t, point M)
{/*<asyxml></code><documentation>Return the trilinear coordinates of 'M' relative to 't'.
   <url href = "http://mathworld.wolfram.com/TrilinearCoordinates.html"/></documentation></function></asyxml>*/
  trilinear ot;
  pair m = locate(M);
  int sameside(pair A, pair B, pair m, pair p)
  {// Return 1 if 'm' and 'p' are same side of line (AB) else return -1.
    pair mil = (A + B)/2;
    pair mA = rotate(90, mil) * A;
    pair mB = rotate(-90, mil) * A;
    return (abs(m - mA) <= abs(m - mB)) == (abs(p - mA) <= abs(p - mB)) ? 1 : -1;
  }
  real det(pair a, pair b) {return a.x * b.y - a.y * b.x;}
  real area(pair a, pair b, pair c){return 0.5 * abs(det(a, b) + det(b, c) + det(c, a));}
  pair A = t.A, B = t.B, C = t.C;
  real t1 = area(B, C, m), t2 = area(C, A, m), t3 = area(A, B, m);
  ot.a = sameside(B, C, A, m) * t1/t.a();
  ot.b = sameside(A, C, B, m) * t2/t.b();
  ot.c = sameside(A, B, C, m) * t3/t.c();
  ot.t = t;
  return ot;
}

/*<asyxml><function type="void" signature="write(trilinear)"><code></asyxml>*/
void write(trilinear tri)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  write(format("%f : ", tri.a) + format("%f : ", tri.b) + format("%f", tri.c));
}

/*<asyxml><function type="point" signature="trilinear(triangle,real,real,real)"><code></asyxml>*/
point point(trilinear tri)
{/*<asyxml></code><documentation>Return the trilinear coordinates relative to 't'.
   <url href = "http://mathworld.wolfram.com/TrilinearCoordinates.html"/></documentation></function></asyxml>*/
  triangle t = tri.t;
  return masscenter(0.5 * t.a() * mass(t.A, tri.a),
                    0.5 * t.b() * mass(t.B, tri.b),
                    0.5 * t.c() * mass(t.C, tri.c));
}

/*<asyxml><function type="int[]" signature="tricoef(side)"><code></asyxml>*/
int[] tricoef(side side)
{/*<asyxml></code><documentation>Return an array of integer (values are 0 or 1) which represents 'side'.
   For example, side = t.BC will be represented by {0, 1, 1}.</documentation></function></asyxml>*/
  int[] oi;
  int n = numarray[abs(side.n) - 1];
  oi.push((n == 1 || n == 3) ? 1 : 0);
  oi.push((n == 1 || n == 2) ? 1 : 0);
  oi.push((n == 2 || n == 3) ? 1 : 0);
  return oi;
}

/*<asyxml><operator type = "point" signature="cast(trilinear)"><code></asyxml>*/
point operator cast(trilinear tri)
{/*<asyxml></code><documentation>Cast trilinear to point.
   One may use the routine 'point(trilinear)' to force the casting.</documentation></operator></asyxml>*/
  return point(tri);
}

/*<asyxml><typedef type = "centerfunction" return = "real" params = "real, real, real"><code></asyxml>*/
typedef real centerfunction(real, real, real);/*<asyxml></code><documentation><url href = "http://mathworld.wolfram.com/TriangleCenterFunction.html"/></documentation></typedef></asyxml>*/

/*<asyxml><function type="trilinear" signature="trilinear(triangle,centerfunction,real,real,real)"><code></asyxml>*/
trilinear trilinear(triangle t, centerfunction f, real a = t.a(), real b = t.b(), real c = t.c())
{/*<asyxml></code><documentation><url href = "http://mathworld.wolfram.com/TriangleCenterFunction.html"/></documentation></function></asyxml>*/
  return trilinear(t, f(a, b, c), f(b, c, a), f(c, a, b));
}

/*<asyxml><function type="point" signature="symmedian(triangle)"><code></asyxml>*/
point symmedian(triangle t)
{/*<asyxml></code><documentation>Return the symmedian point of 't'.</documentation></function></asyxml>*/
  point A, B, C;
  real a = t.a(), b = t.b(), c = t.c();
  A = trilinear(t, 0, b, c);
  B = trilinear(t, a, 0, c);
  return intersectionpoint(line(t.A, A), line(t.B, B));
}

/*<asyxml><function type="point" signature="symmedian(side)"><code></asyxml>*/
point symmedian(side side)
{/*<asyxml></code><documentation>The symmedian point on the side 'side'.</documentation></function></asyxml>*/
  triangle t = side.t;
  int n = numarray[abs(side.n) - 1];
  if(n == 1) return trilinear(t, t.a(), t.b(), 0);
  if(n == 2) return trilinear(t, 0, t.b(), t.c());
  return trilinear(t, t.a(), 0, t.c());
}

/*<asyxml><function type="line" signature="symmedian(vertex)"><code></asyxml>*/
line symmedian(vertex V)
{/*<asyxml></code><documentation>Return the symmedian passing through 'V'.</documentation></function></asyxml>*/
  return line(point(V), symmedian(V.t));
}

/*<asyxml><function type="triangle" signature="cevian(triangle,point)"><code></asyxml>*/
triangle cevian(triangle t, point P)
{/*<asyxml></code><documentation>Return the Cevian triangle with respect of 'P'
   <url href = "http://mathworld.wolfram.com/CevianTriangle.html"/>.</documentation></function></asyxml>*/
  trilinear tri = trilinear(t, locate(P));
  point A = point(trilinear(t, 0, tri.b, tri.c));
  point B = point(trilinear(t, tri.a, 0, tri.c));
  point C = point(trilinear(t, tri.a, tri.b, 0));
  return triangle(A, B, C);
}

/*<asyxml><function type="point" signature="cevian(side,point)"><code></asyxml>*/
point cevian(side side, point P)
{/*<asyxml></code><documentation>Return the Cevian point on 'side' with respect of 'P'.</documentation></function></asyxml>*/
  triangle t = side.t;
  trilinear tri = trilinear(t, locate(P));
  int[] s = tricoef(side);
  return point(trilinear(t, s[0] * tri.a, s[1] * tri.b, s[2] * tri.c));
}

/*<asyxml><function type="line" signature="cevian(vertex,point)"><code></asyxml>*/
line cevian(vertex V, point P)
{/*<asyxml></code><documentation>Return line passing through 'V' and its Cevian image with respect of 'P'.</documentation></function></asyxml>*/
  return line(point(V), cevian(opposite(V), P));
}

/*<asyxml><function type="point" signature="gergonne(triangle)"><code></asyxml>*/
point gergonne(triangle t)
{/*<asyxml></code><documentation>Return the Gergonne point of 't'.</documentation></function></asyxml>*/
  real f(real a, real b, real c){return 1/(a * (b + c - a));}
  return point(trilinear(t, f));
}

/*<asyxml><function type="point[]" signature="fermat(triangle)"><code></asyxml>*/
point[] fermat(triangle t)
{/*<asyxml></code><documentation>Return the Fermat points of 't'.</documentation></function></asyxml>*/
  point[] P;
  real A = t.alpha(), B = t.beta(), C = t.gamma();
  P.push(point(trilinear(t, 1/Sin(A + 60), 1/Sin(B + 60), 1/Sin(C + 60))));
  P.push(point(trilinear(t, 1/Sin(A - 60), 1/Sin(B - 60), 1/Sin(C - 60))));
  return P;
}

/*<asyxml><function type="point" signature="isotomicconjugate(triangle,point)"><code></asyxml>*/
point isotomicconjugate(triangle t, point M)
{/*<asyxml></code><documentation><url href = "http://mathworld.wolfram.com/IsotomicConjugate.html"/></documentation></function></asyxml>*/
  if(!inside(t.Path(), locate(M))) abort("isotomic: the point must be inside the triangle.");
  trilinear tr = trilinear(t, M);
  return point(trilinear(t, 1/(t.a()^2 * tr.a), 1/(t.b()^2 * tr.b), 1/(t.c()^2 * tr.c)));
}

/*<asyxml><function type="line" signature="isotomic(vertex,point)"><code></asyxml>*/
line isotomic(vertex V, point M)
{/*<asyxml></code><documentation><url href = "http://mathworld.wolfram.com/IsotomicConjugate.html"/>.</documentation></function></asyxml>*/
  side op = opposite(V);
  return line(V, rotate(180, midpoint(op)) * cevian(op, M));
}

/*<asyxml><function type="point" signature="isotomic(side,point)"><code></asyxml>*/
point isotomic(side side, point M)
{/*<asyxml></code><documentation><url href = "http://mathworld.wolfram.com/IsotomicConjugate.html"/></documentation></function></asyxml>*/
  return intersectionpoint(isotomic(opposite(side), M), side);
}

/*<asyxml><function type="triangle" signature="isotomic(triangle,point)"><code></asyxml>*/
triangle isotomic(triangle t, point M)
{/*<asyxml></code><documentation><url href = "http://mathworld.wolfram.com/IsotomicConjugate.html"/></documentation></function></asyxml>*/
  return triangle(isotomic(t.BC, M), isotomic(t.CA, M), isotomic(t.AB, M));
}

/*<asyxml><function type="point" signature="isogonalconjugate(triangle,point)"><code></asyxml>*/
point isogonalconjugate(triangle t, point M)
{/*<asyxml></code><documentation><url href = "http://mathworld.wolfram.com/IsogonalConjugate.html"/></documentation></function></asyxml>*/
  trilinear tr = trilinear(t, M);
  return point(trilinear(t, 1/tr.a, 1/tr.b, 1/tr.c));
}

/*<asyxml><function type="point" signature="isogonal(side,point)"><code></asyxml>*/
point isogonal(side side, point M)
{/*<asyxml></code><documentation><url href = "http://mathworld.wolfram.com/IsogonalConjugate.html"/></documentation></function></asyxml>*/
  return cevian(side, isogonalconjugate(side.t, M));
}

/*<asyxml><function type="line" signature="isogonal(vertex,point)"><code></asyxml>*/
line isogonal(vertex V, point M)
{/*<asyxml></code><documentation><url href = "http://mathworld.wolfram.com/IsogonalConjugate.html"/></documentation></function></asyxml>*/
  return line(V, isogonal(opposite(V), M));
}

/*<asyxml><function type="triangle" signature="isogonal(triangle,point)"><code></asyxml>*/
triangle isogonal(triangle t, point M)
{/*<asyxml></code><documentation><url href = "http://mathworld.wolfram.com/IsogonalConjugate.html"/></documentation></function></asyxml>*/
  return triangle(isogonal(t.BC, M), isogonal(t.CA, M), isogonal(t.AB, M));
}

/*<asyxml><function type="triangle" signature="pedal(triangle,point)"><code></asyxml>*/
triangle pedal(triangle t, point M)
{/*<asyxml></code><documentation>Return the pedal triangle of 'M' in 't'.
   <url href = "http://mathworld.wolfram.com/PedalTriangle.html"/></documentation></function></asyxml>*/
  return triangle(projection(t.BC) * M, projection(t.AC) * M, projection(t.AB) * M);
}

/*<asyxml><function type="triangle" signature="pedal(triangle,point)"><code></asyxml>*/
line pedal(side side, point M)
{/*<asyxml></code><documentation>Return the pedal line of 'M' cutting 'side'.
   <url href = "http://mathworld.wolfram.com/PedalTriangle.html"/></documentation></function></asyxml>*/
  return line(M, projection(side) * M);
}

/*<asyxml><function type="triangle" signature="antipedal(triangle,point)"><code></asyxml>*/
triangle antipedal(triangle t, point M)
{/*<asyxml></code><documentation><url href = "http://mathworld.wolfram.com/AntipedalTriangle.html"/></documentation></function></asyxml>*/
  trilinear Tm = trilinear(t, M);
  real a = Tm.a, b = Tm.b, c = Tm.c;
  real CA = Cos(t.alpha()), CB = Cos(t.beta()), CC = Cos(t.gamma());
  point A = trilinear(t, -(b + a * CC) * (c + a * CB), (c + a * CB) * (a + b * CC), (b + a * CC) * (a + c * CB));
  point B = trilinear(t, (c + b * CA) * (b + a * CC), -(c + b * CA) * (a + b * CC), (a + b * CC) * (b + c * CA));
  point C = trilinear(t, (b + c * CA) * (c + a * CB), (a + c * CB) * (c + b * CA), -(a + c * CB) * (b + c * CA));
  return triangle(A, B, C);
}

/*<asyxml><function type="triangle" signature="extouch(triangle)"><code></asyxml>*/
triangle extouch(triangle t)
{/*<asyxml></code><documentation>Return the extouch triangle of the triangle 't'.
   The extouch triangle of 't' is the triangle formed by the points
   of tangency of a triangle 't' with its excircles.</documentation></function></asyxml>*/
  point A, B, C;
  real a = t.a(), b = t.b(), c = t.c();
  A = trilinear(t, 0, (a - b + c)/b, (a + b - c)/c);
  B = trilinear(t, (-a + b + c)/a, 0, (a + b - c)/c);
  C = trilinear(t, (-a + b + c)/a, (a - b + c)/b, 0);
  return triangle(A, B, C);
}

/*<asyxml><function type="triangle" signature="extouch(triangle)"><code></asyxml>*/
triangle incentral(triangle t)
{/*<asyxml></code><documentation>Return the incentral triangle of the triangle 't'.
   It is the triangle whose vertices are determined by the intersections of the
   reference triangle's angle bisectors with the respective opposite sides.</documentation></function></asyxml>*/
  point A, B, C;
  // real a = t.a(), b = t.b(), c = t.c();
  A = trilinear(t, 0, 1, 1);
  B = trilinear(t, 1, 0, 1);
  C = trilinear(t, 1, 1, 0);
  return triangle(A, B, C);
}

/*<asyxml><function type="triangle" signature="extouch(side)"><code></asyxml>*/
triangle extouch(side side)
{/*<asyxml></code><documentation>Return the triangle formed by the points of tangency of the triangle referenced by 'side' with its excircles.
   One vertex of the returned triangle is on the segment 'side'.</documentation></function></asyxml>*/
  triangle t = side.t;
  transform p1 = projection((line)t.AB);
  transform p2 = projection((line)t.AC);
  transform p3 = projection((line)t.BC);
  point EP = excenter(side);
  return triangle(p3 * EP, p2 * EP, p1 * EP);
}

/*<asyxml><function type="point" signature="bisectorpoint(side)"><code></asyxml>*/
point bisectorpoint(side side)
{/*<asyxml></code><documentation>The intersection point of the angle bisector from the
   opposite point of 'side' with the side 'side'.</documentation></function></asyxml>*/
  triangle t = side.t;
  int n = numarray[abs(side.n) - 1];
  if(n == 1) return trilinear(t, 1, 1, 0);
  if(n == 2) return trilinear(t, 0, 1, 1);
  return trilinear(t, 1, 0, 1);
}

/*<asyxml><function type="line" signature="bisector(vertex,real)"><code></asyxml>*/
line bisector(vertex V, real angle = 0)
{/*<asyxml></code><documentation>Return the interior bisector passing through 'V' rotated by angle (in degrees)
   around 'V'.</documentation></function></asyxml>*/
  return rotate(angle, point(V)) * line(point(V), incenter(V.t));
}

/*<asyxml><function type="line" signature="bisector(side)"><code></asyxml>*/
line bisector(side side)
{/*<asyxml></code><documentation>Return the bisector of the line segment 'side'.</documentation></function></asyxml>*/
  return bisector(segment(side));
}

/*<asyxml><function type="point" signature="intouch(side)"><code></asyxml>*/
point intouch(side side)
{/*<asyxml></code><documentation>The point of tangency on the side 'side' of its incircle.</documentation></function></asyxml>*/
  triangle t = side.t;
  real a = t.a(), b = t.b(), c = t.c();
  int n = numarray[abs(side.n) - 1];
  if(n == 1) return trilinear(t, b * c/(-a + b + c), a * c/(a - b + c), 0);
  if(n == 2) return trilinear(t, 0, a * c/(a - b + c), a * b/(a + b - c));
  return trilinear(t, b * c/(-a + b + c), 0, a * b/(a + b - c));
}

/*<asyxml><function type="triangle" signature="intouch(triangle)"><code></asyxml>*/
triangle intouch(triangle t)
{/*<asyxml></code><documentation>Return the intouch triangle of the triangle 't'.
   The intouch triangle of 't' is the triangle formed by the points
   of tangency of a triangle 't' with its incircles.</documentation></function></asyxml>*/
  point A, B, C;
  real a = t.a(), b = t.b(), c = t.c();
  A = trilinear(t, 0, a * c/(a - b + c), a * b/(a + b - c));
  B = trilinear(t, b * c/(-a + b + c), 0, a * b/(a + b - c));
  C = trilinear(t, b * c/(-a + b + c), a * c/(a - b + c), 0);
  return triangle(A, B, C);
}

/*<asyxml><function type="triangle" signature="tangential(triangle)"><code></asyxml>*/
triangle tangential(triangle t)
{/*<asyxml></code><documentation>Return the tangential triangle of the triangle 't'.
   The tangential triangle of 't' is the triangle formed by the lines
   tangent to the circumcircle of the given triangle 't' at its vertices.</documentation></function></asyxml>*/
  point A, B, C;
  real a = t.a(), b = t.b(), c = t.c();
  A = trilinear(t, -a, b, c);
  B = trilinear(t, a, -b, c);
  C = trilinear(t, a, b, -c);
  return triangle(A, B, C);
}

/*<asyxml><function type="triangle" signature="medial(triangle t)"><code></asyxml>*/
triangle medial(triangle t)
{/*<asyxml></code><documentation>Return the triangle whose vertices are midpoints of the sides of 't'.</documentation></function></asyxml>*/
  return triangle(midpoint(t.BC), midpoint(t.AC), midpoint(t.AB));
}

/*<asyxml><function type="line" signature="median(vertex)"><code></asyxml>*/
line median(vertex V)
{/*<asyxml></code><documentation>Return median from 'V'.</documentation></function></asyxml>*/
  return line(point(V), midpoint(segment(opposite(V))));
}

/*<asyxml><function type="line" signature="median(side)"><code></asyxml>*/
line median(side side)
{/*<asyxml></code><documentation>Return median from the opposite vertex of 'side'.</documentation></function></asyxml>*/
  return median(opposite(side));
}

/*<asyxml><function type="triangle" signature="orthic(triangle)"><code></asyxml>*/
triangle orthic(triangle t)
{/*<asyxml></code><documentation>Return the triangle whose vertices are endpoints of the altitudes from each of the vertices of 't'.</documentation></function></asyxml>*/
  return triangle(foot(t.BC), foot(t.AC), foot(t.AB));
}

/*<asyxml><function type="triangle" signature="symmedial(triangle)"><code></asyxml>*/
triangle symmedial(triangle t)
{/*<asyxml></code><documentation>Return the symmedial triangle of 't'.</documentation></function></asyxml>*/
  point A, B, C;
  real a = t.a(), b = t.b(), c = t.c();
  A = trilinear(t, 0, b, c);
  B = trilinear(t, a, 0, c);
  C = trilinear(t, a, b, 0);
  return triangle(A, B, C);
}

/*<asyxml><function type="triangle" signature="anticomplementary(triangle)"><code></asyxml>*/
triangle anticomplementary(triangle t)
{/*<asyxml></code><documentation>Return the triangle which has the given triangle 't' as its medial triangle.</documentation></function></asyxml>*/
  real a = t.a(), b = t.b(), c = t.c();
  real ab = a * b, bc = b * c, ca = c * a;
  point A = trilinear(t, -bc, ca, ab);
  point B = trilinear(t, bc, -ca, ab);
  point C = trilinear(t, bc, ca, -ab);
  return triangle(A, B, C);
}

/*<asyxml><function type="point[]" signature="intersectionpoints(triangle,line,bool)"><code></asyxml>*/
point[] intersectionpoints(triangle t, line l, bool extended = false)
{/*<asyxml></code><documentation>Return the intersection points.
   If 'extended' is true, the sides are lines else the sides are segments.
   intersectionpoints(line, triangle, bool) is also defined.</documentation></function></asyxml>*/
  point[] OP;
  void addpoint(point P)
  {
    if(defined(P)) {
      bool exist = false;
      for (int i = 0; i < OP.length; ++i) {
        if(P == OP[i]) {exist = true; break;}
      }
      if(!exist) OP.push(P);
    }
  }
  if(extended) {
    for (int i = 1; i <= 3; ++i) {
      addpoint(intersectionpoint(t.line(i), l));
    }
  } else {
    for (int i = 1; i <= 3; ++i) {
      addpoint(intersectionpoint((segment)t.line(i), l));
    }
  }
  return OP;
}

point[] intersectionpoints(line l, triangle t, bool extended = false)
{
  return intersectionpoints(t, l, extended);
}

/*<asyxml><function type="vector" signature="dir(vertex)"><code></asyxml>*/
vector dir(vertex V)
{/*<asyxml></code><documentation>The direction (towards the outside of the triangle) of the interior angle bisector of 'V'.</documentation></function></asyxml>*/
  triangle t = V.t;
  if(V.n == 1) return vector(defaultcoordsys, (-dir(t.A--t.B, t.A--t.C)));
  if(V.n == 2) return vector(defaultcoordsys, (-dir(t.B--t.A, t.B--t.C)));
  return vector(defaultcoordsys, (-dir(t.C--t.A, t.C--t.B)));
}

/*<asyxml><function type="void" signature="lvoid label(picture,Label,vertex,pair,real,pen,filltype)"><code></asyxml>*/
void label(picture pic = currentpicture, Label L, vertex V,
           pair align = dir(V),
           real alignFactor = 1,
           pen p = nullpen, filltype filltype = NoFill)
{/*<asyxml></code><documentation>Draw 'L' on picture 'pic' at vertex 'V' aligned by 'alignFactor * align'.</documentation></function></asyxml>*/
  label(pic, L, locate(point(V)), alignFactor * align, p, filltype);
}

/*<asyxml><function type="void" signature="label(picture,Label,Label,Label,triangle,real,real,pen,filltype)"><code></asyxml>*/
void label(picture pic = currentpicture, Label LA = "$A$",
           Label LB = "$B$", Label LC = "$C$",
           triangle t,
           real alignAngle = 0,
           real alignFactor = 1,
           pen p = nullpen, filltype filltype = NoFill)
{/*<asyxml></code><documentation>Draw labels LA, LB and LC aligned in the rotated (by 'alignAngle' in degrees) direction
   (towards the outside of the triangle) of the interior angle bisector of vertices.
   One can  individually modify the alignment by setting the Label parameter 'align'.</documentation></function></asyxml>*/
  Label lla = LA.copy();
  lla.align(lla.align, rotate(alignAngle) * locate(dir(t.VA)));
  label(pic, LA, t.VA, align = lla.align.dir, alignFactor = alignFactor, p, filltype);
  Label llb = LB.copy();
  llb.align(llb.align, rotate(alignAngle) * locate(dir(t.VB)));
  label(pic, llb, t.VB, align = llb.align.dir, alignFactor = alignFactor, p, filltype);
  Label llc = LC.copy();
  llc.align(llc.align, rotate(alignAngle) * locate(dir(t.VC)));
  label(pic, llc, t.VC, align = llc.align.dir, alignFactor = alignFactor, p, filltype);
}

/*<asyxml><function type="void" signature="show(picture,Label,Label,Label,Label,Label,Label,triangle,pen,filltype)"><code></asyxml>*/
void show(picture pic = currentpicture,
          Label LA = "$A$", Label LB = "$B$", Label LC = "$C$",
          Label La = "$a$", Label Lb = "$b$", Label Lc = "$c$",
          triangle t, pen p = currentpen, filltype filltype = NoFill)
{/*<asyxml></code><documentation>Draw triangle and labels of sides and vertices.</documentation></function></asyxml>*/
  pair a = locate(t.A), b = locate(t.B), c = locate(t.C);
  draw(pic, a--b--c--cycle, p);
  label(pic, LA, a, -dir(a--b, a--c), p, filltype);
  label(pic, LB, b, -dir(b--a, b--c), p, filltype);
  label(pic, LC, c, -dir(c--a, c--b), p, filltype);
  pair aligna = I * unit(c - b), alignb = I * unit(c - a), alignc = I * unit(b - a);
  pair mAB = locate(midpoint(t.AB)), mAC = locate(midpoint(t.AC)), mBC = locate(midpoint(t.BC));
  label(pic, La, b--c, align = rotate(dot(a - mBC, aligna) > 0 ? 180 :0) * aligna, p);
  label(pic, Lb, a--c, align = rotate(dot(b - mAC, alignb) > 0 ? 180 :0) * alignb, p);
  label(pic, Lc, a--b, align = rotate(dot(c - mAB, alignc) > 0 ? 180 :0) * alignc, p);
}

/*<asyxml><function type="void" signature="draw(picture,triangle,pen,marker)"><code></asyxml>*/
void draw(picture pic = currentpicture, triangle t, pen p = currentpen, marker marker = nomarker)
{/*<asyxml></code><documentation>Draw sides of the triangle 't' on picture 'pic' using pen 'p'.</documentation></function></asyxml>*/
  draw(pic, t.Path(), p, marker);
}

/*<asyxml><function type="void" signature="draw(picture,triangle[],pen,marker)"><code></asyxml>*/
void draw(picture pic = currentpicture, triangle[] t, pen p = currentpen, marker marker = nomarker)
{/*<asyxml></code><documentation>Draw sides of the triangles 't' on picture 'pic' using pen 'p'.</documentation></function></asyxml>*/
  for(int i = 0; i < t.length; ++i) draw(pic, t[i], p, marker);
}

/*<asyxml><function type="void" signature="drawline(picture,triangle,pen)"><code></asyxml>*/
void drawline(picture pic = currentpicture, triangle t, pen p = currentpen)
{/*<asyxml></code><documentation>Draw lines of the triangle 't' on picture 'pic' using pen 'p'.</documentation></function></asyxml>*/
  draw(t, p);
  draw(pic, line(t.A, t.B), p);
  draw(pic, line(t.A, t.C), p);
  draw(pic, line(t.B, t.C), p);
}

/*<asyxml><function type="void" signature="dot(picture,triangle,pen)"><code></asyxml>*/
void dot(picture pic = currentpicture, triangle t, pen p = currentpen)
{/*<asyxml></code><documentation>Draw a dot at each vertex of 't'.</documentation></function></asyxml>*/
  dot(pic, t.A^^t.B^^t.C, p);
}
// *.......................TRIANGLES.......................*
// *=======================================================*

// *=======================================================*
// *.......................INVERSIONS......................*
/*<asyxml><function type="point" signature="inverse(real k,point,point)"><code></asyxml>*/
point inverse(real k, point A, point M)
{/*<asyxml></code><documentation>Return the inverse point of 'M' with respect to point A and inversion radius 'k'.</documentation></function></asyxml>*/
  return A + k/conj(M - A);
}

/*<asyxml><function type="point" signature="radicalcenter(circle,circle)"><code></asyxml>*/
point radicalcenter(circle c1, circle c2)
{/*<asyxml></code><documentation><url href = "http://fr.wikipedia.org/wiki/Puissance_d'un_point_par_rapport_%C3%A0_un_cercle"/></documentation></function></asyxml>*/
  point[] P = standardizecoordsys(c1.C, c2.C);
  real k = c1.r^2 - c2.r^2;
  pair C1 = locate(c1.C);
  pair C2 = locate(c2.C);
  pair oop = C2 - C1;
  pair K = (abs(oop) == 0) ?
    (infinity, infinity) :
    midpoint(C1--C2) + 0.5 * k * oop/dot(oop, oop);
  return point(P[0].coordsys, K/P[0].coordsys);
}

/*<asyxml><function type="line" signature="radicalline(circle,circle)"><code></asyxml>*/
line radicalline(circle c1, circle c2)
{/*<asyxml></code><documentation><url href = "http://fr.wikipedia.org/wiki/Puissance_d'un_point_par_rapport_%C3%A0_un_cercle"/></documentation></function></asyxml>*/
  if (c1.C == c2.C) abort("radicalline: the centers must be distinct");
  return perpendicular(radicalcenter(c1, c2), line(c1.C, c2.C));
}

/*<asyxml><function type="point" signature="radicalcenter(circle,circle,circle)"><code></asyxml>*/
point radicalcenter(circle c1, circle c2, circle c3)
{/*<asyxml></code><documentation><url href = "http://fr.wikipedia.org/wiki/Puissance_d'un_point_par_rapport_%C3%A0_un_cercle"/></documentation></function></asyxml>*/
  return intersectionpoint(radicalline(c1, c2), radicalline(c1, c3));
}

/*<asyxml><struct signature="inversion"><code></asyxml>*/
struct inversion
{/*<asyxml></code><documentation>http://mathworld.wolfram.com/Inversion.html</documentation></asyxml>*/
  point C;
  real k;
}/*<asyxml></struct></asyxml>*/

/*<asyxml><function type="inversion" signature="inversion(real,point)"><code></asyxml>*/
inversion inversion(real k, point C)
{/*<asyxml></code><documentation>Return the inversion with respect to 'C' having inversion radius 'k'.</documentation></function></asyxml>*/
  inversion oi;
  oi.k = k;
  oi.C = C;
  return oi;
}
/*<asyxml><function type="inversion" signature="inversion(real,point)"><code></asyxml>*/
inversion inversion(point C, real k)
{/*<asyxml></code><documentation>Return the inversion with respect to 'C' having inversion radius 'k'.</documentation></function></asyxml>*/
  return inversion(k, C);
}

/*<asyxml><function type="inversion" signature="inversion(circle,circle)"><code></asyxml>*/
inversion inversion(circle c1, circle c2, real sgn = 1)
{/*<asyxml></code><documentation>Return the inversion which transforms 'c1' to
   . 'c2' and positive inversion radius if 'sgn > 0';
   . 'c2' and negative inversion radius if 'sgn < 0';
   . 'c1' and 'c2' to 'c2' if 'sgn = 0'.</documentation></function></asyxml>*/
  if(sgn == 0) {
    point O = radicalcenter(c1, c2);
    return inversion(O^c1, O);
  }
  real a = abs(c1.r/c2.r);
  if(sgn > 0) {
    point O = c1.C + a/abs(1 - a) * (c2.C - c1.C);
    return inversion(a * abs(abs(O - c2.C)^2 - c2.r^2), O);
  }
  point O = c1.C + a/abs(1 + a) * (c2.C - c1.C);
  return inversion(-a * abs(abs(O - c2.C)^2 - c2.r^2), O);
}

/*<asyxml><function type="inversion" signature="inversion(circle,circle,circle)"><code></asyxml>*/
inversion inversion(circle c1, circle c2, circle c3)
{/*<asyxml></code><documentation>Return the inversion which transform 'c1' to 'c1', 'c2' to 'c2' and 'c3' to 'c3'.</documentation></function></asyxml>*/
  point Rc = radicalcenter(c1, c2, c3);
  return inversion(Rc, Rc^c1);
}

circle operator cast(inversion i){return circle(i.C, sgn(i.k) * sqrt(abs(i.k)));}
/*<asyxml><function type="circle" signature="circle(inversion)"><code></asyxml>*/
circle circle(inversion i)
{/*<asyxml></code><documentation>Return the inversion circle of 'i'.</documentation></function></asyxml>*/
  return i;
}

inversion operator cast(circle c)
{
  return inversion(sgn(c.r) * c.r^2, c.C);
}
/*<asyxml><function type="inversion" signature="inversion(circle)"><code></asyxml>*/
inversion inversion(circle c)
{/*<asyxml></code><documentation>Return the inversion represented by the circle of 'c'.</documentation></function></asyxml>*/
  return c;
}

/*<asyxml><operator type = "point" signature="*(inversion,point)"><code></asyxml>*/
point operator *(inversion i, point P)
{/*<asyxml></code><documentation>Provide inversion * point.</documentation></operator></asyxml>*/
  return inverse(i.k, i.C, P);
}

void lineinversion()
{
  warning("lineinversion", "the inversion of the line is not a circle.
The returned circle has an infinite radius, circle.l has been set.");
}


/*<asyxml><function type="circle" signature="inverse(real,point,line)"><code></asyxml>*/
circle inverse(real k, point A, line l)
{/*<asyxml></code><documentation>Return the inverse circle of 'l' with
   respect to point 'A' and inversion radius 'k'.</documentation></function></asyxml>*/
  if(A @ l) {
    lineinversion();
    circle C = circle(A, infinity);
    C.l = l;
    return C;
  }
  point Ap = inverse(k, A, l.A), Bp = inverse(k, A, l.B);
  return circle(A, Ap, Bp);
}

/*<asyxml><operator type = "circle" signature="*(inversion,line)"><code></asyxml>*/
circle operator *(inversion i, line l)
{/*<asyxml></code><documentation>Provide inversion * line for lines that don't pass through the inversion center.</documentation></operator></asyxml>*/
  return inverse(i.k, i.C, l);
}

/*<asyxml><function type="circle" signature="inverse(real,point,circle)"><code></asyxml>*/
circle inverse(real k, point A, circle c)
{/*<asyxml></code><documentation>Return the inverse circle of 'c' with
   respect to point A and inversion radius 'k'.</documentation></function></asyxml>*/
  if(degenerate(c)) return inverse(k, A, c.l);
  if(A @ c) {
    lineinversion();
    point M = rotate(180, c.C) * A, Mp = rotate(90, c.C) * A;
    circle oc = circle(A, infinity);
    oc.l = line(inverse(k, A, M), inverse(k, A, Mp));
    return oc;
  }
  point[] P = standardizecoordsys(A, c.C);
  real s = k/((P[1].x - P[0].x)^2 + (P[1].y - P[0].y)^2 - c.r^2);
  return circle(P[0] + s * (P[1]-P[0]), abs(s) * c.r);
}

/*<asyxml><operator type = "circle" signature="*(inversion,circle)"><code></asyxml>*/
circle operator *(inversion i, circle c)
{/*<asyxml></code><documentation>Provide inversion * circle.</documentation></operator></asyxml>*/
  return inverse(i.k, i.C, c);
}
// *.......................INVERSIONS......................*
// *=======================================================*

// *=======================================================*
// *........................FOOTER.........................*
/*<asyxml><function type="point[]" signature="intersectionpoints(line,circle)"><code></asyxml>*/
point[] intersectionpoints(line l, circle c)
{/*<asyxml></code><documentation>Note that the line 'l' may be a segment by casting.
   intersectionpoints(circle, line) is also defined.</documentation></function></asyxml>*/
  if(degenerate(c)) return new point[]{intersectionpoint(l, c.l)};
  point[] op;
  coordsys R = samecoordsys(l.A, c.C) ?
    l.A.coordsys : defaultcoordsys;
  coordsys Rp = defaultcoordsys;
  circle cc = circle(changecoordsys(Rp, c.C), c.r);
  point proj = projection(l) * c.C;
  if(proj @ cc) { // The line is a tangente of the circle.
    if(proj @ l) op.push(proj);// line may be a segement...
  } else {
    coordsys Rc = cartesiansystem(c.C, (1, 0), (0, 1));
    line ll = changecoordsys(Rc, l);
    pair[] P = intersectionpoints(ll.A.coordinates, ll.B.coordinates,
                                1, 0, 1, 0, 0, -c.r^2);
    for (int i = 0; i < P.length; ++i) {
      point inter = changecoordsys(R, point(Rc, P[i]));
      if(inter @ l) op.push(inter);
    }
  }
  return op;
}

point[] intersectionpoints(circle c, line l)
{
  return intersectionpoints(l, c);
}

/*<asyxml><function type="point[]" signature="intersectionpoints(line,ellipse)"><code></asyxml>*/
point[] intersectionpoints(line l, ellipse el)
{/*<asyxml></code><documentation>Note that the line 'l' may be a segment by casting.
   intersectionpoints(ellipse, line) is also defined.</documentation></function></asyxml>*/
  if(el.e == 0) return intersectionpoints(l, (circle)el);
  if(degenerate(el)) return new point[]{intersectionpoint(l, el.l)};
  point[] op;
  coordsys R = samecoordsys(l.A, el.C) ? l.A.coordsys : defaultcoordsys;
  coordsys Rp = defaultcoordsys;
  line ll = changecoordsys(Rp, l);
  ellipse ell = changecoordsys(Rp, el);
  circle C = circle(ell.C, ell.a);
  point[] Ip = intersectionpoints(ll, C);
  if (Ip.length > 0 &&
      (perpendicular(ll, line(ell.F1, Ip[0])) ||
       perpendicular(ll, line(ell.F2, Ip[0])))) {
    // http://www.mathcurve.com/courbes2d/ellipse/ellipse.shtml
    //  Définition tangentielle par antipodaire de cercle.
    // 'l' is a tangent of 'el'
    transform t = scale(el.a/el.b, el.F1, el.F2, el.C, rotate(90, el.C) * el.F1);
    point inter = inverse(t) * intersectionpoints(C, t * ll)[0];
    if(inter @ l) op.push(inter);
  } else {
    coordsys Rc = canonicalcartesiansystem(el);
    line ll = changecoordsys(Rc, l);
    pair[] P = intersectionpoints(ll.A.coordinates, ll.B.coordinates,
                                1/el.a^2, 0, 1/el.b^2, 0, 0, -1);
    for (int i = 0; i < P.length; ++i) {
      point inter = changecoordsys(R, point(Rc, P[i]));
      if(inter @ l) op.push(inter);
    }
  }
  return op;
}

point[] intersectionpoints(ellipse el, line l)
{
  return intersectionpoints(l, el);
}

/*<asyxml><function type="point[]" signature="intersectionpoints(line,parabola)"><code></asyxml>*/
point[] intersectionpoints(line l, parabola p)
{/*<asyxml></code><documentation>Note that the line 'l' may be a segment by casting.
   intersectionpoints(parabola, line) is also defined.</documentation></function></asyxml>*/
  point[] op;
  coordsys R = coordsys(p);
  bool tgt = false;
  line ll = changecoordsys(R, l),
    lv = parallel(p.V, p.D);
  point M = intersectionpoint(lv, ll), tgtp;
  if(finite(M)) {// Test if 'l' is tangent to 'p'
    line l1 = bisector(line(M, p.F));
    line l2 = rotate(90, M) * lv;
    point P = intersectionpoint(l1, l2);
    tgtp = rotate(180, P) * p.F;
    tgt = (tgtp @ l);
  }
  if(tgt) {
    if(tgtp @ l) op.push(tgtp);
  } else {
    real[] eq = changecoordsys(defaultcoordsys, equation(p)).a;
    pair[] tp = intersectionpoints(locate(l.A), locate(l.B), eq);
    point inter;
    for (int i = 0; i < tp.length; ++i) {
      inter = point(R, tp[i]/R);
      if(inter @ l) op.push(inter);
    }
  }
  return op;
}

point[] intersectionpoints(parabola p, line l)
{
  return intersectionpoints(l, p);
}

/*<asyxml><function type="point[]" signature="intersectionpoints(line,hyperbola)"><code></asyxml>*/
point[] intersectionpoints(line l, hyperbola h)
{/*<asyxml></code><documentation>Note that the line 'l' may be a segment by casting.
   intersectionpoints(hyperbola, line) is also defined.</documentation></function></asyxml>*/
  point[] op;
  coordsys R = coordsys(h);
  point A = intersectionpoint(l, h.A1), B = intersectionpoint(l, h.A2);
  point M = midpoint(segment(A, B));
  bool tgt = M @ h;
  if(tgt) {
    if(M @ l) op.push(M);
  } else {
    real[] eq = changecoordsys(defaultcoordsys, equation(h)).a;
    pair[] tp = intersectionpoints(locate(l.A), locate(l.B), eq);
    point inter;
    for (int i = 0; i < tp.length; ++i) {
      inter = point(R, tp[i]/R);
      if(inter @ l) op.push(inter);
    }
  }
  return op;
}

point[] intersectionpoints(hyperbola h, line l)
{
  return intersectionpoints(l, h);
}

/*<asyxml><function type="point[]" signature="intersectionpoints(line,conic)"><code></asyxml>*/
point[] intersectionpoints(line l, conic co)
{/*<asyxml></code><documentation>Note that the line 'l' may be a segment by casting.
   intersectionpoints(conic, line) is also defined.</documentation></function></asyxml>*/
  point[] op;
  if(co.e < 1) op = intersectionpoints((ellipse)co, l);
  else
    if(co.e == 1) op = intersectionpoints((parabola)co, l);
    else op = intersectionpoints((hyperbola)co, l);
  return op;
}

point[] intersectionpoints(conic co, line l)
{
  return intersectionpoints(l, co);
}

/*<asyxml><function type="point[]" signature="intersectionpoints(conic,conic)"><code></asyxml>*/
point[] intersectionpoints(conic co1, conic co2)
{/*<asyxml></code><documentation>Return the intersection points of the two conics.</documentation></function></asyxml>*/
  if(degenerate(co1)) return intersectionpoints(co1.l[0], co2);
  if(degenerate(co2)) return intersectionpoints(co1, co2.l[0]);
  return intersectionpoints(equation(co1), equation(co2));
}

/*<asyxml><function type="point[]" signature="intersectionpoints(triangle,conic,bool)"><code></asyxml>*/
point[] intersectionpoints(triangle t, conic co, bool extended = false)
{/*<asyxml></code><documentation>Return the intersection points.
   If 'extended' is true, the sides are lines else the sides are segments.
   intersectionpoints(conic, triangle, bool) is also defined.</documentation></function></asyxml>*/
  if(degenerate(co)) return intersectionpoints(t, co.l[0], extended);
  point[] OP;
  void addpoint(point P[])
  {
    for (int i = 0; i < P.length; ++i) {
      if(defined(P[i])) {
        bool exist = false;
        for (int j = 0; j < OP.length; ++j) {
          if(P[i] == OP[j]) {exist = true; break;}
        }
        if(!exist) OP.push(P[i]);
      }}}
  if(extended) {
    for (int i = 1; i <= 3; ++i) {
      addpoint(intersectionpoints(t.line(i), co));
    }
  } else {
    for (int i = 1; i <= 3; ++i) {
      addpoint(intersectionpoints((segment)t.line(i), co));
    }
  }
  return OP;
}

point[] intersectionpoints(conic co, triangle t, bool extended = false)
{
  return intersectionpoints(t, co, extended);
}

/*<asyxml><function type="point[]" signature="intersectionpoints(ellipse,ellipse)"><code></asyxml>*/
point[] intersectionpoints(ellipse a, ellipse b)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  // if(degenerate(a)) return intersectionpoints(a.l, b);
  // if(degenerate(b)) return intersectionpoints(a, b.l);;
  return intersectionpoints((conic)a, (conic)b);
}
/*<asyxml><function type="point[]" signature="intersectionpoints(ellipse,circle)"><code></asyxml>*/
point[] intersectionpoints(ellipse a, circle b)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  // if(degenerate(a)) return intersectionpoints(a.l, b);
  // if(degenerate(b)) return intersectionpoints(a, b.l);;
  return intersectionpoints((conic)a, (conic)b);
}
/*<asyxml><function type="point[]" signature="intersectionpoints(circle,ellipse)"><code></asyxml>*/
point[] intersectionpoints(circle a, ellipse b)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  return intersectionpoints(b, a);
}
/*<asyxml><function type="point[]" signature="intersectionpoints(ellipse,parabola)"><code></asyxml>*/
point[] intersectionpoints(ellipse a, parabola b)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  // if(degenerate(a)) return intersectionpoints(a.l, b);
  return intersectionpoints((conic)a, (conic)b);
}
/*<asyxml><function type="point[]" signature="intersectionpoints(parabola,ellipse)"><code></asyxml>*/
point[] intersectionpoints(parabola a, ellipse b)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  return intersectionpoints(b, a);
}
/*<asyxml><function type="point[]" signature="intersectionpoints(ellipse,hyperbola)"><code></asyxml>*/
point[] intersectionpoints(ellipse a, hyperbola b)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  // if(degenerate(a)) return intersectionpoints(a.l, b);
  return intersectionpoints((conic)a, (conic)b);
}
/*<asyxml><function type="point[]" signature="intersectionpoints(hyperbola,ellipse)"><code></asyxml>*/
point[] intersectionpoints(hyperbola a, ellipse b)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  return intersectionpoints(b, a);
}

/*<asyxml><function type="point[]" signature="intersectionpoints(circle,parabola)"><code></asyxml>*/
point[] intersectionpoints(circle a, parabola b)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  return intersectionpoints((conic)a, (conic)b);
}
/*<asyxml><function type="point[]" signature="intersectionpoints(parabola,circle)"><code></asyxml>*/
point[] intersectionpoints(parabola a, circle b)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  return intersectionpoints((conic)a, (conic)b);
}
/*<asyxml><function type="point[]" signature="intersectionpoints(circle,hyperbola)"><code></asyxml>*/
point[] intersectionpoints(circle a, hyperbola b)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  return intersectionpoints((conic)a, (conic)b);
}
/*<asyxml><function type="point[]" signature="intersectionpoints(hyperbola,circle)"><code></asyxml>*/
point[] intersectionpoints(hyperbola a, circle b)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  return intersectionpoints((conic)a, (conic)b);
}

/*<asyxml><function type="point[]" signature="intersectionpoints(parabola,parabola)"><code></asyxml>*/
point[] intersectionpoints(parabola a, parabola b)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  return intersectionpoints((conic)a, (conic)b);
}
/*<asyxml><function type="point[]" signature="intersectionpoints(parabola,hyperbola)"><code></asyxml>*/
point[] intersectionpoints(parabola a, hyperbola b)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  return intersectionpoints((conic)a, (conic)b);
}
/*<asyxml><function type="point[]" signature="intersectionpoints(hyperbola,parabola)"><code></asyxml>*/
point[] intersectionpoints(hyperbola a, parabola b)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  return intersectionpoints((conic)a, (conic)b);
}
/*<asyxml><function type="point[]" signature="intersectionpoints(hyperbola,hyperbola)"><code></asyxml>*/
point[] intersectionpoints(hyperbola a, hyperbola b)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  return intersectionpoints((conic)a, (conic)b);
}

/*<asyxml><function type="point[]" signature="intersectionpoints(circle,circle)"><code></asyxml>*/
point[] intersectionpoints(circle c1, circle c2)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  if(degenerate(c1))
    return degenerate(c2) ?
      new point[]{intersectionpoint(c1.l, c2.l)} : intersectionpoints(c1.l, c2);
  if(degenerate(c2)) return intersectionpoints(c1, c2.l);
  return (c1.C == c2.C) ?
    new point[] :
    intersectionpoints(radicalline(c1, c2), c1);
}

/*<asyxml><function type="line" signature="tangent(circle,abscissa)"><code></asyxml>*/
line tangent(circle c, abscissa x)
{/*<asyxml></code><documentation>Return the tangent of 'c' at 'point(c, x)'.</documentation></function></asyxml>*/
  if(c.r == 0) abort("tangent: a circle with a radius equals zero has no tangent.");
  point M = point(c, x);
  return line(rotate(90, M) * c.C, M);
}

/*<asyxml><function type="line[]" signature="tangents(circle,point)"><code></asyxml>*/
line[] tangents(circle c, point M)
{/*<asyxml></code><documentation>Return the tangents of 'c' passing through 'M'.</documentation></function></asyxml>*/
  line[] ol;
  if(inside(c, M)) return ol;
  if(M @ c) {
    ol.push(tangent(c, relabscissa(c, M)));
  } else {
    circle cc = circle(c.C, M);
    point[] inter = intersectionpoints(c, cc);
    for (int i = 0; i < inter.length; ++i)
      ol.push(tangents(c, inter[i])[0]);
  }
  return ol;
}

/*<asyxml><function type="point" signature="point(circle,point)"><code></asyxml>*/
point point(circle c, point M)
{/*<asyxml></code><documentation>Return the intersection point of 'c'
   with the half-line '[c.C M)'.</documentation></function></asyxml>*/
  return intersectionpoints(c, line(c.C, false, M))[0];
}

/*<asyxml><function type="line" signature="tangent(circle,point)"><code></asyxml>*/
line tangent(circle c, point M)
{/*<asyxml></code><documentation>Return the tangent of 'c' at the
   intersection point of the half-line'[c.C M)'.</documentation></function></asyxml>*/
  return tangents(c, point(c, M))[0];
}

/*<asyxml><function type="point" signature="point(circle,explicit vector)"><code></asyxml>*/
point point(circle c, explicit vector v)
{/*<asyxml></code><documentation>Return the intersection point of 'c'
   with the half-line '[c.C v)'.</documentation></function></asyxml>*/
  return point(c, c.C + v);
}

/*<asyxml><function type="line" signature="tangent(circle,explicit vector)"><code></asyxml>*/
line tangent(circle c, explicit vector v)
{/*<asyxml></code><documentation>Return the tangent of 'c' at the
   point M so that vec(c.C M) is collinear to 'v' with the same sense.</documentation></function></asyxml>*/
  line ol = tangent(c, c.C + v);
  return dot(ol.v, v) > 0 ? ol : reverse(ol);
}

/*<asyxml><function type="line" signature="tangent(ellipse,abscissa)"><code></asyxml>*/
line tangent(ellipse el, abscissa x)
{/*<asyxml></code><documentation>Return the tangent of 'el' at 'point(el, x)'.</documentation></function></asyxml>*/
  point M = point(el, x);
  line l1 = line(el.F1, M);
  line l2 = line(el.F2, M);
  line ol = (l1 == l2) ? perpendicular(M, l1) : bisector(l1, l2, 90, false);
  return ol;
}

/*<asyxml><function type="line[]" signature="tangents(ellipse,point)"><code></asyxml>*/
line[] tangents(ellipse el, point M)
{/*<asyxml></code><documentation>Return the tangents of 'el' passing through 'M'.</documentation></function></asyxml>*/
  line[] ol;
  if(inside(el, M)) return ol;
  if(M @ el) {
    ol.push(tangent(el, relabscissa(el, M)));
  } else {
    point Mp = samecoordsys(M, el.F2) ?
      M : changecoordsys(el.F2.coordsys, M);
    circle c = circle(Mp, abs(el.F1 - Mp));
    circle cc = circle(el.F2, 2 * el.a);
    point[] inter = intersectionpoints(c, cc);
    for (int i = 0; i < inter.length; ++i) {
      line tl = line(inter[i], el.F2, false);
      point[] P = intersectionpoints(tl, el);
      ol.push(line(Mp, P[0]));
    }
  }
  return ol;
}

/*<asyxml><function type="line" signature="tangent(parabola,abscissa)"><code></asyxml>*/
line tangent(parabola p, abscissa x)
{/*<asyxml></code><documentation>Return the tangent of 'p' at 'point(p, x)' (use the Wells method).</documentation></function></asyxml>*/
  line lt = rotate(90, p.V) * line(p.V, p.F);
  point P = point(p, x);
  if(P == p.V) return lt;
  point M = midpoint(segment(P, p.F));
  line l = rotate(90, M) * line(P, p.F);
  return line(P, projection(lt) * M);
}

/*<asyxml><function type="line[]" signature="tangents(parabola,point)"><code></asyxml>*/
line[] tangents(parabola p, point M)
{/*<asyxml></code><documentation>Return the tangent of 'p' at 'M' (use the Wells method).</documentation></function></asyxml>*/
  line[] ol;
  if(inside(p, M)) return ol;
  if(M @ p) {
    ol.push(tangent(p, angabscissa(p, M)));
  }
  else {
    point Mt = changecoordsys(coordsys(p), M);
    circle c = circle(Mt, p.F);
    line l = rotate(90, p.V) * line(p.V, p.F);
    point[] R = intersectionpoints(l, c);
    for (int i = 0; i < R.length; ++i) {
      ol.push(line(Mt, R[i]));
    }
    // An other method: http://www.du.edu/~jcalvert/math/parabola.htm
    //   point[] R = intersectionpoints(p.directrix, c);
    //   for (int i = 0; i < R.length; ++i) {
    //     ol.push(bisector(segment(p.F, R[i])));
    //   }
  }
  return ol;
}

/*<asyxml><function type="line" signature="tangent(hyperbola,abscissa)"><code></asyxml>*/
line tangent(hyperbola h, abscissa x)
{/*<asyxml></code><documentation>Return the tangent of 'h' at 'point(p, x)'.</documentation></function></asyxml>*/
  point M = point(h, x);
  line ol = bisector(line(M, h.F1), line(M, h.F2));
  if(sameside(h.F1, h.F2, ol) || ol == line(h.F1, h.F2)) ol = rotate(90, M) * ol;
  return ol;
}

/*<asyxml><function type="line[]" signature="tangents(hyperbola,point)"><code></asyxml>*/
line[] tangents(hyperbola h, point M)
{/*<asyxml></code><documentation>Return the tangent of 'h' at 'M'.</documentation></function></asyxml>*/
  line[] ol;
  if(M @ h) {
    ol.push(tangent(h, angabscissa(h, M, fromCenter)));
  } else {
    coordsys cano = canonicalcartesiansystem(h);
    bqe bqe = changecoordsys(cano, equation(h));
    real a = abs(1/(bqe.a[5] * bqe.a[0])), b = abs(1/(bqe.a[5] * bqe.a[2]));
    point Mp = changecoordsys(cano, M);
    real x0 = Mp.x, y0 = Mp.y;
    if(abs(x0) > epsgeo) {
      real c0 = a * y0^2/(b * x0)^2 - 1/b,
        c1 = 2 * a * y0/(b * x0^2), c2 = a/x0^2 - 1;
      real[] sol = quadraticroots(c0, c1, c2);
      for (real y:sol) {
        point tmp = changecoordsys(coordsys(h), point(cano, (a * (1 + y * y0/b)/x0, y)));
        ol.push(line(M, tmp));
      }
    } else if(abs(y0) > epsgeo) {
      real y = -b/y0, x = sqrt(a * (1 + b/y0^2));
      ol.push(line(M, changecoordsys(coordsys(h), point(cano, (x, y)))));
      ol.push(line(M, changecoordsys(coordsys(h), point(cano, (-x, y)))));
    }}
  return ol;
}

/*<asyxml><function type="point[]" signature="intersectionpoints(conic,arc)"><code></asyxml>*/
point[] intersectionpoints(conic co, arc a)
{/*<asyxml></code><documentation>intersectionpoints(arc, circle) is also defined.</documentation></function></asyxml>*/
  point[] op;
  point[] tp = intersectionpoints(co, (conic)a.el);
  for (int i = 0; i < tp.length; ++i)
    if(tp[i] @ a) op.push(tp[i]);
  return op;
}

point[] intersectionpoints(arc a, conic co)
{
  return intersectionpoints(co, a);
}

/*<asyxml><function type="point[]" signature="intersectionpoints(arc,arc)"><code></asyxml>*/
point[] intersectionpoints(arc a1, arc a2)
{/*<asyxml></code><documentation></documentation></function></asyxml>*/
  point[] op;
  point[] tp = intersectionpoints(a1.el, a2.el);
  for (int i = 0; i < tp.length; ++i)
    if(tp[i] @ a1 && tp[i] @ a2) op.push(tp[i]);
  return op;
}


/*<asyxml><function type="point[]" signature="intersectionpoints(line,arc)"><code></asyxml>*/
point[] intersectionpoints(line l, arc a)
{/*<asyxml></code><documentation>intersectionpoints(arc, line) is also defined.</documentation></function></asyxml>*/
  point[] op;
  point[] tp = intersectionpoints(a.el, l);
  for (int i = 0; i < tp.length; ++i)
    if(tp[i] @ a && tp[i] @ l) op.push(tp[i]);
  return op;
}

point[] intersectionpoints(arc a, line l)
{
  return intersectionpoints(l, a);
}

/*<asyxml><function type="point" signature="arcsubtendedcenter(point,point,real)"><code></asyxml>*/
point arcsubtendedcenter(point A, point B, real angle)
{/*<asyxml></code><documentation>Return the center of the arc retuned
   by the 'arcsubtended' routine.</documentation></function></asyxml>*/
  point OM;
  point[] P = standardizecoordsys(A, B);
  angle = angle%(sgnd(angle) * 180);
  line bis = bisector(P[0], P[1]);
  line AB = line(P[0], P[1]);
  return intersectionpoint(bis, rotate(90 - angle, A) * AB);
}

/*<asyxml><function type="arc" signature="arcsubtended(point,point,real)"><code></asyxml>*/
arc arcsubtended(point A, point B, real angle)
{/*<asyxml></code><documentation>Return the arc circle from which the segment AB is saw with
   the angle 'angle'.
   If the point 'M' is on this arc, the oriented angle (MA, MB) is
   equal to 'angle'.</documentation></function></asyxml>*/
  point[] P = standardizecoordsys(A, B);
  line AB = line(P[0], P[1]);
  angle = angle%(sgnd(angle) * 180);
  point C = arcsubtendedcenter(P[0], P[1], angle);
  real BC = degrees(B - C)%360;
  real AC = degrees(A - C)%360;
  return arc(circle(C, abs(B - C)), BC, AC, angle > 0 ? CCW : CW);
}

/*<asyxml><function type="arc" signature="arccircle(point,point,point)"><code></asyxml>*/
arc arccircle(point A, point M, point B)
{/*<asyxml></code><documentation>Return the CCW arc circle 'AB' passing through 'M'.</documentation></function></asyxml>*/
  circle tc = circle(A, M, B);
  real a = degrees(A - tc.C);
  real b = degrees(B - tc.C);
  real m = degrees(M - tc.C);

  arc oa = arc(tc, a, b);
  // TODO : use cross product to determine CWW or CW
  if (!(M @ oa)) {
    oa.direction = !oa.direction;
  }

  return oa;
}

/*<asyxml><function type="arc" signature="arc(ellipse,abscissa,abscissa,bool)"><code></asyxml>*/
arc arc(ellipse el, explicit abscissa x1, explicit abscissa x2, bool direction = CCW)
{/*<asyxml></code><documentation>Return the arc from 'point(c, x1)' to 'point(c, x2)' in the direction 'direction'.</documentation></function></asyxml>*/
  real a = degrees(point(el, x1) - el.C);
  real b = degrees(point(el, x2) - el.C);
  arc oa = arc(el, a - el.angle, b - el.angle, fromCenter, direction);
  return oa;
}

/*<asyxml><function type="arc" signature="arc(ellipse,point,point,bool)"><code></asyxml>*/
arc arc(ellipse el, point M, point N, bool direction = CCW)
{/*<asyxml></code><documentation>Return the arc from 'M' to 'N' in the direction 'direction'.
   The points 'M' and 'N' must belong to the ellipse 'el'.</documentation></function></asyxml>*/
  return arc(el, relabscissa(el, M), relabscissa(el, N), direction);
}

/*<asyxml><function type="arc" signature="arccircle(point,point,real,bool)"><code></asyxml>*/
arc arccircle(point A, point B, real angle, bool direction = CCW)
{/*<asyxml></code><documentation>Return the arc circle centered on A
   from B to rotate(angle, A) * B in the direction 'direction'.</documentation></function></asyxml>*/
  point M = rotate(angle, A) * B;
  return arc(circle(A, abs(A - B)), B, M, direction);
}

/*<asyxml><function type="arc" signature="arc(explicit arc,abscissa,abscissa)"><code></asyxml>*/
arc arc(explicit arc a, abscissa x1, abscissa x2)
{/*<asyxml></code><documentation>Return the arc from 'point(a, x1)' to 'point(a, x2)' traversed in the direction of the arc direction.</documentation></function></asyxml>*/
  real a1 = angabscissa(a.el, point(a, x1), a.polarconicroutine).x;
  real a2 = angabscissa(a.el, point(a, x2), a.polarconicroutine).x;
  return arc(a.el, a1, a2, a.polarconicroutine, a.direction);
}

/*<asyxml><function type="arc" signature="arc(explicit arc,point,point)"><code></asyxml>*/
arc arc(explicit arc a, point M, point N)
{/*<asyxml></code><documentation>Return the arc from 'M' to 'N'.
   The points 'M' and 'N' must belong to the arc 'a'.</documentation></function></asyxml>*/
  return arc(a, relabscissa(a, M), relabscissa(a, N));
}

/*<asyxml><function type="arc" signature="inverse(real,point,segment)"><code></asyxml>*/
arc inverse(real k, point A, segment s)
{/*<asyxml></code><documentation>Return the inverse arc circle of 's'
   with respect to point A and inversion radius 'k'.</documentation></function></asyxml>*/
  point Ap = inverse(k, A, s.A), Bp = inverse(k, A, s.B),
    M = inverse(k, A, midpoint(s));
  return arccircle(Ap, M, Bp);
}

/*<asyxml><operator type = "arc" signature="*(inversion,segment)"><code></asyxml>*/
arc operator *(inversion i, segment s)
{/*<asyxml></code><documentation>Provide
   inversion * segment.</documentation></operator></asyxml>*/
  return inverse(i.k, i.C, s);
}

/*<asyxml><operator type = "path" signature="*(inversion,triangle)"><code></asyxml>*/
path operator *(inversion i, triangle t)
{/*<asyxml></code><documentation>Provide inversion * triangle.</documentation></operator></asyxml>*/
  return (path)(i * segment(t.AB))--
    (path)(i * segment(t.BC))--
    (path)(i * segment(t.CA))&cycle;
}

/*<asyxml><function type="path" signature="compassmark(pair,pair,real,real)"><code></asyxml>*/
path compassmark(pair O, pair A, real position, real angle = 10)
{/*<asyxml></code><documentation>Return an arc centered on O with the angle 'angle' so that the position
   of 'A' on this arc makes an angle 'position * angle'.</documentation></function></asyxml>*/
  real a = degrees(A - O);
  real pa = (a - position * angle)%360,
    pb = (a - (position - 1) * angle)%360;
  real t1 = intersect(unitcircle, (0, 0)--2 * dir(pa))[0];
  real t2 = intersect(unitcircle, (0, 0)--2 * dir(pb))[0];
  int n = length(unitcircle);
  if(t1 >= t2) t1 -= n;
  return shift(O) * scale(abs(O - A)) * subpath(unitcircle, t1, t2);
}

/*<asyxml><function type="line" signature="tangent(explicit arc,abscissa)"><code></asyxml>*/
line tangent(explicit arc a, abscissa x)
{/*<asyxml></code><documentation>Return the tangent of 'a' at 'point(a, x)'.</documentation></function></asyxml>*/
  abscissa ag = angabscissa(a, point(a, x));
  return tangent(a.el, ag + a.angle1 + (a.el.e == 0 ? a.angle0 : 0));
}

/*<asyxml><function type="line" signature="tangent(explicit arc,point)"><code></asyxml>*/
line tangent(explicit arc a, point M)
{/*<asyxml></code><documentation>Return the tangent of 'a' at 'M'.
   The points 'M' must belong to the arc 'a'.</documentation></function></asyxml>*/
  return tangent(a, angabscissa(a, M));
}

// *=======================================================*
// *.......Routines for compatibility with original geometry module........*

path square(pair z1, pair z2)
{
  pair v = z2 - z1;
  pair z3 = z2 + I * v;
  pair z4 = z3 - v;
  return z1--z2--z3--z4--cycle;
}

// Draw a perpendicular symbol at z aligned in the direction align
// relative to the path z--z + dir.
void perpendicular(picture pic = currentpicture, pair z, pair align,
                   pair dir = E, real size = 0, pen p = currentpen,
                   margin margin = NoMargin, filltype filltype = NoFill)
{
  perpendicularmark(pic, (point) z, align, dir, size, p, margin, filltype);
}


// Draw a perpendicular symbol at z aligned in the direction align
// relative to the path z--z + dir(g, 0)
void perpendicular(picture pic = currentpicture, pair z, pair align, path g,
                   real size = 0, pen p = currentpen, margin margin = NoMargin,
                   filltype filltype = NoFill)
{
  perpendicularmark(pic, (point) z, align, dir(g, 0), size, p, margin, filltype);
}

// Return an interior arc BAC of triangle ABC, given a radius r > 0.
// If r < 0, return the corresponding exterior arc of radius |r|.
path arc(explicit pair B, explicit pair A, explicit pair C, real r)
{
  return arc(A, r, degrees(B - A), degrees(C - A));
}

// *.......End of compatibility routines........*
// *=======================================================*

// *........................FOOTER.........................*
// *=======================================================*