summaryrefslogtreecommitdiff
path: root/Master/texmf-dist/tex/generic/pgfplots/pgfplots.scaling.code.tex
blob: 283650ac54dd8f0811f099ff8837341d473eff7c (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
%--------------------------------------------
%
% Package pgfplots
%
% Provides a user-friendly interface to create function plots (normal
% plots, semi-logplots and double-logplots).
% 
% It is based on Till Tantau's PGF package.
%
% Copyright 2007-2012 by Christian Feuersänger.
%
% This program is free software: you can redistribute it and/or modify
% it under the terms of the GNU 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 General Public License for more details.
% 
% You should have received a copy of the GNU General Public License
% along with this program.  If not, see <http://www.gnu.org/licenses/>.
%
%--------------------------------------------

% PRECONDITION: 
% 	- final axis limits are given in transformed range
% 	-  \pgfplots@set@default@size@options has been invoked before
% POSTCONDITION: 
% 	- the current x,y and z unit vectors are defined properly;
% 	- the fast-access registers are initialised for the axis limits,
%	- the following macros are assigned:
%		\pgfplots@[xyz]@veclength
%		\pgfplots@[xyz]@inverseveclength
%		\pgfplotspointxaxis
%		\pgfplotspointyaxis
%		\pgfplotspointzaxis
%		\pgfplotspointcenter
%		\pgfplotspointminminmin
%
\def\pgfplots@initsizes{%
	% INIT.
	%
	%
	\pgfplots@xmin@reg=\pgfplots@xmin pt %
	\pgfplots@xmax@reg=\pgfplots@xmax pt %
	\pgfplots@ymin@reg=\pgfplots@ymin pt %
	\pgfplots@ymax@reg=\pgfplots@ymax pt %
	\ifpgfplots@threedim
		\pgfplots@zmin@reg=\pgfplots@zmin pt %
		\pgfplots@zmax@reg=\pgfplots@zmax pt %
	\fi
	%
	%-----------------------------------------
	% PROCESS THE 'width' and 'height' options
	%-----------------------------------------
	%
	%
	\pgfkeysgetvalue{/pgfplots/view/az}{\pgfplots@view@az}%
	\pgfkeysgetvalue{/pgfplots/view/el}{\pgfplots@view@el}%
	\ifpgfplots@threedim
	\else
		\let\pgfplots@view@el=\pgfutil@empty
		\let\pgfplots@view@az=\pgfutil@empty
	\fi
	\ifx\pgfplots@view@az\pgfutil@empty
		% Note that in presence of "x,y,z" options, the
		% \pgfplots@set@default@size@options RESETS /pgfplots/view/az.
		%
		%\let\pgfplots@rectangle@width=\pgfutil@empty
		%\let\pgfplots@rectangle@height=\pgfutil@empty
		\pgfplotsmathvectorfromstring{0,0,1}{default}%
		\let\pgfplots@view@dir@threedim=\pgfplotsretval
		%
		\ifx\pgfplots@x\pgfutil@empty
			\ifx\pgfplots@width\pgfutil@empty
				\pgfplots@error{INTERNAL LOGIC ERROR! WIDTH NOT SET}%
			\fi
		\fi
		%
		\ifx\pgfplots@y\pgfutil@empty
			\ifx\pgfplots@height\pgfutil@empty
				\pgfplots@error{INTERNAL LOGIC ERROR! HEIGHT NOT SET}%
			\fi
		\fi
		\ifpgfplots@threedim
			\def\pgfplots@tmp@Zscale{1}%
		\else
			\def\pgfplots@tmp@Zscale{0}%
		\fi
		%
		\pgfplots@initsizes@setunitvector{x}{0}{1}{\pgfplots@tmp@xisaxisparallel}%
		\pgfplots@initsizes@setunitvector{y}{1}{1}{\pgfplots@tmp@yisaxisparallel}%
		\pgfplots@initsizes@setunitvector{z}{2}{\pgfplots@tmp@Zscale}{\pgfplots@loc@TMPc}%
		%
		\pgfplots@scaling@changewidthheight@for@enlargelimits@by@dimension%
		%
		\pgfplots@scale@plotbox@to@widthheight
	\else
		% 3D case by `view':
		\let\pgfplots@x=\pgfutil@empty
		\let\pgfplots@y=\pgfutil@empty
		\let\pgfplots@z=\pgfutil@empty
		\pgfplotssetaxesfromazel{\pgfplots@view@az}{\pgfplots@view@el}{\pgfplots@tmp@xisaxisparallel}%
		%
		\pgfplots@scaling@changewidthheight@for@enlargelimits@by@dimension%
		%
		\pgfplots@scale@plotbox@to@widthheight
		\if1\pgfplots@tmp@xisaxisparallel%
			\def\pgfplots@tmp@yisaxisparallel{1}%
		\fi
	\fi
	\pgfplots@computeunitvectorlengths
	%
	\pgfplots@scaling@apply@enlargelimits@by@dimension{x}%
	\pgfplots@scaling@apply@enlargelimits@by@dimension{y}%
	\ifpgfplots@threedim
		\pgfplots@scaling@apply@enlargelimits@by@dimension{z}%
	\fi
	%
	\ifpgfplots@threedim
		\pgfplotsgetnormalforcurrentview
	\fi
%\message{Pgfplots debug: initialised unit vectors to x=(\the\pgf@xx,\the\pgf@xy), y=(\the\pgf@yx,\the\pgf@yy), z=(\the\pgf@zx,\the\pgf@zy), n = (\pgfplots@view@dir@threedim). Limits are x=\pgfplots@xmin:\pgfplots@xmax, y=\pgfplots@ymin:\pgfplots@ymax^^J }%
	%
	\let\pgfplotsmathfloatviewdepthxyz@=\pgfplotsmathfloatviewdepthxyz@infigure
	\let\pgfplotsmathviewdepthxyz@=\pgfplotsmathviewdepthxyz@infigure
	%
	\pgfplotsmath@ifzero{\pgfplots@x@veclength}{\pgfplots@hide@xtrue\pgfplots@shownothingof@xtrue}{}%
	\pgfplotsmath@ifzero{\pgfplots@y@veclength}{\pgfplots@hide@ytrue\pgfplots@shownothingof@ytrue}{}%
	\ifpgfplots@threedim
		\pgfplotsmath@ifzero{\pgfplots@z@veclength}{\pgfplots@hide@ztrue\pgfplots@shownothingof@ztrue}{}%
	\else
		\if1\pgfplots@tmp@xisaxisparallel%
			\if1\pgfplots@tmp@yisaxisparallel%
				% Optimize for axis-parallel case!
				% puh. Did not make any measureable difference!? Ok...
				\let\pgfplotsqpointxy=\pgfplotsqpointxy@orthogonal
			\fi
		\fi
	\fi
}

% Defines \pgfmathresult to be the desired width without axis labels.
%
\def\pgfplots@initsizes@get@width@withoutlabels{%
	\pgfplots@initsizes@handle@label@const{\pgfplots@width}{45pt}{width}%
}

% Defines \pgfmathresult to be the desired height without axis labels.
\def\pgfplots@initsizes@get@height@withoutlabels{%
	\pgfplots@initsizes@handle@label@const{\pgfplots@height}{45pt}{height}%
}

\def\pgfplots@initsizes@handle@label@const#1#2#3{%
	\begingroup
	\pgf@xa=#1\relax
	% EXPECTED WIDTH = X = \pgfplots@width
	% ACTUAL WIDTH = c + x * (xmax-xmin)
	% where c is a CONSTANT (for the axis labels/tick labels).
	% -> \pgfplots@tmpXscale = (X - c) / (x *(xmax-xmin))
	%
	% \pgf@xa := X-c:
	\ifpgfplots@scale@only@axis
	\else
		\advance\pgf@xa by-#2 % FIXME determine 'c' correctly!
	\fi
	\ifdim\pgf@xa<0pt
		\pgfplots@error{Error: Plot #3 `#1' is too small. This cannot be implemented while maintaining constant size for labels. Sorry, label sizes are only approximate. You will need to adjust your #3.}%
		\pgf@xa=0pt
	\fi
	\edef\pgfmathresult{\the\pgf@xa}%
	\pgfmath@smuggleone\pgfmathresult
	\endgroup
}%

% #1: axis
\def\pgfplots@axis@apply@post@scale#1{%
	%
	\pgfkeysgetvalue{/pgfplots/#1 post scale}\pgfplots@loc@TMPd
	\ifx\pgfplots@loc@TMPd\pgfutil@empty
	\else
		\pgfmathparse{\csname pgfplots@target@unit@scale@#1x\endcsname*\pgfplots@loc@TMPd}%
		\expandafter\let\csname pgfplots@target@unit@scale@#1x\endcsname=\pgfmathresult
		%
		\pgfmathparse{\csname pgfplots@target@unit@scale@#1y\endcsname*\pgfplots@loc@TMPd}%
		\expandafter\let\csname pgfplots@target@unit@scale@#1y\endcsname=\pgfmathresult
	\fi
}

% Takes azimuth (horizontal angle) '#1' and elongation (vertical
% angle) '#2' (both in degrees) and computes 
% x,y and z vectors which define the view in the direction
% defined by '#1' and '#2'.
%
% 'azimuth' means a rotation around the viewport's x axis. 'elongation' means
% a rotation around the original coordinate system's z axis.
%
% The method works by computing 
% Az = [ cos(azimuth) -sin(azimuth) 0; ...
%     sin(azimuth) cos(azimuth) 0; ...
%     0 0 1 ];
% 
%
% Ax = [ 1 0 0; ...
%     0 cos(elevation) -sin(elevation) ;...
%     0 sin(elevation) cos(elevation) ];
%
% v= Ax * Az;
% = [ ...
% 	cosaz  -sinaz cosel  sinaz sinel; ...
% 	sinaz  cosaz cosel   -sinel cosaz; ...
% 	0	  sinel         cosel ];
%
% Then, we use the rotated XZ plane as viewport, that means 
%   xvec = v * [1 0 0]' = <first column of v>
%   zvec = v * [0 0 1]' = <third column of v>
% and we define the projection onto the twodimensional surface
% spanned by 'xvec' and 'zvec' as
%   P( q ) = [ q^T xvec,  q^T zvec ]'
% for q in R^3.
% As a consequence, we compute the three unit vectors as
%  x = P( [1 0 0] )
%  	 = [ cosaz,  sinaz sinel ]'
%  y = P( [0 1 0] )
%    = [ sinaz,  -sinel cosaz ]'
%  z = P( [0 0 1] )
%    = [ 0,  cosel]'
%
% Furthermore, the 3D view vector which points into the direction of the view
% is
%   n = v * [0 1 0 ]' = <second column of v> =  [-sinaz cosel,  cosaz cosel,  sinel]' 
% because the normal view point was the XZ plane with y as its normal
% vector.
% The 3D vector n is returned by this routine as well - it is
% necessary for some kind of z buffering (determining what is
% foreground and what is background).
%
% INPUT:
% - #1 : azimuth ("yaw")
% - #2 : elevation ("pitch")
% OUTPUT:
% - #3 : a macro which will be set to '1' if and only if 
%      the viewport is the standard XY axis (i.e. azimuth=0, elevation=90).
% - [xyz] vectors,
%   \pgfplots@view@dir@threedim will contain the three components
%   of 'n' (without the suffix 'pt', but in units of 'pt') (see
%   \pgfplotsmathvectorfromstring).
\def\pgfplotssetaxesfromazel#1#2#3{%
	\begingroup
	\pgfmathparse{#1}%
	\let\pgfplots@az=\pgfmathresult
	\pgfmathparse{#2}%
	\edef\pgfplots@el{-\pgfmathresult}%
	\pgfmathsin@{\pgfplots@az}%
	\let\sinaz=\pgfmathresult
	\pgfmathcos@{\pgfplots@az}%
	\let\cosaz=\pgfmathresult
	\pgfmathsin@{\pgfplots@el}%
	\let\sinel=\pgfmathresult
	\pgfmathcos@{\pgfplots@el}%
	\let\cosel=\pgfmathresult
	% x:
	\pgfmathmultiply@{\sinaz}{\sinel}%
	\xdef\pgfplots@glob@TMPa{\noexpand\pgfqpoint{\cosaz pt}{\pgfmathresult pt}}%
	% y:
	\pgfmathmultiply@{-\sinel}{\cosaz}%
	\xdef\pgfplots@glob@TMPb{\noexpand\pgfqpoint{\sinaz pt}{\pgfmathresult pt}}%
	% z:
	\xdef\pgfplots@glob@TMPc{\noexpand\pgfqpoint{0pt}{\cosel pt}}%
	%
	\pgfkeysgetvalue{/pgfplots/x dir/value}\pgfplots@loc@dirvalue@x
	\pgfkeysgetvalue{/pgfplots/y dir/value}\pgfplots@loc@dirvalue@y
	\pgfkeysgetvalue{/pgfplots/z dir/value}\pgfplots@loc@dirvalue@z
	\if r\pgfplots@loc@dirvalue@x
		\t@pgfplots@toka=\expandafter{\pgfplots@glob@TMPa}%
		\xdef\pgfplots@glob@TMPa{\noexpand\pgfqpointscale{-1}{\the\t@pgfplots@toka}}%
	\fi
	\if r\pgfplots@loc@dirvalue@y
		\t@pgfplots@toka=\expandafter{\pgfplots@glob@TMPb}%
		\xdef\pgfplots@glob@TMPb{\noexpand\pgfqpointscale{-1}{\the\t@pgfplots@toka}}%
	\fi
	\if r\pgfplots@loc@dirvalue@z
		\t@pgfplots@toka=\expandafter{\pgfplots@glob@TMPc}%
		\xdef\pgfplots@glob@TMPc{\noexpand\pgfqpointscale{-1}{\the\t@pgfplots@toka}}%
	\fi
	%
	% Process 'plot box ratio':
	\def\pgfplots@extract@plot@box@ratio##1##2##3##4\pgfplots@EOI{%
		\pgfmathparse{##1}\let\pgfplots@plotboxratio@x=\pgfmathresult
		\pgfmathparse{##2}\let\pgfplots@plotboxratio@y=\pgfmathresult
		\pgfmathparse{##3}\let\pgfplots@plotboxratio@z=\pgfmathresult
	}%
	\def\pgfplots@extract@plot@box@ratio@spaces##1 ##2 ##3 ##4\pgfplots@EOI{%
		\pgfplots@extract@plot@box@ratio{##1}{##2}{##3}{##4}\pgfplots@EOI
	}%
	\pgfkeysgetvalue{/pgfplots/plot box ratio}\pgfplots@loc@TMPa
	% Auto-determine input format which is either '{x}{y}{z}' or 'x y z'
	\def\pgfplots@loc@TMPb{%
		\pgfutil@ifnextchar\bgroup{%
			\pgfplots@loc@tmptrue
			\pgfplots@gobble@until@EOI
		}{%
			\pgfplots@loc@tmpfalse
			\pgfplots@gobble@until@EOI
		}%
	}%
	\expandafter\pgfplots@loc@TMPb\pgfplots@loc@TMPa\pgfplots@EOI
	\ifpgfplots@loc@tmp
		% Ah- braces format.
		\edef\pgfplots@loc@TMPa{\pgfplots@loc@TMPa{1}{1}{1}}%
		\expandafter\pgfplots@extract@plot@box@ratio\pgfplots@loc@TMPa\pgfplots@EOI
	\else
		% Ah- space-separated
		\edef\pgfplots@loc@TMPa{\pgfplots@loc@TMPa\space 1 1 1}%
		\expandafter\pgfplots@extract@plot@box@ratio@spaces\pgfplots@loc@TMPa\pgfplots@EOI
	\fi
	%
	% process it:
	\ifdim\pgfplots@plotboxratio@x pt=1pt
	\else
		\t@pgfplots@toka=\expandafter{\pgfplots@glob@TMPa}%
		\xdef\pgfplots@glob@TMPa{\noexpand\pgfqpointscale{\pgfplots@plotboxratio@x}{\the\t@pgfplots@toka}}%
	\fi
	\ifdim\pgfplots@plotboxratio@y pt=1pt
	\else
		\t@pgfplots@toka=\expandafter{\pgfplots@glob@TMPb}%
		\xdef\pgfplots@glob@TMPb{\noexpand\pgfqpointscale{\pgfplots@plotboxratio@y}{\the\t@pgfplots@toka}}%
	\fi
	\ifdim\pgfplots@plotboxratio@z pt=1pt
	\else
		\t@pgfplots@toka=\expandafter{\pgfplots@glob@TMPc}%
		\xdef\pgfplots@glob@TMPc{\noexpand\pgfqpointscale{\pgfplots@plotboxratio@z}{\the\t@pgfplots@toka}}%
	\fi
	%
	% n (3D!)
	\pgfmathmultiply@{-\sinaz}{\cosel}%
	\let\pgfmathresultNx=\pgfmathresult
	\pgfmathmultiply@{\cosaz}{\cosel}%
	\xdef\pgfplots@glob@TMPd{{\pgfmathresultNx}{\pgfmathresult}{\sinel}}%
	\endgroup
%\message{Setting x,y and z from {#1}{#2} to^^J x = \meaning\pgfplots@glob@TMPa,^^J y = \meaning\pgfplots@glob@TMPb,^^J z = \meaning\pgfplots@glob@TMPc,^^J n = \pgfplots@glob@TMPd.^^J}%
	\pgfsetxvec{\pgfplots@glob@TMPa}%
	\pgfsetyvec{\pgfplots@glob@TMPb}%
	\pgfsetzvec{\pgfplots@glob@TMPc}%
	\def\pgfplots@loc@TMPa##1##2##3{%
		\pgfplotsmathvectorfromstring{##1,##2,##3}{default}%
		\let\pgfplots@view@dir@threedim=\pgfplotsretval
	}%
	\expandafter\pgfplots@loc@TMPa\pgfplots@glob@TMPd\relax
	\def#3{0}%
}%

% Takes the current plot box, defined by the actual PGF x,y and z unit
% vectors, and re-scales it such that it fits into the 
% width and height of the axis (as they have been provided by the
% user).
%
% @see \pgfplots@scale@axisbox@to@widthheight
% @see\pgfplots@scaleaxes@to@BB
\def\pgfplots@scale@plotbox@to@widthheight{%
	\pgfplots@initsizes@get@width@withoutlabels
	\let\pgfplots@loc@TMPa=\pgfmathresult
	\pgfplots@initsizes@get@height@withoutlabels
	%
	\edef\pgfplots@loc@TMPa{{\pgfplots@loc@TMPa}{\pgfmathresult}}%
	\expandafter\pgfplots@scaleaxes@to@BB\pgfplots@loc@TMPa
}


% Takes the current PGF x,y and z unit vectors and scales them such
% that the bounding box of the final image has width #1 and height #2.
%
% The relative length of the input vectors is important for the 3D case: it
% will be scaled as-is. 
%
% PRECONDITION
% 	- the x, y and z unit vectors have been set to the proper
% 	DIRECTIONS. Their relative vector lengths are set-up properly
% 	(i.e. y is twice as large as x and half as large as z or so).
%	- the \ifpgfplots@threedim boolean is set.
%	- the data limits have been initialised and transformed according
%	to the data transformation.
%	- the data transformation has ONLY been applied to the axis limits
%	(not other axis inputs). It may be changed by this method.
% 
% POSTCONDITION
% 	- the unit vectors have been re-scaled such that the final plot
% 	has the desired dimensions.
% 	- the @veclength and @inverseveclength have been initialized
\def\pgfplots@scaleaxes@to@BB#1#2{%
	\if0\pgfplots@scale@mode@choice
		% scale mode=auto
		\def\pgfplots@scale@mode@choice{2}% stretch to fill
	\fi
	\pgfplots@scaleaxes@to@BB@{#1}{#2}%
	% NOTE: we have not yet computed the lengths of unit vectors. In
	% addition, we have not yet updated the normal vector.
}%

% DEPRECATED:
\def\pgfplots@rescale@view@dir{%
	\expandafter\ifx\csname pgfplots@view@dir@threedim\endcsname\relax
	\else
		% At this point, we ALREADY HAVE a normal vector. However, it
		% might be skewed due to the scaling.
		%
		% -> recompute normal vector. In earlier versions, I tried to
		% rescale it - but that was too complicated (for me). This
		% here produces correct results, and it is a correct approach
		% anyway.
		\pgfplotsgetnormalforcurrentview
	\fi
}%

% \pgfplots@BB@for@plotbox@get@unit@scales@for@limits{#1}{#2}{#3}:
% a helper tool which computes individual unit vector scales in order
% to respect the limits.
%
% This method ignores width/height; its purpose is only to make sure
% that [xmin,xmax] fits into the CURRENT plot box. 
%
% In this context, each unit vector is supposed to be scaled such that
% width/height fit if xmin=0 and xmax=1.
%
% #1 [output] a macro name which will contain the INVERSE scale for x
% #2 [output] a macro name which will contain the INVERSE scale for y
% #3 [output] a macro name which will contain the INVERSE scale for z
% 
\def\pgfplots@BB@for@plotbox@get@unit@scales@for@limits#1#2#3{%
	\if1\b@pgfplots@plotbox@xisunit
		% Consequently, we have to multiply  with 1/(max-min):
		% compute 1/(xmax - xmin) in float for more recent versions (see /pgfplots/compat/scaling).
		% I observed that it is much more accurate 
		\pgfmathsubtract@{\pgfplots@xmax}{\pgfplots@xmin}%
	\else
		\def\pgfmathresult{1}%
	\fi
	\let#1=\pgfmathresult
	%
	\if1\b@pgfplots@plotbox@yisunit
		\pgfmathsubtract@{\pgfplots@ymax}{\pgfplots@ymin}%
	\else
		\def\pgfmathresult{1}%
	\fi
	\let#2=\pgfmathresult
	%
	\ifpgfplots@threedim
		\if1\b@pgfplots@plotbox@zisunit
			\pgfmathsubtract@{\pgfplots@zmax}{\pgfplots@zmin}%
		\else
			\def\pgfmathresult{1}%
		\fi
	\else
		\def\pgfmathresult{1}%
	\fi
	\let#3=\pgfmathresult
}%
\def\pgfplots@BB@for@plotbox{%
	\ifpgfplots@threedim
		\pgfpathmoveto{\pgfqpointxyz\pgfplots@plotbox@xmin\pgfplots@plotbox@ymin\pgfplots@plotbox@zmin}%
		\pgfpathmoveto{\pgfqpointxyz\pgfplots@plotbox@xmin\pgfplots@plotbox@ymin\pgfplots@plotbox@zmax}%
		\pgfpathmoveto{\pgfqpointxyz\pgfplots@plotbox@xmin\pgfplots@plotbox@ymax\pgfplots@plotbox@zmin}%
		\pgfpathmoveto{\pgfqpointxyz\pgfplots@plotbox@xmin\pgfplots@plotbox@ymax\pgfplots@plotbox@zmax}%
		\pgfpathmoveto{\pgfqpointxyz\pgfplots@plotbox@xmax\pgfplots@plotbox@ymin\pgfplots@plotbox@zmin}%
		\pgfpathmoveto{\pgfqpointxyz\pgfplots@plotbox@xmax\pgfplots@plotbox@ymin\pgfplots@plotbox@zmax}%
		\pgfpathmoveto{\pgfqpointxyz\pgfplots@plotbox@xmax\pgfplots@plotbox@ymax\pgfplots@plotbox@zmin}%
		\pgfpathmoveto{\pgfqpointxyz\pgfplots@plotbox@xmax\pgfplots@plotbox@ymax\pgfplots@plotbox@zmax}%
	\else
		\pgfpathmoveto{\pgfqpointxy\pgfplots@plotbox@xmin\pgfplots@plotbox@ymin}%
		\pgfpathmoveto{\pgfqpointxy\pgfplots@plotbox@xmin\pgfplots@plotbox@ymax}%
		\pgfpathmoveto{\pgfqpointxy\pgfplots@plotbox@xmax\pgfplots@plotbox@ymin}%
		\pgfpathmoveto{\pgfqpointxy\pgfplots@plotbox@xmax\pgfplots@plotbox@ymax}%
	\fi
}%



% Returns width and height of the current plot box
% (the path produced by \pgfplots@BB@for@plotbox).
%
% PRECONDITION: \pgfplots@BB@for@plotbox is defined to produce a path
% for the plot box
%
% POSTCONDITION: \pgfplotsretval contains the with and
% \pgfplotsretvalb contains the height
\def\pgfplots@get@dimension@of@BB{%
	\begingroup
	\pgfinterruptboundingbox
	%
	% the result of this call will be used to scale to target
	% dimensions. If we omit \pgftransformreset here, we might
	% accidentally UNDO the PGF transformation matrix (compare by 
	% writing \tikzpicture[scale=0.5] before the axis).
	\pgftransformreset
	%
	% STEP 1: compute the bounding box for the plot box.
	\pgfplots@BB@for@plotbox
	%
	% TMPa = width
	\pgf@xa=\pgf@pathmaxx
	\advance\pgf@xa by-\pgf@pathminx
	% TMPb = height
	\pgf@xb=\pgf@pathmaxy
	\advance\pgf@xb by-\pgf@pathminy
	\xdef\pgfplots@glob@TMPa{%
		\def\noexpand\pgfplotsretval{\the\pgf@xa}%
		\def\noexpand\pgfplotsretvalb{\the\pgf@xb}%
	}%
	\pgfusepath{discard}%
	\endpgfinterruptboundingbox
	\endgroup
	\pgfplots@glob@TMPa
}%

\def\pgfplots@scaleaxes@to@BB@prepare@plotbox@limits{%
	\def\b@pgfplots@rescale@x{1}%
	\def\b@pgfplots@rescale@y{1}%
	\def\b@pgfplots@rescale@z{1}%
	%
	\pgfplots@scaleaxes@to@BB@prepare@plotbox@limits@ x%
	\pgfplots@scaleaxes@to@BB@prepare@plotbox@limits@ y%
	\pgfplots@scaleaxes@to@BB@prepare@plotbox@limits@ z%
}%

\def\pgfplots@scaleaxes@to@BB@prepare@plotbox@limits@#1{%
	\expandafter\ifx\csname pgfplots@#1\endcsname\pgfutil@empty
		% Ah - we have no unit vector in this direction. 
		\expandafter\def\csname pgfplots@plotbox@#1min\endcsname{0}%
		\expandafter\def\csname pgfplots@plotbox@#1max\endcsname{1}%
		\expandafter\def\csname b@pgfplots@plotbox@#1isunit\endcsname{1}%
	\else
		% we have a unit vector... prepare for limits. Note that the
		% unit vector has been prepared for data scaling already.
		\pgfutil@namelet{pgfplots@plotbox@#1min}{pgfplots@#1min}%
		\pgfutil@namelet{pgfplots@plotbox@#1max}{pgfplots@#1max}%
		\expandafter\def\csname b@pgfplots@plotbox@#1isunit\endcsname{0}%
		\if2\pgfplots@scale@mode@choice
			% scale mode=stretch to fill
			% do NOT rescale unit vectors for stretch to fill :
			\expandafter\def\csname b@pgfplots@rescale@#1\endcsname{0}%
		\fi
	\fi
	\expandafter\def\csname b@pgfplots@unitvec@is@zero@#1\endcsname{0}%
	\ifdim\csname pgf@#1x\endcsname=0pt %
		\ifdim\csname pgf@#1y\endcsname=0pt %
			\expandafter\def\csname b@pgfplots@unitvec@is@zero@#1\endcsname{1}%
		\fi
	\fi
}


\def\pgfplots@scaleaxes@to@BB@#1#2{%
	\begingroup
%\message{SCALING: scale mode choice = \pgfplots@scale@mode@choice^^J}%
	%
	\pgfplots@scaleaxes@to@BB@prepare@plotbox@limits
	\def\pgfplots@target@limitrescale@x{1}%
	\def\pgfplots@target@limitrescale@y{1}%
	\def\pgfplots@target@limitrescale@z{1}%
	\if1\pgfplots@scale@mode@choice
		% scale mode=none
		\def\xscale{1}%
		\def\yscale{1}%
		\def\pgfplots@target@unit@scale@inv@x{1}%
		\def\pgfplots@target@unit@scale@inv@y{1}%
		\def\pgfplots@target@unit@scale@inv@z{1}%
	\else
		%
		% This here CAN cause anisotropic (different) scaling factors.
		\pgfplots@BB@for@plotbox@get@unit@scales@for@limits
			{\pgfplots@target@unit@scale@inv@x}
			{\pgfplots@target@unit@scale@inv@y}
			{\pgfplots@target@unit@scale@inv@z}%
		%
%\message{got scales to fit limits into BB: x=1/\pgfplots@target@unit@scale@inv@x, y=1/\pgfplots@target@unit@scale@inv@y, z=1/\pgfplots@target@unit@scale@inv@z^^J}%
		%
		\if3\pgfplots@scale@mode@choice
			% scale mode=scale uniformly
			% 
			% We need to recompensate in case the previous method chose
			% different unit scaling scalings:
			\pgfplots@BB@for@plotbox@get@unit@scales@compensated@axis@limits
				{\pgfplots@target@unit@scale@inv@x}
				{\pgfplots@target@unit@scale@inv@y}
				{\pgfplots@target@unit@scale@inv@z}
				{\pgfplots@target@limitrescale@x}{\pgfplots@target@limitrescale@y}{\pgfplots@target@limitrescale@z}%
			%
			%\pgfplots@BB@update@cumulative@limit@compensations
		\fi
%\message{adjusted scales for 'scale mode': x=1/\pgfplots@target@unit@scale@inv@x, y=1/\pgfplots@target@unit@scale@inv@y, z=1/\pgfplots@target@unit@scale@inv@z;  ^^J   axis limit componsation scales x=\pgfplots@target@limitrescale@x, y=\pgfplots@target@limitrescale@y, z=\pgfplots@target@limitrescale@z^^J}%
		%
		% ATTENTION: this MODIFIES \pgfplots@target@limitrescale@x and its
		% variants directly - and it needs the input values.
		\pgfplots@get@scale@horiz@and@vert
			{#1}%
			{#2}%
			{\xscale}%
			{\yscale}% yscale
			{\pgfplots@target@limitrescale@x}%
			{\pgfplots@target@limitrescale@y}%
			{\pgfplots@target@limitrescale@z}%
%\message{Got W/H scale for all x components: \xscale; for all y components: \xscale; ^^J   axis limit componsation scales x=1/\pgfplots@target@limitrescale@x, y=1/\pgfplots@target@limitrescale@y, z=1/\pgfplots@target@limitrescale@z^^J}%
		% Ok, we know the W,H scalings now.
		%
		%
		%
		\pgfplots@apply@unit@ratio
			{\pgfplots@target@unit@scale@inv@x}
			{\pgfplots@target@unit@scale@inv@y}
			{\pgfplots@target@unit@scale@inv@z}
			{\pgfplots@target@limitrescale@x@}{\pgfplots@target@limitrescale@y@}{\pgfplots@target@limitrescale@z@}%
		\pgfplots@BB@update@cumulative@limit@compensations
		%
%\message{adjusted scales for 'unit vector ratio': x=1/\pgfplots@target@unit@scale@inv@x, y=1/\pgfplots@target@unit@scale@inv@y, z=1/\pgfplots@target@unit@scale@inv@z;  ^^J   axis limit componsation scales x=\pgfplots@target@limitrescale@x, y=\pgfplots@target@limitrescale@y, z=\pgfplots@target@limitrescale@z^^J}%
	\fi
	%
	%
	\pgfplots@scaling@minimize@limitrescale%
	%
	%
	\pgfplots@scaling@adjust@datascaling%
%\message{adjusted scales for data scale trafo: x=1/\pgfplots@target@unit@scale@inv@x, y=1/\pgfplots@target@unit@scale@inv@y, z=1/\pgfplots@target@unit@scale@inv@z;  ^^J   axis limit componsation scales x=\pgfplots@target@limitrescale@x, y=\pgfplots@target@limitrescale@y, z=\pgfplots@target@limitrescale@z;^^J  data scale trafo exponents x=\pgfplots@target@datascaletrafo@x@exponent@old -> \pgfplots@target@datascaletrafo@x@exponent, y=\pgfplots@target@datascaletrafo@y@exponent@old -> \pgfplots@target@datascaletrafo@y@exponent, z=\pgfplots@target@datascaletrafo@z@exponent@old -> \pgfplots@target@datascaletrafo@z@exponent^^J}%
	%
	\pgfplots@scaling@compute@final@scales%
			{\xscale}{\yscale}%
			{\pgfplots@target@unit@scale@inv@x}%
			{\pgfplots@target@unit@scale@inv@y}%
			{\pgfplots@target@unit@scale@inv@z}%
	%
	\pgfplots@axis@apply@post@scale{x}%
	\pgfplots@axis@apply@post@scale{y}%
	\ifpgfplots@threedim
		\pgfplots@axis@apply@post@scale{z}%
	\fi
	%
	% and finally, resize limits appropriately and add all cumulative limit compensations:
	\xdef\pgfplots@glob@TMPa{%
		%
		\pgf@xx=\pgfplots@target@unit@scale@xx\pgf@xx
		\pgf@xy=\pgfplots@target@unit@scale@xy\pgf@xy
		%
		\pgf@yx=\pgfplots@target@unit@scale@yx\pgf@yx
		\pgf@yy=\pgfplots@target@unit@scale@yy\pgf@yy
		%
		\ifpgfplots@threedim
			\pgf@zx=\pgfplots@target@unit@scale@zx\pgf@zx
			\pgf@zy=\pgfplots@target@unit@scale@zy\pgf@zy
		\fi
		%
		\noexpand\pgfplots@apply@datascaletrafo@change@{x}{\pgfplots@target@datascaletrafo@x@exponent}%
		\noexpand\pgfplots@apply@datascaletrafo@change@{y}{\pgfplots@target@datascaletrafo@y@exponent}%
		\noexpand\pgfplots@apply@datascaletrafo@change@{z}{\pgfplots@target@datascaletrafo@z@exponent}%
		%
		\noexpand\pgfplots@apply@unit@vector@rescale@keep@size{x}{\pgfplots@target@limitrescale@x}%
		\noexpand\pgfplots@apply@unit@vector@rescale@keep@size{y}{\pgfplots@target@limitrescale@y}%
		\noexpand\pgfplots@apply@unit@vector@rescale@keep@size{z}{\pgfplots@target@limitrescale@z}%
		%
		\noexpand\pgfplots@notify@final@scalings{%
			x unit scale=\pgfplots@target@unit@scale@x,%
			y unit scale=\pgfplots@target@unit@scale@y,%
			z unit scale=\pgfplots@target@unit@scale@z,%
			x datatrafo exponent=\pgfplots@target@datascaletrafo@x@exponent,%
			y datatrafo exponent=\pgfplots@target@datascaletrafo@y@exponent,%
			z datatrafo exponent=\pgfplots@target@datascaletrafo@z@exponent,%
			x limit rescale=\pgfplots@target@limitrescale@x,%
			y limit rescale=\pgfplots@target@limitrescale@y,%
			z limit rescale=\pgfplots@target@limitrescale@z,%
		}%
	}%
	\endgroup
	\pgfplots@glob@TMPa
}%

% Checks for the case the ALL (visible) limit compensation scales are
% bigger than one (for example x = 1.22, y = 2). In such a case, we
% want to MINIMIZE the rescaling. This can happen if unit vector ratio
% is active.
%
% In our example, we want to use limit rescaling factors x = 1, y = 2/1.22
% and, consequently, unit rescaling factors x *= 1.22, y *= 1.22 .
%
% This method checks for the case and applies the rescaling if
% necessary.
%
\def\pgfplots@scaling@minimize@limitrescale{%
	% boolean allLimitScalesAreBiggerThanOne;
	\pgfplots@loc@tmptrue
	\if0\b@pgfplots@unitvec@is@zero@x
		\ifdim\pgfplots@target@limitrescale@x pt<1.002pt %
			\pgfplots@loc@tmpfalse
		\fi
	\fi
	\if0\b@pgfplots@unitvec@is@zero@y
		\ifdim\pgfplots@target@limitrescale@y pt<1.002pt %
			\pgfplots@loc@tmpfalse
		\fi
	\fi
	\if0\b@pgfplots@unitvec@is@zero@z
		\ifdim\pgfplots@target@limitrescale@z pt<1.002pt %
			\pgfplots@loc@tmpfalse
		\fi
	\fi
	%
	\ifpgfplots@loc@tmp
		\begingroup
		% Ah -- all non-vanishing limit rescaling factors are BIGGER
		% THAN ONE.
		% In this case, we can save some rescalings!
		%
		% Search for the smallest rescaling factor.
		\let\pgfplots@smallest=\pgf@x
		\pgfplots@smallest=16000pt %
		\def\pgfplots@smallest@arg{}%
		\if0\b@pgfplots@unitvec@is@zero@x
			\pgf@xa=\pgfplots@target@limitrescale@x pt %
			\ifdim\pgf@xa<\pgfplots@smallest%
				\pgfplots@smallest=\pgf@xa
				\def\pgfplots@smallest@arg{x}%
			\fi
		\fi
		\if0\b@pgfplots@unitvec@is@zero@y
			\pgf@xa=\pgfplots@target@limitrescale@y pt %
			\ifdim\pgf@xa<\pgfplots@smallest%
				\pgfplots@smallest=\pgf@xa
				\def\pgfplots@smallest@arg{y}%
			\fi
		\fi
		\if0\b@pgfplots@unitvec@is@zero@z
			\pgf@xa=\pgfplots@target@limitrescale@z pt %
			\ifdim\pgf@xa<\pgfplots@smallest%
				\pgfplots@smallest=\pgf@xa
				\def\pgfplots@smallest@arg{z}%
			\fi
		\fi
		%
		% OK. We have the smallest scaling factor. It is > 1.
		\pgfplotscoordmath{default}{parsenumber}{\pgfplots@target@limitrescale@x}%
		\let\pgfplots@target@limitrescale@x=\pgfmathresult
		\pgfplotscoordmath{default}{parsenumber}{\pgfplots@target@limitrescale@y}%
		\let\pgfplots@target@limitrescale@y=\pgfmathresult
		%
		%
		\pgfplotscoordmath{default}{parsenumber}{\pgfplots@target@unit@scale@inv@x}%
		\let\pgfplots@target@unit@scale@inv@x=\pgfmathresult
		\pgfplotscoordmath{default}{parsenumber}{\pgfplots@target@unit@scale@inv@y}%
		\let\pgfplots@target@unit@scale@inv@y=\pgfmathresult
		%
		\if0\b@pgfplots@unitvec@is@zero@z
			\pgfplotscoordmath{default}{parsenumber}{\pgfplots@target@limitrescale@z}%
			\let\pgfplots@target@limitrescale@z=\pgfmathresult
			\pgfplotscoordmath{default}{parsenumber}{\pgfplots@target@unit@scale@inv@z}%
			\let\pgfplots@target@unit@scale@inv@z=\pgfmathresult
		\fi
		%
		\pgfplotscoordmath{default}{op}{reciprocal}{{\csname pgfplots@target@limitrescale@\pgfplots@smallest@arg\endcsname}}%
		\let\scale=\pgfmathresult
		%
		\pgfplotsforeachentryinCSV\value{%
			\pgfplots@target@unit@scale@inv@x,%
			\pgfplots@target@unit@scale@inv@y,%
			\pgfplots@target@limitrescale@x,%
			\pgfplots@target@limitrescale@y%
		}{%
			\pgfplotscoordmath{default}{op}{multiply}{{\scale}{\value}}%
			\pgfplotscoordmath{default}{tofixed}{\pgfmathresult}%
			\expandafter\let\value=\pgfmathresult
		}%
		\if0\b@pgfplots@unitvec@is@zero@z
			\pgfplotsforeachentryinCSV\value{%
				\pgfplots@target@unit@scale@inv@z,%
				\pgfplots@target@limitrescale@z%
			}{%
				\pgfplotscoordmath{default}{op}{multiply}{{\scale}{\value}}%
				\pgfplotscoordmath{default}{tofixed}{\pgfmathresult}%
				\expandafter\let\value=\pgfmathresult
			}%
		\fi
		%
		\xdef\pgfplots@glob@TMPa{%
			\noexpand\def\noexpand\pgfplots@target@unit@scale@inv@x{\pgfplots@target@unit@scale@inv@x}%
			\noexpand\def\noexpand\pgfplots@target@unit@scale@inv@y{\pgfplots@target@unit@scale@inv@y}%
			\noexpand\def\noexpand\pgfplots@target@unit@scale@inv@z{\pgfplots@target@unit@scale@inv@z}%
			\noexpand\def\noexpand\pgfplots@target@limitrescale@x{\pgfplots@target@limitrescale@x}%
			\noexpand\def\noexpand\pgfplots@target@limitrescale@y{\pgfplots@target@limitrescale@y}%
			\noexpand\def\noexpand\pgfplots@target@limitrescale@z{\pgfplots@target@limitrescale@z}%
		}%
		\endgroup
		\pgfplots@glob@TMPa
%
%\message{adjusted scales by minimizing common scaling factors: x=1/\pgfplots@target@unit@scale@inv@x, y=1/\pgfplots@target@unit@scale@inv@y, z=1/\pgfplots@target@unit@scale@inv@z;  ^^J   axis limit componsation scales x=\pgfplots@target@limitrescale@x, y=\pgfplots@target@limitrescale@y, z=\pgfplots@target@limitrescale@z;^^J}%
	\fi
}%

% Defines 
% \pgfplots@target@unit@scale@xx
% \pgfplots@target@unit@scale@xy
% \pgfplots@target@unit@scale@yx
% \pgfplots@target@unit@scale@yy
% \pgfplots@target@unit@scale@zx
% \pgfplots@target@unit@scale@zy
% %
% \pgfplots@target@unit@scale@x
% \pgfplots@target@unit@scale@y
% \pgfplots@target@unit@scale@z
% by combining the input args.
%
% #1: the scale to be applied to ALL x components
% #2: the scale to be applied to ALL y components
% #3: the scale to be applied to x unit
% #4: the scale to be applied to y unit
% #5: the scale to be applied to z unit
\def\pgfplots@scaling@compute@final@scales#1#2#3#4#5{%
	% ##1: the axis (x,y,or z)
	% ##2: the horizontal scale
	% ##3: the vertical scale
	% ##4: the inverse unit scale for this axis
	\def\pgfplots@loc@TMPa##1##2##3##4{%
		\pgfplotscoordmath{\pgfplots@compat@scaling@coordmath}{parsenumber}{##2}%
		\let\xscale@@=\pgfmathresult
		\pgfplotscoordmath{\pgfplots@compat@scaling@coordmath}{parsenumber}{##3}%
		\let\yscale@@=\pgfmathresult
		\pgfplotscoordmath{\pgfplots@compat@scaling@coordmath}{parsenumber}{##4}%
		\let\unitscale@inv@@=\pgfmathresult
		%
		% NOTE : it *would* be more efficient to use
		% 1/\unitscale@inv@@ in the routines above. BUT THAT IS NOT BACKWARDS COMPATIBLE.
		% Leave it this way!
		\pgfplotscoordmath{\pgfplots@compat@scaling@coordmath}{op}{reciprocal}{{\unitscale@inv@@}}%
		\let\unitscale@@=\pgfmathresult
		\pgfplotscoordmath{\pgfplots@compat@scaling@coordmath}{tofixed}{\pgfmathresult}%
		\expandafter\let\csname pgfplots@target@unit@scale@##1\endcsname=\pgfmathresult
		%
		%
		\ifx\pgfplots@compat@scaling@coordmath@final\pgfplots@compat@scaling@coordmath
		\else
			% backwards compatibility is such a burden.... :-(
			%
			% earlier versions relied on TeX's dimen arithmetics to
			% multiply the final scales. Make sure we do the same -
			% rounding errors on unit vectors are instable, i.e. the
			% errors add up considerably.
			\pgfplotscoordmath{\pgfplots@compat@scaling@coordmath@final}{parsenumber}{\xscale@@}%
			\let\xscale@@=\pgfmathresult
			\pgfplotscoordmath{\pgfplots@compat@scaling@coordmath@final}{parsenumber}{\yscale@@}%
			\let\yscale@@=\pgfmathresult
			\pgfplotscoordmath{\pgfplots@compat@scaling@coordmath@final}{parsenumber}{\unitscale@@}%
			\let\unitscale@@=\pgfmathresult
			\pgfplotscoordmath{\pgfplots@compat@scaling@coordmath@final}{parsenumber}{\unitscale@inv@@}%
			\let\unitscale@inv@@=\pgfmathresult
		\fi
		%
		\ifpgfplots@threedim
			% backw. compatibility: this is how it used to be in 3d
			% axes:
			\pgfplotscoordmath{\pgfplots@compat@scaling@coordmath@final}{op}{multiply}{{\xscale@@}{\unitscale@@}}%
			\pgfplotscoordmath{\pgfplots@compat@scaling@coordmath@final}{tofixed}{\pgfmathresult}%
			\expandafter\let\csname pgfplots@target@unit@scale@##1x\endcsname=\pgfmathresult
			%
			\pgfplotscoordmath{\pgfplots@compat@scaling@coordmath@final}{op}{multiply}{{\yscale@@}{\unitscale@@}}%
			\pgfplotscoordmath{\pgfplots@compat@scaling@coordmath@final}{tofixed}{\pgfmathresult}%
			\expandafter\let\csname pgfplots@target@unit@scale@##1y\endcsname=\pgfmathresult
		\else
			% backw. compatibility: 2d axes used divide in earlier
			% versions, not reciprocal. Believe it or not; for
			% \pgfplots@compat@scaling@coordmath=pgfbasic, it makes a
			% visible difference of about 2-3pt in the complete figure
			% size.
			\pgfplotscoordmath{\pgfplots@compat@scaling@coordmath@final}{op}{divide}{{\xscale@@}{\unitscale@inv@@}}%
			\pgfplotscoordmath{\pgfplots@compat@scaling@coordmath@final}{tofixed}{\pgfmathresult}%
			\expandafter\let\csname pgfplots@target@unit@scale@##1x\endcsname=\pgfmathresult
			%
			\pgfplotscoordmath{\pgfplots@compat@scaling@coordmath@final}{op}{divide}{{\yscale@@}{\unitscale@inv@@}}%
			\pgfplotscoordmath{\pgfplots@compat@scaling@coordmath@final}{tofixed}{\pgfmathresult}%
			\expandafter\let\csname pgfplots@target@unit@scale@##1y\endcsname=\pgfmathresult
		\fi
		%
	}%
	\if1\b@pgfplots@rescale@x
		\pgfplots@loc@TMPa{x}{\xscale}{\yscale}{#3}%
	\else
		\pgfplots@loc@TMPa{x}{1}{1}{#3}%
	\fi
	%
	\if1\b@pgfplots@rescale@y
		\pgfplots@loc@TMPa{y}{\xscale}{\yscale}{#4}%
	\else
		\pgfplots@loc@TMPa{y}{1}{1}{#4}%
	\fi
	%
	\ifpgfplots@threedim
		\if1\b@pgfplots@rescale@z
			\pgfplots@loc@TMPa{z}{\xscale}{\yscale}{#5}%
		\else
			\pgfplots@loc@TMPa{z}{1}{1}{#5}%
		\fi
	\else
		\def\pgfplots@target@unit@scale@z{0}%
		\def\pgfplots@target@unit@scale@zx{0}%
		\def\pgfplots@target@unit@scale@zy{0}%
		\def\pgfplots@target@unit@scale@inv@z{inf}%
	\fi
	%
}%

\def\pgfplots@notify@final@scalings#1{%
	\pgfkeys{/pgfplots/scaling/.cd,	
		.unknown/.code={%
%\message{setting key '\pgfkeyscurrentkey' to {##1}^^J}
			\pgfkeyssetvalue{\pgfkeyscurrentkey}{##1}%
		},
		#1%
	}%
}%

% #1: either x,y, or z
% #2: the new exponent
\def\pgfplots@apply@datascaletrafo@change@#1#2{%
	\pgfplots@if{pgfplots@apply@datatrafo@#1}{%
		\pgfplotscoordmath{#1}{datascaletrafo get params}%
		\edef\pgfplots@loc@TMPa{\expandafter\pgfutil@firstoftwo\pgfmathresult}%
		\edef\pgfplots@loc@TMPb{#2}%
		\ifx\pgfplots@loc@TMPa\pgfplots@loc@TMPb
			% ok; the data scale trafo did not change at all - we
			% still have the same exponent.
		\else
			% Ah - we have a new data scale trafo!
			\pgfplotscoordmath{#1}{datascaletrafo inverse}{\csname pgfplots@#1min\endcsname}%
			\let\pgfplots@loc@TMPa=\pgfmathresult
			\pgfplotscoordmath{#1}{datascaletrafo inverse}{\csname pgfplots@#1max\endcsname}%
			\let\pgfplots@loc@TMPb=\pgfmathresult
			%
			% first: determine the optimal shift (which is the
			% transformed lower limit):
			\pgfplotscoordmath{#1}{datascaletrafo set params}{#2}{0}%
			\pgfplotscoordmath{#1}{datascaletrafo}{\pgfplots@loc@TMPa}%
			%
			% ok, finalize the data trafo:
			\pgfplotscoordmath{#1}{datascaletrafo set params}{#2}{\pgfmathresult}%
			%
			% ... and recompute axis limits:
			\pgfplotscoordmath{#1}{datascaletrafo}{\pgfplots@loc@TMPa}%
			\expandafter\let\csname pgfplots@#1min\endcsname=\pgfmathresult
			\pgfplotscoordmath{#1}{datascaletrafo}{\pgfplots@loc@TMPb}%
			\expandafter\let\csname pgfplots@#1max\endcsname=\pgfmathresult
		\fi
	}{}%
}%

% Inspects the limit enlargement factors and reinitializes the data
% scale transformations.
%
% The purpose of this method is to avoid "dimension too large" if the
% factors exceed certain limits.
%
% INPUT:
%  \pgfplots@target@limitrescale@x and its variants for y and z
%  \pgfplots@target@unit@scale@inv@x and its variants for y and z
%
% OUTPUT:
%  \pgfplots@target@datascaletrafo@x@exponent and its variants for y and z
%     -> contains NEW datascaletrafo exponents 
%  \pgfplots@target@datascaletrafo@x@exponent@old and its variants for y and z
%     -> contains OLD datascaletrafo exponents
%  \pgfplots@target@unit@scale@inv@x and its variants for y and z
%     -> contains (modified) unit vector scales
\def\pgfplots@scaling@adjust@datascaling{%
	\pgfplots@scaling@adjust@datascaling@for x%
	\pgfplots@scaling@adjust@datascaling@for y%
	\pgfplots@scaling@adjust@datascaling@for z%
}
\def\pgfplots@scaling@adjust@datascaling@for#1{%
	\pgfplots@if{pgfplots@apply@datatrafo@#1}{%
		\pgfplotscoordmath{#1}{datascaletrafo get params}%
		\def\pgfplots@loc@TMPa##1##2{%
			\expandafter\def\csname pgfplots@target@datascaletrafo@#1@exponent\endcsname{##1}%
			\expandafter\def\csname pgfplots@target@datascaletrafo@#1@exponent@old\endcsname{##1}%
		}%
		\expandafter\pgfplots@loc@TMPa\pgfmathresult
		\pgf@xa=\csname pgfplots@target@limitrescale@#1\endcsname pt
		\ifdim\pgf@xa>5pt %
			% We want to enlarge axis limits considerably! 
			%
			\pgfplots@scaling@adjust@datascaling@for@get@compensation{\pgf@xa}%
			%
			% Ok, make sure that we do not get "dimension too large"
			% by adjusting the data scale trafo.
			%
			% Note that the data scale trafo has (only) been applied
			% to axis limits, so we have to reapply it before these
			% changes can take effect:
			\pgf@xa=\csname pgfplots@target@unit@scale@inv@#1\endcsname pt
			\divide\pgf@xa by\pgfplotsretval\relax %
			\expandafter\edef\csname pgfplots@target@unit@scale@inv@#1\endcsname{\pgf@sys@tonumber\pgf@xa}%
			%
			\c@pgf@countd=\csname pgfplots@target@datascaletrafo@#1@exponent\endcsname\relax
			\advance\c@pgf@countd by-\pgfplotsretvalb\relax %
			\expandafter\edef\csname pgfplots@target@datascaletrafo@#1@exponent\endcsname{\the\c@pgf@countd}%
		\fi
	}{%
		\expandafter\def\csname pgfplots@target@datascaletrafo@#1@exponent\endcsname{0}%
		\expandafter\def\csname pgfplots@target@datascaletrafo@#1@exponent@old\endcsname{0}%
	}%
}

% Returns 
%  \pgfplotsretval -> the absolute scaling 
%  \pgfplotsretvalb -> the log10 of the scaling
\def\pgfplots@scaling@adjust@datascaling@for@get@compensation#1{
	\ifdim#1<100pt %
		\def\pgfplotsretval{10}%
		\def\pgfplotsretvalb{1}%
	\else
		\ifdim#1<1000pt %
			\def\pgfplotsretval{100}%
			\def\pgfplotsretvalb{2}%
		\else
			\ifdim#1<10000pt %
				\def\pgfplotsretval{1000}%
				\def\pgfplotsretvalb{3}%
			\else
				% too much for this approach anyway... and probably no
				% use-case at all.
				\def\pgfplotsretval{1000}%
				\def\pgfplotsretvalb{3}%
			\fi
		\fi
	\fi
}%

% Computes the initial scale from a plot box of unit size to the
% desired with and height.
%
% #1 the desired width
% #2 the desired height
% #3 [output] a macro which will contain the horizontal (x) scale
% #4 [output] a macro which will contain the vertical (y) scale
% #5 [input/output] a macro which, on input, contains the x axis limit compensation scale
%    which is required to select a single unit vector scale without
%    reducing the plots dimension (without actually respecting the
%    final dimension). On output, the input has been multiplied by
%    some additional x limit componensation scale (selected by scale
%    uniformly strategy).
% #6 [input/output] a macro which will contain a y axis limit
% compensation scale; it works in the same way as #5
% #7 [input/output] a macro which will contain a z axis limit
% compensation scale; it works in the same way as #5
\def\pgfplots@get@scale@horiz@and@vert#1#2#3#4#5#6#7{%
	\begingroup
	\edef\pgfplots@target@limitrescale@x{#5}%
	\edef\pgfplots@target@limitrescale@y{#6}%
	\edef\pgfplots@target@limitrescale@z{#7}%
	\pgfplots@get@dimension@of@BB
	\pgf@xa=\pgfplotsretval\relax
	\pgf@xb=\pgfplotsretvalb\relax
	\pgf@ya=#1\relax
	\pgf@yb=#2\relax
	\edef\w{\pgf@sys@tonumber\pgf@xa}%
	\edef\h{\pgf@sys@tonumber\pgf@xb}%
	\edef\W{\pgf@sys@tonumber\pgf@ya}%
	\edef\H{\pgf@sys@tonumber\pgf@yb}%
%\message{PGFPLOTS: the current unit vectors result in a UNIT BB of (\the\pgf@xa,\the\pgf@xb). Scaling it to (\the\pgf@ya,\the\pgf@yb)...^^J}%
	\ifcase\pgfplots@scale@mode@choice
		% scale mode=auto  does not happen here
	\or
		% scale mode=none  does not happen here
	\or
		% scale mode=stretch to fill
		% 
		% This is very simple:
		%
		% Compute individual scaling factors for X and Y
		% such that the UNIT-BB will have size #1,#2. Keep limits.
		\pgfmathdivide@{\W}{\w}%
		\let\scalex=\pgfmathresult
		%
		\pgfmathdivide@{\H}{\h}%
		\let\scaley=\pgfmathresult
		% 
		% no changes to the axis limits - we only rescale units.
		\def\pgfplots@target@limitrescale@x@{1}%
		\def\pgfplots@target@limitrescale@y@{1}%
		\def\pgfplots@target@limitrescale@z@{1}%
		\pgfplots@BB@update@cumulative@limit@compensations
	\or
		% scale mode=scale uniformly
		% compute ONE common scale for both, X and Y - and satisfy
		% width/height constraints by adjusting the axis limits.
		%
		% The idea is as follows:
		% we WANT to have width W and height H.
		% The constraint is that each unit vector must get the same
		% scale -- but the axis limits can receive individual
		% compensation scales. But it should "look reasonable well".
		%
		% currently, we have 
		% w = r_x e_xx  + r_y e_yx + rz e_zx  (with e_zx = 0 typically)
		% h = r_x e_xy  + r_y e_yy + rz e_zy
		%
		% where r_x, r_y, r_z are the maximal range of the data in
		% x,y,z respectively. Depending on the context of this method,
		% they are either 1 (relative coords) or
		% (xmax-xmin) (absolute coords).
		%
		% Now, search for a set of real numbers 
		% Rx, Ry, Rz, s
		% such that
		% W = (Rx r_x) (s e_xx)  + (Ry r_y) (s e_yx)  + (Rz r_z) (s e_zx)
		% H = (Rx r_x) (s e_xy)  + (Ry r_y) (s e_yy)  + (Rz r_z) (s e_zy)
		%
		% clearly, the solution is not unique.
		% ONE choice is to employ the fact that e_zx = 0  (or, for 2d
		% plots, e_zx=0, e_zy=0 and e_yx=0):
		%
		% in that case, we can compute s such that the equation for W
		% is satisfied and compensate only the limit r_z, i.e. to
		% choose
		% s := W / w,    (scale to satisfy width constraint)
		% Rx := Ry := 1  (keep limits in X and Y)
		% Rz = ( H - s (w - r_z e_zy) ) / (s r_z e_zy)    (adjust z limit to satisfy height constraint)
		%
		% This approach works well if W < H . If W > H, it will look
		% bad: Rz will be less than 1, causing the limit to become
		% smaller. This, in turn, will clip away parts of the image.
		%
		% 
		%
		% Another solution is to make it the other way: to keep the
		% limit r_z, but to reduce the size and enlarge the other
		% limits to satisfy the size constraints. This solution is
		% considerably more involved; it requires to solve a nonlinear
		% set of equations.
		%
		% Formally, this second solution uses
		% Rz := 1       (no limit componensation scale for z -- keep z limit)
		% R:= Rx := Ry  (same limit componensation scale for both X and Y)
		% R and s still need to be determined from the two equations for W
		% and H.
		%
		% Substituting the given choices into the equations for W and H, we find
		%
		% R = W / (s w)
		%
		% s = H * (R * (h-r_z e_zy) + r_z e_zy)^-1
		%
		% Here, we employed the definition of 'h', see above. The
		% equations are non-linear.
		%
		% ATTENTION: we assume that the datascaletrafo set params
		% method has been called with THE SAME SCALE IN EACH
		% DIRECTION.
		\if0\pgfplots@scaleuniformly@choice
			% scale uniformly strategy=auto
			\pgfplots@get@scale@horiz@and@vert@scaleuniformly@of@optimal@strategy
		\else
			\pgfplots@get@scale@horiz@and@vert@scaleuniformly
			\pgfplots@BB@update@cumulative@limit@compensations
		\fi
	\fi
	%
	\xdef\pgfplots@glob@TMPa{%
		\noexpand\def\noexpand#3{\scalex}%
		\noexpand\def\noexpand#4{\scaley}%
		\noexpand\def\noexpand#5{\pgfplots@target@limitrescale@x}%
		\noexpand\def\noexpand#6{\pgfplots@target@limitrescale@y}%
		\noexpand\def\noexpand#7{\pgfplots@target@limitrescale@z}%
	}%
	\endgroup
	\pgfplots@glob@TMPa
}%

% This is the implementation for 'scale uniformly strategy=auto'.
%
% It works by finding the strategy which involves the minimal scaling
% overhead.
%
% To this end, it computes the result for each 'scale uniformly
% strategy', and computes a cost function. The one with optimal cost
% function wins, and its results are returned.
%
% The cost function is the overal scaling which is applied to AXIS
% LIMITS. It works as follows:
% 1. if a choice requires to REDUCE the axis limits in order to
% fulfill all constraints, it is neglected (using maximal cost 16000).
% Reducing axis limits may clip away information.
% 
% 2. if a choice requires to ENLARGE some axis limits, its cost is the
% sum of the individual scaling factors (even if they are are one -
% who cares).
%
% Note that this method *is* relevant and the optimization appears to
% be necessary. 
% Examples are
%   unittest_scalemode_2d_standard_1.tex
% and perhaps
%   unittest_scalemode_2d_standard_0.tex 
% and more involved 3d examples are also available.
%
% My first guess was that it is sufficient to decide the optimal
% strategy in advance by comparing the target width and the target
% height - but that proved to be insufficient: it leads to correct
% results, but wastes too much space (i.e. enlarges limits too much).
%
% ATTENTION: the cost function INCLUDES RESULTS OF 
%	\pgfplots@BB@for@plotbox@get@unit@scales@for@limits and its
%	corrector
%	\pgfplots@BB@for@plotbox@get@unit@scales@compensated@axis@limits.
%
% More precisely, it relies on already computes limit compensation
% factors which do not depend on the target width/target height: both
% \pgfplots@BB@for@plotbox@get@unit@scales@compensated@axis@limits and
% this implementation of  'scale uniformly strategy' can be used to compute 
% the cost of a strategy.
% 
\def\pgfplots@get@scale@horiz@and@vert@scaleuniformly@of@optimal@strategy{%
	\begingroup
		\def\mathclass{default}%
		\pgfplotscoordmath{\mathclass}{max limit}%
		\let\pgfplots@cost@for@choice@superhigh=\pgfmathresult%
		%
		% private helpers to compute the cost. 
		\def\pgfplots@scalestrategy@compute@cost{%
			\begingroup
			% ATTENTION: this call changes
			% '\pgfplots@target@limitrescale@x' and its variants.
			% Restore its value after the iteration:
			\pgfplots@BB@update@cumulative@limit@compensations
			\pgfplotscoordmath{\mathclass}{one}%
			\let\ONE=\pgfmathresult
			\pgfplotscoordmath{\mathclass}{parsenumber}{\pgfplots@target@limitrescale@x}%
			\let\X=\pgfmathresult
			\pgfplotscoordmath{\mathclass}{parsenumber}{\pgfplots@target@limitrescale@y}%
			\let\Y=\pgfmathresult
			\ifpgfplots@threedim
				\pgfplotscoordmath{\mathclass}{parsenumber}{\pgfplots@target@limitrescale@z}%
				\let\Z=\pgfmathresult
			\else
				\let\Z=\pgfplots@target@limitrescale@z
			\fi
			%
			% If one of the resulting limit compensation scales is
			% less than 1, we can immediately skip it - we do not want
			% to risk to clip away image content.
			\pgfplotscoordmath{\mathclass}{if less than}{\X}{\ONE}{%
				\let\pgfplots@cost@for@choice=\pgfplots@cost@for@choice@superhigh
			}{%
				\pgfplotscoordmath{\mathclass}{if less than}{\Y}{\ONE}{%
					\let\pgfplots@cost@for@choice=\pgfplots@cost@for@choice@superhigh
				}{%
					\ifpgfplots@threedim
						\pgfplotscoordmath{\mathclass}{if less than}{\Z}{\ONE}{%
							\let\pgfplots@cost@for@choice=\pgfplots@cost@for@choice@superhigh
						}{%
							% ah - 3 limit scales >= 1. Good, assign cost:
							\pgfplots@scalestrategy@compute@cost@
						}%
					\else
						% ah - all limit scales >=1. Good, assign cost:
						\pgfplots@scalestrategy@compute@cost@
					\fi
				}%
			}%
%\message{scale uniformly strategy=auto: '\pgfplots@tostring@scaleuniformlystrategy{\pgfplots@scaleuniformly@choice}' has cost \pgfplots@cost@for@choice\space(limit rescaling factors x=\X, y=\Y, z=\Z)^^J}%
			\xdef\pgfplots@glob@TMPa{%
				\noexpand\def\noexpand\pgfplots@scaleuniformly@choice{\pgfplots@scaleuniformly@choice}%
				\noexpand\def\noexpand\scalex{\scalex}%
				\noexpand\def\noexpand\scaley{\scaley}%
				\noexpand\def\noexpand\pgfplots@target@limitrescale@x{\pgfplots@target@limitrescale@x}%
				\noexpand\def\noexpand\pgfplots@target@limitrescale@y{\pgfplots@target@limitrescale@y}%
				\noexpand\def\noexpand\pgfplots@target@limitrescale@z{\pgfplots@target@limitrescale@z}%
			}%
			\pgfmath@smuggleone\pgfplots@cost@for@choice
			% keep in mind that this scope IS NECESSARY: we have
			% changed the target quantities
			% \pgfplots@target@limitrescale@x and its variants!
			\endgroup
			\let\pgfplots@scalestrategy@values=\pgfplots@glob@TMPa
		}%
		\def\pgfplots@scalestrategy@compute@cost@{%
			\pgfplotscoordmath{\mathclass}{op}{add}{{\X}{\Y}}%
			\ifpgfplots@threedim
				\pgfplotscoordmath{\mathclass}{op}{add}{{\pgfmathresult}{\Z}}%
			\fi
			\let\pgfplots@cost@for@choice=\pgfmathresult
		}%
		%
		% compute initial cost:
		\def\pgfplots@scaleuniformly@choice{3}% change horizontal limits
		\pgfplots@get@scale@horiz@and@vert@scaleuniformly
		\pgfplots@scalestrategy@compute@cost
		%
		% init minimum:
		\let\pgfplots@cost@for@choice@arg=\pgfplots@scalestrategy@values
		\let\pgfplots@cost@for@choice@sofar=\pgfplots@cost@for@choice%
		%
		% compute cost of next strategy:
		\def\pgfplots@scaleuniformly@choice{2}% change vertical limits
		\pgfplots@get@scale@horiz@and@vert@scaleuniformly
		\pgfplots@scalestrategy@compute@cost
		%
		% update minimum:
		\pgfplotscoordmath{\mathclass}{if less than}{\pgfplots@cost@for@choice}{\pgfplots@cost@for@choice@sofar}{%
			\let\pgfplots@cost@for@choice@arg=\pgfplots@scalestrategy@values
			\let\pgfplots@cost@for@choice@sofar=\pgfplots@cost@for@choice%
		}{%
		}%
		%
		\ifx\pgfplots@cost@for@choice@sofar\pgfplots@cost@for@choice@superhigh
			% the algorithm discarded every available strategy.
			\def\pgfplots@scaleuniformly@choice{1}% fall back to 'units only'
			\pgfplots@get@scale@horiz@and@vert@scaleuniformly
			\pgfplots@scalestrategy@compute@cost
			\let\pgfplots@cost@for@choice@arg=\pgfplots@scalestrategy@values
			\let\pgfplots@cost@for@choice@sofar=\pgfplots@cost@for@choice%
		\fi
		%
		%
	\global\let\pgfplots@glob@TMPa=\pgfplots@cost@for@choice@arg
	\endgroup
	\pgfplots@glob@TMPa
%\message{scale uniformly strategy=auto: choosing '\pgfplots@tostring@scaleuniformlystrategy{\pgfplots@scaleuniformly@choice}'^^J}%
}

\def\pgfplots@tostring@scaleuniformlystrategy#1{%
	% scale uniformly strategy:
	\ifcase#1\relax 
		auto
	\or
		units only
	\or
		change vertical limits
	\or
		change horizontal limits
	\fi
}
% Does the work for 'scale mode=scale uniformly' inside of
% \pgfplots@get@scale@horiz@and@vert.
%
% It returns its result into \pgfplots@target@limitrescale@x@ (i.e.
% with an extra '@')
\def\pgfplots@get@scale@horiz@and@vert@scaleuniformly{%
	\ifcase\pgfplots@scaleuniformly@choice\relax
		% scale uniformly strategy=auto does not happen here.
	\or
		% scale uniformly strategy=units only
		\pgfplots@scaleuniformly@onlyunits
	\or
		% scale uniformly strategy=change vertical limits
		%
		% first, scale to the width ...
		\pgfplots@scaleuniformly@onlyunits@{\w}{\W}%
		% ... and change (only) vertical limits to get the "correct"
		% height:
		\ifdim\pgf@zy=0pt
			\ifdim\pgf@yx=0pt
				\pgfplots@prepare@vertical@rescaling@for@scale@uniformly@in@dir{y}\returninto\pgfplots@target@limitrescale@y@
			\else
				\pgfplots@scale@uniformly@fallback
			\fi
		\else
			\ifdim\pgf@zx=0pt
				\pgfplots@prepare@vertical@rescaling@for@scale@uniformly@in@dir{z}\returninto\pgfplots@target@limitrescale@z@
			\else
				\pgfplots@scale@uniformly@fallback
			\fi
		\fi
	\or
		% scale uniformly strategy=change horizontal limits
		\ifdim\pgf@zy=0pt
			\ifdim\pgf@yx=0pt
				\ifdim\pgf@xy=0pt
					% special 2d routine with explicit solution
					\pgfplots@scaleuniformly@change@horizontal@limits@twodim
						{\scalex}
						{\pgfplots@target@limitrescale@x@}
						{\pgfplots@target@limitrescale@y@}
						{\pgfplots@target@limitrescale@z@}%
				\else
					\pgfplots@scale@uniformly@fallback
				\fi
			\else
				\pgfplots@scale@uniformly@fallback
			\fi
		\else
			\ifdim\pgf@zx=0pt
				\pgfplots@scaleuniformly@change@horizontal@limits
					{\scalex}
					{\pgfplots@target@limitrescale@x@}
					{\pgfplots@target@limitrescale@y@}
					{\pgfplots@target@limitrescale@z@}%
			\else
				\pgfplots@scale@uniformly@fallback
			\fi
		\fi
		\let\scaley=\scalex
	\fi
}

\def\pgfplots@scaleuniformly@onlyunits{%
	% scale to the smaller target dimension:
	\ifdim\W pt<\H pt %
		\pgfplots@scaleuniformly@onlyunits@{\w}{\W}%
	\else
		\pgfplots@scaleuniformly@onlyunits@{\h}{\H}%
	\fi
}%

% #1 : the actual dimension
% #2 : the target dimension
\def\pgfplots@scaleuniformly@onlyunits@#1#2{%
	\def\pgfplots@target@limitrescale@x@{1}%
	\def\pgfplots@target@limitrescale@y@{1}%
	\def\pgfplots@target@limitrescale@z@{1}%
	\pgfmathdivide@{#2}{#1}%
	\let\scalex=\pgfmathresult
	\let\scaley=\scalex % we *need* the same unit scale.
}%

% Computes 'scale uniformly strategy=change horizontal limits'.
% This is a complicated solution, see the documentation in the
% implementation for 
% 'scale mode=scale uniformly'
%
% #1 [output] a macro which will contain the (uniform) scale for the
% unit vectors
% #2 [output] a macro which will contain a x axis limit compensation scale
% #3 [output] a macro which will contain a x axis limit compensation scale
% #4 [output] a macro which will contain a x axis limit compensation scale
\def\pgfplots@scaleuniformly@change@horizontal@limits#1#2#3#4{%
	\begingroup
	%
	\pgfplots@BB@for@plotbox@getunitheight{\pgf@xc}{z}%
	%
	% compute the rest in floating point - intermediate results may
	% become too huge for TeX.
	\pgfplotscoordmath{default}{parsenumber}{\expandafter\pgf@sys@tonumber\csname pgf@xc\endcsname}%
	\let\M=\pgfmathresult
	%
	\pgfplotscoordmath{default}{parsenumber}{\w}%
	\let\w=\pgfmathresult
	\pgfplotscoordmath{default}{parsenumber}{\W}%
	\let\W=\pgfmathresult
	\pgfplotscoordmath{default}{parsenumber}{\h}%
	\let\h=\pgfmathresult
	\pgfplotscoordmath{default}{parsenumber}{\H}%
	\let\H=\pgfmathresult
	\pgfplotscoordmath{default}{op}{divide}{{\W}{\w}}%
	\let\Wwinv=\pgfmathresult
	\pgfplotscoordmath{default}{op}{subtract}{{\h}{\M}}%
	\let\hminusM=\pgfmathresult
	%
	\pgfplotscoordmath{default}{one}%
	\let\S=\pgfmathresult%
	\let\R=\pgfmathresult%
	\let\Rx=\pgfmathresult
	\def\Rz{1}%
	%
	\def\pgfplots@hold@S@get@R{%
		\pgfplotscoordmath{default}{op}{divide}{{\Wwinv}{\S}}%
		\let\R=\pgfmathresult
%\message{updated R = \R\space ( S = \S ) ^^J}%
	}%
	\def\pgfplots@hold@R@get@S{%
		\pgfplotscoordmath{default}{op}{multiply}{{\R}{\hminusM}}%
		\pgfplotscoordmath{default}{op}{add}{{\pgfmathresult}{\M}}%
		\pgfplotscoordmath{default}{op}{divide}{{\H}{\pgfmathresult}}%
		\let\S=\pgfmathresult
%\message{updated S = \S\space ( R = \R ) ^^J}%
	}%
	%
	% This is the (most stupid) nonlinear method which is at hand:
	% fix point iteration.
	% choose R arbitrarily (R=1 seems adequate), solve for s. 
	% Then, fix s and solve for R. Then, fix R and
	% solve for s until convergence.
	\c@pgf@countc=0
	\pgfplotsloop{%
		\ifnum\c@pgf@countc<\pgfkeysvalueof{/pgfplots/scale uniformly strategy iter} %
			\pgfplotsloopcontinuetrue
		\else
			\pgfplotsloopcontinuefalse
		\fi
	}{%
		\pgfplots@hold@R@get@S \pgfplots@hold@S@get@R
		\advance\c@pgf@countc by1 %
	}%
	%
	\pgfplotscoordmath{default}{tofixed}{\R}\let\R=\pgfmathresult
	\pgfplotscoordmath{default}{tofixed}{\S}\let\S=\pgfmathresult
	\xdef\pgfplots@glob@TMPa{%
		\noexpand\def\noexpand#1{\S}%
		\noexpand\def\noexpand#2{\R}%
		\noexpand\def\noexpand#3{\R}%
		\noexpand\def\noexpand#4{\Rz}%
	}%
	\endgroup
	%
	\pgfplots@glob@TMPa
}%

% Computes 'scale uniformly strategy=change horizontal limits'.
% 
% This is a simplified closed solution assuming that e_xy=0 and e_yx = 0
%
% #1 [output] a macro which will contain the (uniform) scale for the
% unit vectors
% #2 [output] a macro which will contain a x axis limit compensation scale
% #3 [output] a macro which will contain a x axis limit compensation scale
% #4 [output] a macro which will contain a x axis limit compensation scale
\def\pgfplots@scaleuniformly@change@horizontal@limits@twodim#1#2#3#4{%
	\begingroup
	% Assuming that we have a standard 2d axis, i.e. 
	% e_zx = e_zy = 0,  e_xy = 0, and e_yx =0,
	% we can immediately compute a solution.
	%
	% In this case, we have the actual width
	% w = r_x e_xx  + r_y e_yx + rz e_zx
	%   = r_x e_xx
	% and actual height
	% h = r_x e_xy  + r_y e_yy + rz e_zy
	%   = r_y e_yy
	% and, consequently, desired width
	% W = (Rx r_x) (s e_xx)  + (Ry r_y) (s e_yx)  + (Rz r_z) (s e_zx)
	%   = (Rx r_x) (s e_xx)
	% and desired height
	% H = (Rx r_x) (s e_xy)  + (Ry r_y) (s e_yy)  + (Rz r_z) (s e_zy)
	%   = (Ry r_y) (s e_yy).
	% since this strategy changes horizontal limits (only), we have
	% Ry := 1.
	% We find
	% s : = H/h 
	% and
	% Rx : = W/w /s .
	%
	\pgfplotscoordmath{default}{parsenumber}{\w}%
	\let\w=\pgfmathresult
	\pgfplotscoordmath{default}{parsenumber}{\W}%
	\let\W=\pgfmathresult
	\pgfplotscoordmath{default}{parsenumber}{\h}%
	\let\h=\pgfmathresult
	\pgfplotscoordmath{default}{parsenumber}{\H}%
	\let\H=\pgfmathresult
	\pgfplotscoordmath{default}{op}{divide}{{\H}{\h}}%
	\let\S=\pgfmathresult
	\pgfplotscoordmath{default}{op}{divide}{{\W}{\w}}%
	\pgfplotscoordmath{default}{op}{divide}{{\pgfmathresult}{\S}}%
	\let\Rx=\pgfmathresult
	\def\Ry{1}%
	\def\Rz{1}%
	%
	\pgfplotscoordmath{default}{tofixed}{\Rx}\let\Rx=\pgfmathresult
	\pgfplotscoordmath{default}{tofixed}{\S}\let\S=\pgfmathresult
	\xdef\pgfplots@glob@TMPa{%
		\noexpand\def\noexpand#1{\S}%
		\noexpand\def\noexpand#2{\Rx}%
		\noexpand\def\noexpand#3{\Ry}%
		\noexpand\def\noexpand#4{\Rz}%
	}%
	\endgroup
	%
	\pgfplots@glob@TMPa
}%

\def\pgfplots@BB@update@cumulative@limit@compensations{%
%\message{  -> additional limit componensation scales x=\pgfplots@target@limitrescale@x@, y=\pgfplots@target@limitrescale@y@, z=\pgfplots@target@limitrescale@z@^^J}%
	% add limit compensation to what we have from earlier
	% operations:
	\pgfplotscoordmath{pgfbasic}{op}{multiply}{{\pgfplots@target@limitrescale@x@}{\pgfplots@target@limitrescale@x}}%
	\let\pgfplots@target@limitrescale@x=\pgfmathresult
	\pgfplotscoordmath{pgfbasic}{op}{multiply}{{\pgfplots@target@limitrescale@y@}{\pgfplots@target@limitrescale@y}}%
	\let\pgfplots@target@limitrescale@y=\pgfmathresult
	\pgfplotscoordmath{pgfbasic}{op}{multiply}{{\pgfplots@target@limitrescale@z@}{\pgfplots@target@limitrescale@z}}%
	\let\pgfplots@target@limitrescale@z=\pgfmathresult
}%

\def\pgfplots@scale@uniformly@fallback{%
	\ifpgfplots@scaleuniformly@warning
		\pgfplotswarning{scale uniformly unsupported}\pgfeov%
	\fi
	\pgfplots@scaleuniformly@onlyunits
}%

% This is part of the implementation of 'scale mode=scale uniformly'.
%
% Its purpose it to set up the initial scaling such that 
% 1. each unit vector gets the same scale
% 2. the axis limits are resized (enlarged) to keep the plot box ratio
% (as far as possible)
%
% It repairs the outcome of
% \pgfplots@BB@for@plotbox@get@unit@scales@for@limits .
%
% The assumption is that on input #1, #2, and #3 are the factors which
% would be used by stretch-to-fill in order to squeze the axis limits
% into the plot box defined by e_x, e_y, and e_z (the unit vectors).
%
% On output, #1, #2, and #3 will be modified such that *each has the
% same value*. The value will be chosen with care. More precisely, it
% is the *minimum* of {#1,#2,#3}.
%
% Clearly, 'scale mode=scale uniformly' has less freedom than
% strech-to-fill. In order to keep the plot box ratio intact (as far
% as possible), the axis limits will be rescaled to componsate for the
% ignored scaling factors. More precisely, if direction i is not the
% extremal value (as discussed in the last paragraph), the axis limits
% will be rescaled by #i/extremum .
%
%
%
% #1: on input, it contains the x unit scale which would be taken without the
% compensation. On output, it contains the x unit scale which *will* be
% used.
% #2: same as #1, but for y
% #3: same as #1, but for z
% #4: [output] a scale for use as argument of \pgfplots@apply@unit@vector@rescale@keep@size{x}{<arg>}
% #5: [output] a scale for use as argument of \pgfplots@apply@unit@vector@rescale@keep@size{y}{<arg>}
% #6: [output] a scale for use as argument of \pgfplots@apply@unit@vector@rescale@keep@size{z}{<arg>}
%
% The output arguments need to be applied before they take effect.
\def\pgfplots@BB@for@plotbox@get@unit@scales@compensated@axis@limits#1#2#3#4#5#6{%
	\begingroup
	% ATTENTION : this code ASSUMES that the datascaling trafo is
	% initialized with THE SAME SCALE IN EACH DIRECTION.
	% The data scaling also leads to (potentially non-uniform) scaling per component.
	%
	% Note that we could handle the datascaling here -- but we would
	% leave the supported number range easily. That's why that part of
	% the 'scale mode=scale uniformly' implementation has been moved
	% to \pgfplots@set@optimal@datatrafos@allaxes
	%
	% This here handles the limits only.
	\edef\pgfplots@scale@unitx{#1}%
	\edef\pgfplots@scale@unity{#2}%
	\edef\pgfplots@scale@unitz{#3}%
	%
	% compute extreme + arg extreme of these scales:
	\def\pgfplots@extreme@scale{-16300}%
	\def\pgfplots@extreme@scale@arg{NONE}%
	%
	\if0\b@pgfplots@unitvec@is@zero@x
		\ifdim\pgfplots@extreme@scale pt<\pgfplots@scale@unitx pt
			\let\pgfplots@extreme@scale=\pgfplots@scale@unitx
			\def\pgfplots@extreme@scale@arg{x}%
		\fi
	\fi
	\if0\b@pgfplots@unitvec@is@zero@y
		\ifdim\pgfplots@extreme@scale pt<\pgfplots@scale@unity pt
			\let\pgfplots@extreme@scale=\pgfplots@scale@unity
			\def\pgfplots@extreme@scale@arg{y}%
		\fi
	\fi
	\if0\b@pgfplots@unitvec@is@zero@z
		\ifdim\pgfplots@extreme@scale pt<\pgfplots@scale@unitz pt
			\let\pgfplots@extreme@scale=\pgfplots@scale@unitz
			\def\pgfplots@extreme@scale@arg{z}%
		\fi
	\fi
	%
	% Now, adjust axis limits to compensate for the effect: we still
	% want to have a plot box which is as close as possible to the
	% target plot box.
	\def\pgfplots@loc@TMPa##1##2{%
		\if0\csname b@pgfplots@unitvec@is@zero@##1\endcsname
			\if1\pgfplots@scaleuniformly@choice % FIXME : this appears to be too much. Disable this!?
				% ok, nothing to do for this direction.
				\pgfplotscoordmath{pgfbasic}{one}%
				\let##2=\pgfmathresult
			\else
				\if\pgfplots@extreme@scale@arg ##1%
					% ok, nothing to do for this direction.
					\pgfplotscoordmath{pgfbasic}{one}%
					\let##2=\pgfmathresult
				\else
					\pgfplotscoordmath{pgfbasic}{op}{divide}{{\pgfplots@extreme@scale}{\csname pgfplots@scale@unit##1\endcsname}}%
					% do not call apply@unit@rescale immediately because the
					% unit vectors may not be in their final state. Postpone until
					% they are final.
					\edef##2{\pgfmathresult}%
				\fi
			\fi
		\else
			\def##2{1}%
		\fi
	}%
	\pgfplots@loc@TMPa{x}{#4}%
	\pgfplots@loc@TMPa{y}{#5}%
	\pgfplots@loc@TMPa{z}{#6}%
	%
	\toks0=\expandafter{#4}%
	\toks1=\expandafter{#5}%
	\toks2=\expandafter{#6}%
	\xdef\pgfplots@glob@TMPa{%
		% same scale in each dir:
		\def\noexpand#1{\pgfplots@extreme@scale}%
		\def\noexpand#2{\pgfplots@extreme@scale}%
		\def\noexpand#3{\pgfplots@extreme@scale}%
		\def\noexpand#4{\the\toks0}%
		\def\noexpand#5{\the\toks1}%
		\def\noexpand#6{\the\toks2}%
	}%
	\endgroup
	\pgfplots@glob@TMPa
}

% #1 : a dimen register
% #2 : x, y, or z
\def\pgfplots@BB@for@plotbox@getunitheight#1#2{%
	#1=\csname pgfplots@plotbox@#2max\endcsname\csname pgf@#2y\endcsname
	\advance#1 by -\csname pgfplots@plotbox@#2min\endcsname\csname pgf@#2y\endcsname
	\ifdim#1<0pt %
		% we want to return a height. It is also bigger than 0.
		% the difference above may be negative if the unit points
		% downward (special combinations of view/h and view/v)
		#1=-#1\relax
	\fi
}%


% Modifies the AXIS LIMITS to ensure that a suitable width/height is
% achieved.
%
% This does NOT introduce a further scale to the unit vectors.
%
% #1: a direction (x,y, or z)
% #2: a macro name. It will be assigned globally. It will contain
% EXECUTABLE instructions which will modify the axis limits to fit the
% scaling.
%
% PRECONDITION: 
% - \pgfplots@glob@TMPa contains the already computed
% scaling factor for 'scale uniformly'
% - \pgf@xb is the actual height and \pgf@yb is the desired height
%   (set as in the scaling routine)
%
% POSTCONDITION:
% #2 will contain the argument <arg> for \pgfplots@apply@unit@vector@rescale@keep@size{#1}{<arg>}
\def\pgfplots@prepare@vertical@rescaling@for@scale@uniformly@in@dir#1\returninto#2{%
	% The strategy is as follows:
	% 1. I want to fit the axis into width #1 (\pgf@ya) and
	% height #1 (\pgf@yb).
	% 2. I want to MAINTAIN the unit vector ratio. 
	% 3. I want to MAINTAIN the unit vector directions.
	%
	% I already know the scaling factor to fit the width (it
	% is stored in \scalex = \scaley).
	% Let's call it "s".
	%
	% Consequently, a uniform scaling by "s" leads to the image
	% height
	% h = s* (r_x * e_xy + r_y * e_yy + r_z * e_zy)
	% where r_i = (imax - imin). This here is essentially the
	% same as the bounding box computation above (at least for
	% standart orthographic 3D axes).
	%
	% What I want now is to enlarge the limits such that I
	% have BOTH, width #1 AND height #2, without obscuring the
	% unit vector ratio. Recall that width #1 is already
	% given.
	%
	% This strategy achieves this goal by
	% modifying axis limits for an axis whose unit vector is
	% parallel to the canvas y axis, i.e. e_i = (0,*). 
	%
	% That means I have to introduce a SECOND scale s_z which
	% applies only to the Z unit vector (since e_z = (0,*) ).
	% If H = #2 is the desired height, I find the target
	% equation for s_z,
	%
	% H = s* r_x e_xy + s * r_y e_yy + s_z * s * r_z * e_zy
	% =>
	% s_z = ( H- s*r_x e_xy - s*r_y e_yy) / ( s * r_z * e_zy).
	%
	% Remember that 
	% s = \scalex
	% H = \H
	% h = r_x * e_xy + r_y * e_yy + r_z * e_zy  = \h
	% =>
	% s_z = ( H- s*( h - r_z * e_zy) ) / ( s * r_z * e_zy).
	% 
	\begingroup
	\pgfplots@BB@for@plotbox@getunitheight{\pgf@xc}{#1}%
	%
	% compute the rest in floating point - intermediate results may
	% become too huge for TeX.
	\pgfplotscoordmath{default}{parsenumber}{\expandafter\pgf@sys@tonumber\csname pgf@xc\endcsname}%
	\let\pgfplots@diff=\pgfmathresult
	%
	\pgfplotscoordmath{default}{parsenumber}{\scalex}%
	\let\pgfplots@s=\pgfmathresult
	%
	% this is a precondition of this method:
	\pgfplotscoordmath{default}{parsenumber}{\h}%
	\let\h=\pgfmathresult
	\pgfplotscoordmath{default}{parsenumber}{\H}%
	\let\H=\pgfmathresult
	%
	% compute counter := H - s * (h - (max-min))
	\pgfplotscoordmath{default}{op}{subtract}{{\h}{\pgfplots@diff}}%
	\pgfplotscoordmath{default}{op}{multiply}{{\pgfplots@s}{\pgfmathresult}}%
	\pgfplotscoordmath{default}{op}{subtract}{{\H}{\pgfmathresult}}%
	\let\pgfplots@counter=\pgfmathresult
	%
	% computer denom := s * (max-min)
	\pgfplotscoordmath{default}{op}{multiply}{{\pgfplots@s}{\pgfplots@diff}}%
	\let\pgfplots@denom=\pgfmathresult
	%
	\pgfplotscoordmath{default}{op}{divide}{{\pgfplots@counter}{\pgfplots@denom}}%
	\pgfplotscoordmath{default}{tofixed}{\pgfmathresult}%
	%
	% Now, s_z = \pgfmathresult .
	%
	% Now, adjust the z limits.
	% Note that \pgfplots@apply@unit@vector@rescale@keep@size
	% has a slightly different context; it assumes that the
	% unit vector has been rescaled, not the axis limits.
	% Consequently, the inverse of the scaling factor enters.
	% Since \pgfplots@apply@unit@vector@rescale@keep@size
	% expects the inverse of the scale, we can provide
	% \pgfmathresult:
	\pgfmath@smuggleone\pgfmathresult
	\endgroup
	\let#2=\pgfmathresult
}

\def\pgfplots@computeunitvectorlengths{%
	\pgfplotsutil@edef@invoke\pgfmathveclen@{%
		{\pgf@sys@tonumber\pgf@xx}%
		{\pgf@sys@tonumber\pgf@xy}%
	}%
	\let\pgfplots@x@veclength=\pgfmathresult
	\pgfplotsmath@ifzero{\pgfplots@x@veclength}{%
		\def\pgfmathresult{infty}%
		% this case will be caught in \pgfplots@initsizes
	}{%
		\expandafter\pgfmath@basic@reciprocal@\expandafter{\pgfmathresult}%
	}%
	\let\pgfplots@x@inverseveclength=\pgfmathresult
	%
	\pgfplotsutil@edef@invoke\pgfmathveclen@{%
		{\pgf@sys@tonumber\pgf@yx}%
		{\pgf@sys@tonumber\pgf@yy}%
	}%
	\let\pgfplots@y@veclength=\pgfmathresult
	\pgfplotsmath@ifzero{\pgfplots@y@veclength}{%
		\def\pgfmathresult{infty}%
		% this case will be caught in \pgfplots@initsizes
	}{%
		\expandafter\pgfmath@basic@reciprocal@\expandafter{\pgfmathresult}%
	}%
	\let\pgfplots@y@inverseveclength=\pgfmathresult
	%
	\ifpgfplots@threedim
		\pgfplotsutil@edef@invoke\pgfmathveclen@{%
			{\pgf@sys@tonumber\pgf@zx}%
			{\pgf@sys@tonumber\pgf@zy}%
		}%
		\let\pgfplots@z@veclength=\pgfmathresult
		\pgfplotsmath@ifzero{\pgfplots@z@veclength}{%
			\def\pgfmathresult{infty}%
			% this case will be caught in \pgfplots@initsizes
		}{%
			\expandafter\pgfmath@basic@reciprocal@\expandafter{\pgfmathresult}%
		}%
		\let\pgfplots@z@inverseveclength=\pgfmathresult
	\else
		\def\pgfplots@z@veclength{0}%
		\def\pgfplots@z@inverseveclength{infty}%
	\fi
}%

% Defines \pgfplots@view@dir@threedim according to the actual
% configuration of x,y,z (2d) unit vectors, assuming the associated
% unit vectors form a right-handed-system.
%
% The algorithm works for standard three dimensional axes. It works as
% follows:
%
% First, observe that we have a normal direction N if all its
% multiples are mapped onto the same point in 2D canvas
% coordinates. In other words: all 3D coordinates which are mapped
% onto an arbitrary point in 2D canvas coordinates (take, for example,
% the origin (0,0)  ) are on a line in direction of N.
%
% We use this observation to compute the normal axis, i.e. we search
% for all points which are mapped onto the 2D canvas coordinate (0,0):
%  N_x e_xx + N_y e_yx + N_z e_zx = 0
%  N_x e_xy + N_y e_yy + N_z e_zy = 0.
% All solutions make up a linear space of dimension 1 (up to special
% cases). In the general case, we can chose an arbitrary  N_z != 0
% and reduce the linear system to
%  N_x e_xx + N_y e_yx = - N_z e_zx
%  N_x e_xy + N_y e_yy = - N_z e_zy.
% Choosing *any* N_z != 0, say, N_z=-1 (which corresponds to view
% from above) will lead to a vector parallel to the normal direction.
% But it might have the wrong sign.
%
% FIXME : this fails if one of e_x or e_y is zero.
%
% To find the correct sign for N, I have made several case
% distinctions to identify the cases when we have to multiply with -1.
% The key idea is to assume a right-handed-system of unit vectors;
% this is the condition which allows to determine the sign.
%
% Furthermore, I assume that e_z points to the top, i.e. that e_zy >0.
% Then, there are (mainly) four conditions on the signs of e_x and e_y
% which indicate that we are viewing from below and should switch the
% sign of N (keep in mind that our initial choice was N_z =-1, see above).
%
% The conditions can be identified by drawing a 3D box and
% identifying the corner which represents the lower left 3D limits.
%
% You can visualize these cases using
%--------------------------------------------------
% \pgfplotsset{
% 		separate axis lines,
% 		every outer x axis line/.append style= {-stealth},
% 		every outer y axis line/.append style= {-stealth},
% 		every outer z axis line/.append style= {-stealth},
% 		samples=2,shader=interp,title={view=\h,\v},
% 		domain=0:1,
% 		enlargelimits=false,
% 		view=\h\v,xlabel=x,ylabel=y,
% 		extra description/.code={%
% 			\node[draw,fill=white] at (axis cs:0,0,0) {};
% 		},
% }
% 
% \def\v{30}
% \foreach \h in {30,120,210,300} {
% \message{VIEW={\h}{\v}^^J}
% \begin{tikzpicture}
% 	\begin{axis}
% 	\addplot3[surf] {x};
% 	\end{axis}
% \end{tikzpicture}
% 
% }
% 
% \def\v{-30}
% \foreach \h in {30,120,210,300} {
% \message{VIEW={\h}{\v}^^J}
% \begin{tikzpicture}
% 	\begin{axis}
% 	\addplot3[surf] {x};
% 	\end{axis}
% \end{tikzpicture}
% 
% }
%-------------------------------------------------- 
%  The precise formulas can be found below in the source code.
%
% You can override this function by the /pgfplots/view dir key.
\def\pgfplotsgetnormalforcurrentview{%
	\pgfkeysgetvalue{/pgfplots/view dir}\pgfplots@loc@TMPc
	\ifx\pgfplots@loc@TMPc\pgfutil@empty
	\begingroup	
		% temporarily undo the effects of reversed axes -- we *really*
		% need a right-handed-coordinate system here:
		\if r\pgfkeysvalueof{/pgfplots/x dir/value}%
			\pgf@xx=-\pgf@xx
			\pgf@xy=-\pgf@xy
		\fi
		\if r\pgfkeysvalueof{/pgfplots/y dir/value}%
			\pgf@yx=-\pgf@yx
			\pgf@yy=-\pgf@yy
		\fi
		\if r\pgfkeysvalueof{/pgfplots/z dir/value}%
			\pgf@zx=-\pgf@zx
			\pgf@zy=-\pgf@zy
		\fi
		% FIRST: check for special cases.
		\let\pgfplots@view@dir@threedim=\pgfutil@empty%
		% Special case:
		% e_xx = e_xy = 0 
		%
		% i.e.:
		%
		%   ^
		%   | |---|
		%   z |   |
		%     |---|
		%       y->
		% 
		% In this case, N must be the x axis.
		\ifdim\pgf@xx=0pt %
			\ifdim\pgf@xy=0pt %
				\def\pgfplots@view@dir@threedim{-1,0,0}%
			\fi
		\fi
		% Special case:
		% e_yx = e_yy = 0 
		%
		% i.e.:
		%
		%   ^
		%   | |---|
		%   z |   |
		%     |---|
		%       x->
		% 
		% In this case, N must be the y axis.
		\ifdim\pgf@yx=0pt %
			\ifdim\pgf@yy=0pt %
				\def\pgfplots@view@dir@threedim{0,1,0}%
			\fi
		\fi
		% Special case:
		% e_xy = e_yy = 0  (i.e. one row)
		%
		% that is hard to draw, use view={30}{0} to see it. 
		% 
		% In this case, N_z must be 0 and we have a different system.
		\ifdim\pgf@xy=0pt %
			\ifdim\pgf@yy=0pt %
				\ifx\pgfplots@view@dir@threedim\pgfutil@empty
					% we have N_x e_xx + N_y e_yx = 0
					% Note that e_xx != 0 and e_yx != 0 (otherwise one
					% of our other special cases above would have
					% caught the case)
					% -> we have N_x = -N_y e_yx / e_xx and N_y
					%  arbitrary. only the sign needs to be fixed.
					\def\pgfplots@view@dir@threedim@z{0}%
					\def\pgfplots@view@dir@threedim@y{1}% fix it somehow. We correct the sign later.
					\edef\pgfplots@loc@TMPa{-(\pgfplots@view@dir@threedim@y) * \pgf@sys@tonumber\pgf@yx / (\pgf@sys@tonumber\pgf@xx)}%
					\pgfmathparse{\pgfplots@loc@TMPa}%
					\let\pgfplots@view@dir@threedim@x=\pgfmathresult
					%
					\def\pgfplots@scale{1}%
					% I identified these cases by comparing the
					% results with \pgfplots@scale{1} with those of
					% the view dir generated by
					% \pgfplotssetaxesfromazel (which has the correct quality of solution)
					\ifdim\pgf@zy>0pt %
						\ifdim\pgf@xx<0pt %
							\def\pgfplots@scale{-1}%
						\fi
					\else
						\ifdim\pgf@xx>0pt %
							\def\pgfplots@scale{-1}%
						\fi
					\fi
					\pgfmathmultiply@{\pgfplots@scale}{\pgfplots@view@dir@threedim@x}%
					\let\pgfplots@view@dir@threedim@x\pgfmathresult
					\pgfmathmultiply@{\pgfplots@scale}{\pgfplots@view@dir@threedim@y}%
					\let\pgfplots@view@dir@threedim@y\pgfmathresult
					\pgfmathmultiply@{\pgfplots@scale}{\pgfplots@view@dir@threedim@z}%
					\let\pgfplots@view@dir@threedim@z\pgfmathresult
					%
					\edef\pgfplots@view@dir@threedim{\pgfplots@view@dir@threedim@x,\pgfplots@view@dir@threedim@y,\pgfplots@view@dir@threedim@z}%
				\else
					% Ah - we already caught that special case above.
				\fi
			\fi
		\fi
		%
		% NOTE : the case e_xx = e_yx = 0 IS NO USE-CASE (would
		% require a rotated z axis which is forbidden currently)
		%
		\ifx\pgfplots@view@dir@threedim\pgfutil@empty
			\def\pgfplots@view@dir@threedim@z{-1}% hold it at some arbitrary value
			\pgf@xa=-\pgfplots@view@dir@threedim@z\pgf@zx
			\pgf@ya=-\pgfplots@view@dir@threedim@z\pgf@zy
			\edef\pgfplots@loc@TMPa{%
				{%
					{\pgf@sys@tonumber\pgf@xx}{\pgf@sys@tonumber\pgf@yx}%
					{\pgf@sys@tonumber\pgf@xy}{\pgf@sys@tonumber\pgf@yy}%
				}%
				{%
					{\pgf@sys@tonumber\pgf@xa}{\pgf@sys@tonumber\pgf@ya}%
				}%
			}%
			\expandafter\pgfutilsolvetwotwoleq\pgfplots@loc@TMPa
			\def\pgfplots@loc@TMPb##1##2{%
				\def\pgfplots@view@dir@threedim@x{##1}%
				\def\pgfplots@view@dir@threedim@y{##2}%
			}%
			\expandafter\pgfplots@loc@TMPb\pgfmathresult
			%
			% Identify if we need to switch the sign.
			% To verify that these cases are useful, I suggest visualizing
			% that stuff using the TeX code from above...
			%
			% I guess it is correct up to collapsing views (as you see, I
			% did not properly identify the cases with "=0" )
			\def\pgfplots@scale{1}%
			\ifdim\pgf@xx>0pt
				\ifdim\pgf@yx<0pt
				\else
					%
					\ifdim\pgf@xy<0pt
					\else
						\ifdim\pgf@yy<0pt
							\def\pgfplots@scale{-1}%
						\fi
					\fi
					%
				\fi
			\else
				\ifdim\pgf@xx<0pt
					\ifdim\pgf@yx>0pt
					\else
						%
						\ifdim\pgf@xy>0pt
						\else
							\ifdim\pgf@yy>0pt
								\def\pgfplots@scale{-1}%
							\fi
						\fi
						%
					\fi
				\fi
			\fi
			\ifdim\pgf@xy>0pt
				\ifdim\pgf@yy<0pt
				\else
					%
					\ifdim\pgf@xx>0pt
					\else
						\ifdim\pgf@yx>0pt
							\def\pgfplots@scale{-1}%
						\fi
					\fi
					%
				\fi
			\else
				\ifdim\pgf@xy<0pt
					\ifdim\pgf@yy>0pt
					\else
						%
						\ifdim\pgf@xx<0pt
						\else
							\ifdim\pgf@yx<0pt
								\def\pgfplots@scale{-1}%
							\fi
						\fi
						%
					\fi
				\fi
			\fi
			\pgfmathmultiply@{\pgfplots@scale}{\pgfplots@view@dir@threedim@x}%
			\let\pgfplots@view@dir@threedim@x\pgfmathresult
			\pgfmathmultiply@{\pgfplots@scale}{\pgfplots@view@dir@threedim@y}%
			\let\pgfplots@view@dir@threedim@y\pgfmathresult
			\pgfmathmultiply@{\pgfplots@scale}{\pgfplots@view@dir@threedim@z}%
			\let\pgfplots@view@dir@threedim@z\pgfmathresult
			%
			\pgfplotsmathvectorfromstring{\pgfplots@view@dir@threedim@x,\pgfplots@view@dir@threedim@y,\pgfplots@view@dir@threedim@z}{default}%
			\let\pgfplots@view@dir@threedim=\pgfplotsretval
			% normalize. This is not absolutely required -- but it is used
			% to accumulate point depth (for the mesh handler) in pgfmath
			% arithmetics. At least \pgfplotsmathviewdepthxyz should use
			% a properly scaled view dir.
			\pgfplotsmathvectorlength{\pgfplotsretval}{default}%
			\pgfplotscoordmath{default}{op}{reciprocal}{{\pgfplotsretval}}%
			\pgfplotsmathvectorscale{\pgfplots@view@dir@threedim}{\pgfmathresult}{default}%
		\else
			\pgfplotsmathvectorfromstring{\pgfplots@view@dir@threedim}{default}%
		\fi
		%
		\pgfmath@smuggleone\pgfplotsretval
	\endgroup
		\let\pgfplots@view@dir@threedim=\pgfplotsretval
	\else
		\def\pgfplots@loc@TMPb##1##2##3{%
			\pgfplotsmathvectorfromstring{##1,##2,##3}{default}%
			\let\pgfplots@view@dir@threedim=\pgfplotsretval
		}%
		\expandafter\pgfplots@loc@TMPb\pgfplots@loc@TMPc
	\fi
}%

% PRECONDITION:
% 	none
% POSTCONDITION:
% 	\pgfplots@default@aspect@ratio is set.
\def\pgfplots@compute@default@aspect@ratio{%
	\expandafter\pgfmath@x\axisdefaultwidth
	\expandafter\pgfmath@y\axisdefaultheight
	\pgfmathlog@invoke@expanded\pgfmathdivide@{%
		{\pgf@sys@tonumber{\pgfmath@x}}%
		{\pgf@sys@tonumber{\pgfmath@y}}%
	}%
	\let\pgfplots@default@aspect@ratio=\pgfmathresult
}

\def\pgfplots@ifneeds@one@uniform@datascale#1#2{%
	\if3\pgfplots@scale@mode@choice
		% scale mode=scale uniformly
		\def\pgfplots@loc@TMPa{1}%
		%
		% if we have at least one unit vector given explicitly, the
		% meaning changes: in that case, we can (and probably should)
		% use different data scale factors in each direction.
		\ifx\pgfplots@x\pgfutil@empty
		\else
			\def\pgfplots@loc@TMPa{0}%
		\fi
		\ifx\pgfplots@y\pgfutil@empty
		\else
			\def\pgfplots@loc@TMPa{0}%
		\fi
		\ifx\pgfplots@z\pgfutil@empty
		\else
			\def\pgfplots@loc@TMPa{0}%
		\fi
	\else
		\def\pgfplots@loc@TMPa{0}%
	\fi
	\if1\pgfplots@loc@TMPa
		#1%
	\else
		#2%
	\fi
}%

\def\pgfplots@set@default@size@options{%
	% The axes 'x' and 'y' vectors will be scaled such that the total
	% size is (\axisdefaultwidth, \axisdefaultheight).
	%
	% If the user specifies ONE of width OR height, 
	% the plot will be resized; keeping the aspect ratio.
	%
	\let\pgfplots@default@aspect@ratio=\pgfutil@empty
	\pgfkeysgetvalue{/pgfplots/x}{\pgfplots@x}%
	\pgfkeysgetvalue{/pgfplots/y}{\pgfplots@y}%
	\pgfkeysgetvalue{/pgfplots/z}{\pgfplots@z}%
	%\pgfkeysgetvalue{/pgfplots/viewdir}{\pgfplots@viewdir}%
	\pgfkeysgetvalue{/pgfplots/width}{\pgfplots@width}%
	\pgfkeysgetvalue{/pgfplots/height}{\pgfplots@height}%
	\ifx\pgfplots@width\pgfutil@empty
		\def\pgfplots@user@provided@width{0}%
	\else
		\def\pgfplots@user@provided@width{1}%
		\pgfmathparse{\pgfplots@width}%
		\edef\pgfplots@width{\pgfmathresult pt}%
	\fi
	\ifx\pgfplots@height\pgfutil@empty
		\def\pgfplots@user@provided@height{0}%
	\else
		\def\pgfplots@user@provided@height{1}%
		\pgfmathparse{\pgfplots@height}%
		\edef\pgfplots@height{\pgfmathresult pt}%
	\fi
	%
	% CASES:
	% W := 'width' option non-empty
	% H := 'height' option non-empty
	%
	% 		W H
	% 		0 0 -> \axisdefaultwidth 
	% 		0 1 -> determine width out of H and the default aspect ratio
	% 		1 X -> ok, use the user parameter.
	% -> KEEP ASPECT RATIO if just one W, or H is given!
	\ifx\pgfplots@width\pgfutil@empty
		\ifx\pgfplots@height\pgfutil@empty
			% The case W=0 H=0:
			\let\pgfplots@width=\axisdefaultwidth
			\let\pgfplots@height=\axisdefaultheight
		\else
			% The case W=0 H=1:
			\pgfplots@compute@default@aspect@ratio
			\expandafter\pgfmath@y\pgfplots@height
			\pgfmathlog@invoke@expanded\pgfmathmultiply@{%
				{\pgf@sys@tonumber{\pgfmath@y}}%
				{\pgfplots@default@aspect@ratio}%
			}%
			\edef\pgfplots@width{\pgfmathresult pt}%
		\fi
	\else
		\ifx\pgfplots@height\pgfutil@empty
			% The case W=1 H=0:
			\pgfplots@compute@default@aspect@ratio
			\expandafter\pgfmath@x\pgfplots@width
			\pgfmathlog@invoke@expanded\pgfmathdivide@{%
				{\pgf@sys@tonumber{\pgfmath@x}}%
				{\pgfplots@default@aspect@ratio}%
			}%
			\edef\pgfplots@height{\pgfmathresult pt}%
		\else
			% The case W=1 H=1:
		\fi
	\fi
	\pgfkeyslet{/pgfplots/width}{\pgfplots@width}%
	\pgfkeyslet{/pgfplots/height}{\pgfplots@height}%
	%
	\ifpgfplots@threedim
		\pgfplots@set@default@size@options@threedim
	\fi
	%
	\pgfplots@set@scale@mode
}

% This method must be called BEFORE THE DATASCALING is initialized.
\def\pgfplots@set@scale@mode{%
	\pgfkeysgetvalue{/pgfplots/unit vector ratio}\pgfplots@loc@TMPb
	\ifx\pgfplots@loc@TMPb\pgfutil@empty
	\else
		\ifcase\pgfplots@scale@mode@choice
			% 'scale mode'=auto
			\def\pgfplots@scale@mode@choice{3}% set to 'scale uniformly'
			%
			\if1\pgfplots@compat@scale@mode@compatible@mode
				% backwards compatibility mode...
				\ifpgfplots@threedim
					% ... for 3d: there is no backwards compatibility
					% mode here; it was plain wrong for 3d axes:
					% neither lengths nor angles have been correct.
					\pgfplots@compat@scale@mode@compatible@mode@warning
				\fi
			\fi
		\or
			% scale mode=none: keep it this way.
			\immediate\write-1{PGFPlots: scale mode=none  and unit vector ratio is incompatible. Ignoring unit vector ratio.^^J}%
		\or
			% scale mode=stretch to fill
			\immediate\write-1{PGFPlots: scale mode=stretch to fill  and unit vector ratio might produce unexpected results. Consider using scale mode=auto^^J}%
		\fi
	\fi
	%
}

\def\pgfplots@compat@scale@mode@compatible@mode@warning{%
	\pgfplotswarning{axis equal incompatible change}\pgfeov%
}%

\def\pgfplots@set@default@size@options@threedim{%
	\pgfplots@loc@tmpfalse
	\ifx\pgfplots@x\pgfutil@empty
	\else
		\pgfplots@loc@tmptrue
	\fi
	\ifx\pgfplots@y\pgfutil@empty
	\else
		\pgfplots@loc@tmptrue
	\fi
	\ifx\pgfplots@z\pgfutil@empty
	\else
		\pgfplots@loc@tmptrue
	\fi
	\ifpgfplots@loc@tmp
		% oh - we have at least one of the [xyz] unit vectors!
		% make sure all of them are there
		\ifx\pgfplots@x\pgfutil@empty
			\pgfplots@set@default@size@options@threedim@{x}{(1pt,0pt)}%
		\fi
		\ifx\pgfplots@y\pgfutil@empty
			\pgfplots@set@default@size@options@threedim@{y}{(0pt,1pt)}%
		\fi
		\ifx\pgfplots@z\pgfutil@empty
			\pgfplots@set@default@size@options@threedim@{z}{(0pt,1pt)}%
		\fi
		\pgfkeyslet{/pgfplots/view/az}\pgfutil@empty
		\pgfkeyslet{/pgfplots/view/el}\pgfutil@empty
	\fi
}
\def\pgfplots@set@default@size@options@threedim@#1#2{%
	\pgfplots@error{Sorry, a 3D axis needs either NONE or ALL of "x,y,z". I found partial information, but (at least) '#1' is lacking... please add '#1'}%
	\expandafter\def\csname pgfplots@#1\endcsname{#2}%
}

% A helper method for \pgfplots@initsizes which
% - applies the data scaling trafo to user arguments
% - sets calls pgfset#1vec
%
% #1: the vector to set (either 'x' or 'y')
% #2: the index of the vector to set (either 0 or 1)
% #3: the already precomputed temporary scale (see pgfplots@initsizes)
% #4: an output argument. It is a macro name which will be defined to
% '1' if and only if the finally set vector is parallel to the #1 axis
% of PGF, that means (x,0) for #1=x and (0,y) for #2=y.
\def\pgfplots@initsizes@setunitvector#1#2#3#4{%
	\pgfkeysgetvalue{/pgfplots/#1 dir/value}\pgfplots@loc@dirvalue
	\expandafter\let\expandafter\pgfplots@loc@TMPb\csname pgfplots@#1\endcsname
	\ifx\pgfplots@loc@TMPb\pgfutil@empty
		\def#4{1}% we have (#1,0) or (0,#1)
		%
%\message{Setting unitvector(#1) to auto-computed multiple of e_#2 ...}%
		\edef\pgfplots@loc@TMPa{#3}%
		\if r\pgfplots@loc@dirvalue
			\edef\pgfplots@loc@TMPa{-#3}%
		\fi
		\ifcase#2\relax
			\pgfsetxvec{\pgfqpoint{\pgfplots@loc@TMPa pt}{0pt}}%
		\or
			\pgfsetyvec{\pgfqpoint{0pt}{\pgfplots@loc@TMPa pt}}%
		\or
			\pgfsetzvec{\pgfqpoint{\pgfplots@loc@TMPa pt}{\pgfplots@loc@TMPa pt}}%
		\fi
	\else
		% Ok, we have a user-defined unit vector.
		%
		% That means we also need to apply the scaling trafo!
		%
		% 1. Check whether we have a complete vector of type (x,y):
		\expandafter\pgfutil@in@\expandafter(\expandafter{\pgfplots@loc@TMPb}%
		\ifpgfutil@in@
			% YES: we have (x,y):
			%
			\def#4{0}% we DON'T have (#1,0) or (0,#1). At least I think so.
			%
%\message{Setting unitvector(#1) to non-standard \csname pgfplots@#1\endcsname ...}%
			\def\pgfplots@loc@TMPa(##1,##2){%
				\pgfplotscoordmath{default}{parse}{##1}%
				\pgfplotscoordmath{default}{tofixed}{\pgfmathresult}%
				\let\pgfplots@loc@TMPb=\pgfmathresult
				\pgfplotscoordmath{default}{parse}{##2}%
				\pgfplotscoordmath{default}{tofixed}{\pgfmathresult}%
				\let\pgfplots@loc@TMPc=\pgfmathresult
				%
				\pgfplots@if{pgfplots@apply@datatrafo@#1}{%
					\pgfplotscoordmath{#1}{datascaletrafo noshift inverse to fixed}{\pgfplots@loc@TMPb}%
					\let\pgfplots@loc@TMPb=\pgfmathresult
					\pgfplotscoordmath{#1}{datascaletrafo noshift inverse to fixed}{\pgfplots@loc@TMPc}%
					\let\pgfplots@loc@TMPc=\pgfmathresult
				}{}%
				\csname pgfset#1vec\endcsname{%
					\pgfqpoint
						{\if r\pgfplots@loc@dirvalue -\fi\pgfplots@loc@TMPb pt}
						{\if r\pgfplots@loc@dirvalue -\fi\pgfplots@loc@TMPc pt}}%
			}%
			\expandafter\pgfplots@loc@TMPa\pgfplots@loc@TMPb%
			%
		\else
			% NO we simply have a scalar value.
			\def#4{1}% we have (#1,0) or (0,#1)
%\message{Setting unitvector(#1) to \csname pgfplots@#1\endcsname * e_{#2}...}%
			\pgfplots@if{pgfplots@apply@datatrafo@#1}{%
				\pgfplotscoordmath{default}{parse}{\csname pgfplots@#1\endcsname}%
				\pgfplotscoordmath{default}{tofixed}{\pgfmathresult}%
				\pgfplotscoordmath{#1}{datascaletrafo noshift inverse to fixed}{\pgfmathresult}%
				\edef\pgfplots@loc@TMPb{\pgfmathresult pt}%
			}{\relax}%
			\edef\pgfplots@loc@TMPb{\if r\pgfplots@loc@dirvalue -\fi\pgfplots@loc@TMPb}%
			\begingroup
				\pgf@xa=\pgfplots@loc@TMPb\relax
				\xdef\pgfplots@glob@TMPb{\pgf@sys@tonumber{\pgf@xa}}%
			\endgroup
			\ifcase#2\relax
				\pgfsetxvec{\pgfqpoint{\pgfplots@loc@TMPb}{0pt}}%
			\or
				\pgfsetyvec{\pgfqpoint{0pt}{\pgfplots@loc@TMPb}}%
			\or
				\pgfsetzvec{\pgfqpoint{\pgfplots@loc@TMPb}{\pgfplots@loc@TMPb}}%
			\fi
		\fi
	\fi
%\message{-> got unitvector(#1) = (\the\csname pgf@#1x\endcsname, \the\csname pgf@#1y\endcsname).^^J}%
}%

% Applies the 'axis equal' feature.
%
% PRECONDITION:
% 	- #1, #2, #3 contains the current scaling
% 	factors in x,y, z, resp. which are to be applied to unit vectors
% 	- neither unit vectors nor limits are in their final shape
% 	-  \pgfplots@set@default@size@options has been invoked before
%
% POSTCONDITION:
%   - #1, #2, #3 have been changed to accomodate unit vector ratio
%   - #4, #5, #6 [output] contain axis limit compensation scales
%
% There is just one algorithmic difficulty: the data scaling
% transformation. All unit vector length above are only meaningful in
% the UNTRANSFORMED range, so we have to mingle with the scaling
% transformation.
\def\pgfplots@apply@unit@ratio#1#2#3#4#5#6{%
	\begingroup
	\edef\pgfplots@target@unit@scale@inv@x{#1}%
	\edef\pgfplots@target@unit@scale@inv@y{#2}%
	\edef\pgfplots@target@unit@scale@inv@z{#3}%
	\def\pgfplots@target@limitrescale@x@{1}%
	\def\pgfplots@target@limitrescale@y@{1}%
	\def\pgfplots@target@limitrescale@z@{1}%
	%
	\pgfkeysgetvalue{/pgfplots/unit vector ratio}\pgfplots@unit@vector@ratio
	\ifx\pgfplots@unit@vector@ratio\pgfutil@empty
	\else
		\edef\pgfplots@unit@vector@ratio{\pgfplots@unit@vector@ratio\space1 1 }%
		%
		\expandafter\pgfplots@unit@vector@ratio@check@nop\pgfplots@unit@vector@ratio\pgfplots@EOI
		\ifpgfplots@loc@tmp
			%
			% Step 1: compute the unit vector which STAYS CONSTANT.
			%
			\pgfkeysgetvalue{/pgfplots/unit vector ratio axis}\pgfplots@apply@unit@ratio@reference
			\ifx\pgfplots@apply@unit@ratio@reference\pgfutil@empty
				\pgfplots@apply@unit@ratio@find@reference%
			\fi
			%
			% FIXME : I could spent some attention here to save work: 
			% both, unit ratios and the resulting scales are computed at
			% least twice (once in \pgfplots@apply@unit@ratio@find@reference and once in the
			% following).
			\expandafter\pgfplots@apply@unit@ratio@prepareratios\pgfplots@unit@vector@ratio\pgfplots@EOI
			%
%\message{USING REFERENCE UNIT VECTOR FROM \pgfplots@apply@unit@ratio@reference; ratio \pgfplots@unit@ratio@x\space \pgfplots@unit@ratio@y\space \pgfplots@unit@ratio@z.^^J}%
			%
			% Step 2: apply the scaling:
			\pgfplots@rescale@unit@vector@reltoreference{x}{\pgfplots@unit@ratio@x}%
			\pgfplots@rescale@unit@vector@reltoreference{y}{\pgfplots@unit@ratio@y}%
			\ifpgfplots@threedim
				\pgfplots@rescale@unit@vector@reltoreference{z}{\pgfplots@unit@ratio@z}%
			\fi
			%
		\else
%\message{Skipped application of 'unit vector ratio=\pgfkeysvalueof{/pgfplots/unit vector ratio}': it is already done by 'scale uniformly'.^^J}%
		\fi
	\fi
	\xdef\pgfplots@glob@TMPa{%
		\noexpand\def\noexpand#1{\pgfplots@target@unit@scale@inv@x}%
		\noexpand\def\noexpand#2{\pgfplots@target@unit@scale@inv@y}%
		\noexpand\def\noexpand#3{\pgfplots@target@unit@scale@inv@z}%
		\noexpand\def\noexpand#4{\pgfplots@target@limitrescale@x@}%
		\noexpand\def\noexpand#5{\pgfplots@target@limitrescale@y@}%
		\noexpand\def\noexpand#6{\pgfplots@target@limitrescale@z@}%
	}%
	\endgroup
	\pgfplots@glob@TMPa
}%

\def\pgfplots@appy@unit@ratio@reciprocal#1{%
	\pgfplotscoordmath{default}{parsenumber}{#1}%
	\pgfplotscoordmath{default}{op}{reciprocal}{{\pgfmathresult}}%
	\pgfplotscoordmath{default}{tofixed}{\pgfmathresult}%
}%

% Defines \ifpgfplots@loc@tmp := need to modify scaling factors
\def\pgfplots@unit@vector@ratio@check@nop#1 #2 #3 #4\pgfplots@EOI{%
	\pgfplots@loc@tmptrue
	\if3\pgfplots@scale@mode@choice
		% scale mode=scale uniformly
		\ifpgfplots@threedim
			\ifdim#1pt=#2pt
				\ifdim#1pt=#3pt
					% 'axis equal' is implicitly done by 'scale mode=scale
					% uniformly' anyway
					\pgfplots@loc@tmpfalse
				\fi
			\fi
		\else
			\ifdim#1pt=#2pt
				% 'axis equal' is implicitly done by 'scale mode=scale
				% uniformly' anyway
				\pgfplots@loc@tmpfalse
			\fi
		\fi
	\fi
	% activate the following line to deactivate optimization: [FIXME]
	%\pgfplots@loc@tmpfalse
}%

% This macro determines the reference axis for unit vector rescaling.
% The reference axis remains unscaled (it gets scaling factor 1 if you
% want it this way).
% 
% The other axes are scaled such that the desired unit vector ratios
% are fulfilled.
%
% The idea to select a reference axis is as follows:
% 1. Every unit vector scaling factor s should fulfill  s  <= 1.
% 2. Choose the reference axis such that the minimal amount of scaling
% is performed.
%
% The motivation for (1) is: if all involved scaling factors are at
% most 1, the resulting picture will only become *smaller*.
% Consequently, we can simply enlarge axis limits to restore the
% original width/height!
%
% The motivation for (2) is: a huge amount of scaling might reduce the
% size of the image too much. Of course, the figure will be enlarged
% to fit the original width/height, but most of it will be empty. So,
% use the smallest scaling.
%
% @POSTCONDITION The reference axis is stored in
% \pgfplots@apply@unit@ratio@reference .
%
% @see the key 'unit vector ratio axis=y' which allows to manually
% select the reference axis. This will illustrate what happens here.
\def\pgfplots@apply@unit@ratio@find@reference{%
	%
	\begingroup
	\let\pgfplots@ONE=\pgf@x
	\global\pgfplots@ONE=1.002pt
	%
	\def\pgfplots@optimum@sofar@axis{}%
	\let\pgfplots@optimum@sofar@value=\pgf@y
	\global\pgfplots@optimum@sofar@value=16000pt
	%
	%\pgfplots@apply@unit@ratio@find@reference@checkexplicitlimits
	%
	\ifx\pgfplots@optimum@sofar@axis\pgfutil@empty
		% set \pgfplots@loc@TMPa := 1 if and only if the axis is 3d
		\def\pgfplots@loc@TMPa{0}%
		\if0\b@pgfplots@unitvec@is@zero@z
			% ah, it IS 3d!
			\def\pgfplots@loc@TMPa{1}%
		\else
			% ok, 2d mode (includes view={0}{90})
			\def\pgfplots@loc@TMPa{0}%
		\fi
		\if1\pgfplots@loc@TMPa
			% 3D is more complicated than 2D:
			% for every fixed reference axis, we have to check *two*
			% scaling factors.
			%
			% Furthermore, the optimality condition (2) needs to be
			% performed on the maximum max{1-s_a, 1-s_b} provided both of
			% these numbers are positive.
			%
			\def\pgfplots@check@##1##2{%
				% PRECONDITION: \pgfplots@apply@unit@ratio@reference is defined.
				%
				% renormalize \pgfplots@unit@[xyz] :
				\expandafter\pgfplots@apply@unit@ratio@prepareratios\pgfplots@unit@vector@ratio\pgfplots@EOI
				%
				% compute s_a :
				\pgfplots@getscale@unit@vector@reltoreference ##1{\csname pgfplots@unit@ratio@##1\endcsname}%
				\let\pgfplots@scale@a=\pgfmathresult
				%
				% compute s_b :
				\pgfplots@getscale@unit@vector@reltoreference ##2{\csname pgfplots@unit@ratio@##2\endcsname}%
				\let\pgfplots@scale@b=\pgfmathresult
				%
				% check if the actual choice of
				% \pgfplots@apply@unit@ratio@reference is FEASIBLE.
				% That is the case if s_a <= 1  && s_b <= 1.
				% We check
				%  (1 - s_a >= 0 )  && ( 1 - s_b >= 0 )
				% instead, since I need the value 
				%   max( 1-s_a, 1-s_b )
				% anyway.
				\def\pgfplots@ref@is@feasible{1}%
				\pgf@xa=\pgfplots@ONE \advance\pgf@xa by-\pgfplots@scale@a pt
				\ifdim\pgf@xa<0sp
					\def\pgfplots@ref@is@feasible{0}%
				\else
					\pgf@xb=\pgfplots@ONE \advance\pgf@xb by-\pgfplots@scale@b pt
					\ifdim\pgf@xb<0sp
						\def\pgfplots@ref@is@feasible{0}%
					\fi
				\fi
				% compute max(1-s_a,1-s_b) into \pgf@xa:
				% pgf@xa= max(pgf@xa,pgf@xb):
				\ifdim\pgf@xb>\pgf@xa	\pgf@xa=\pgf@xb	\fi
				\if1\pgfplots@ref@is@feasible
					\ifdim\pgf@xa<\pgfplots@optimum@sofar@value
						% Ah, ok. The actual choice is BETTER as it
						% involves less scaling.
						%
						% Remember it!
						\let\pgfplots@optimum@sofar@axis=\pgfplots@apply@unit@ratio@reference
						\global\pgfplots@optimum@sofar@value=\pgf@xa
					\fi
				\fi
%\message{^^Junit vector ratio 3D searching reference: checking \pgfplots@apply@unit@ratio@reference. feasable=\pgfplots@ref@is@feasible. \if1\pgfplots@ref@is@feasible max=\the\pgf@xa. \fi Optimum so far: value =\the\pgfplots@optimum@sofar@value\space for axis \pgfplots@optimum@sofar@axis.^^J}%
			}%
			%
			% Check 'x' as reference :
			\def\pgfplots@apply@unit@ratio@reference{x}%
			\pgfplots@check@ yz%
			%
			% Check 'y' as reference :
			\def\pgfplots@apply@unit@ratio@reference{y}%
			\pgfplots@check@ xz%
			%
			% Check 'z' as reference :
			\def\pgfplots@apply@unit@ratio@reference{z}%
			\pgfplots@check@ xy%
			%
		\else
			% 2D is much simpler: find the scale s which fulfills s <= 1.
			% One of them MUST fulfill it.
			% 
			% try 'x' axis as reference:
			\def\pgfplots@apply@unit@ratio@reference{x}%
			%
			% renormalize:
			\expandafter\pgfplots@apply@unit@ratio@prepareratios\pgfplots@unit@vector@ratio\pgfplots@EOI
			%
			% compute scaling factor: 
			\pgfplots@getscale@unit@vector@reltoreference y\pgfplots@unit@ratio@y%
			% 
%\message{^^Junit vector ratio 2D searching reference: checking \pgfplots@apply@unit@ratio@reference. feasable=\pgfmathresult < 1: \ifdim\pgfmathresult pt <\pgfplots@ONE YES-> use x\else NO->use y\fi^^J}%
			% and check (1). The condition (2) is irrelevant; it is met
			% anyway.
			\ifdim\pgfmathresult pt<\pgfplots@ONE
				\def\pgfplots@optimum@sofar@axis{x}%
			\else
				\def\pgfplots@optimum@sofar@axis{y}%
			\fi
		\fi
	\else
%\message{^^Junit vector ratio chose \pgfplots@optimum@sofar@axis\space to fulfill explicitly provided limits (at least partially).^^J}%
	\fi
	%
	\ifx\pgfplots@optimum@sofar@axis\pgfutil@empty
		\if1\b@pgfplots@unitvec@is@zero@z
			\def\pgfplots@optimum@sofar@axis{y}%
		\else
			\def\pgfplots@optimum@sofar@axis{z}%
		\fi
		\pgfplotswarning{unit vector ratio axis undetermined}{\pgfplots@optimum@sofar@axis}\pgfeov%
	\fi
	\let\pgfplots@apply@unit@ratio@reference=\pgfplots@optimum@sofar@axis
	\pgfmath@smuggleone\pgfplots@apply@unit@ratio@reference
	\endgroup
}%
\def\pgfplots@apply@unit@ratio@find@reference@checkexplicitlimits{%
	\ifpgfplots@autocompute@ymax \else \def\pgfplots@optimum@sofar@axis{y}\fi
	\ifpgfplots@autocompute@ymin \else \def\pgfplots@optimum@sofar@axis{y}\fi
	\ifpgfplots@autocompute@xmax \else \def\pgfplots@optimum@sofar@axis{x}\fi
	\ifpgfplots@autocompute@xmin \else \def\pgfplots@optimum@sofar@axis{x}\fi
	\ifpgfplots@threedim
		\ifpgfplots@autocompute@zmax \else \def\pgfplots@optimum@sofar@axis{z}\fi
		\ifpgfplots@autocompute@zmin \else \def\pgfplots@optimum@sofar@axis{z}\fi
	\fi
}%

% This is ONLY applied to the value of 'unit vector ratio'. It does
% not touch the current axis scaling factors.
\def\pgfplots@apply@unit@ratio@prepareratios#1 #2 #3 #4\pgfplots@EOI{%
	\def\pgfplots@unit@ratio@x{#1}%
	\def\pgfplots@unit@ratio@y{#2}%
	\def\pgfplots@unit@ratio@z{#3}%
	%
	% 'unit vector ratio' is measured relative to the y axis for 2d
	% and relative to the z axis for 3d plots.
	% renormalize such that it is relative to
	% \pgfplots@apply@unit@ratio@reference.
	%
	% Furthermore, renormalize such that
	% unit@ratio@\pgfplots@apply@unit@ratio@reference is 1.
	\pgfmathreciprocal@{\csname pgfplots@unit@ratio@\pgfplots@apply@unit@ratio@reference\endcsname}%
	\let\pgfplots@loc@TMPa=\pgfmathresult
	\ifpgfplots@threedim
		\if z\pgfplots@apply@unit@ratio@reference
		\else
			\pgfmathmultiply@{\pgfplots@loc@TMPa}{\pgfplots@unit@ratio@z}%
			\let\pgfplots@loc@TMPa=\pgfmathresult
		\fi
		%
		\pgfmathmultiply@{\pgfplots@loc@TMPa}{\pgfplots@unit@ratio@x}%
		\let\pgfplots@unit@ratio@x=\pgfmathresult
		%
		\pgfmathmultiply@{\pgfplots@loc@TMPa}{\pgfplots@unit@ratio@y}%
		\let\pgfplots@unit@ratio@y=\pgfmathresult
		%
		\pgfmathmultiply@{\pgfplots@loc@TMPa}{\pgfplots@unit@ratio@z}%
		\let\pgfplots@unit@ratio@z=\pgfmathresult
	\else
		\if y\pgfplots@apply@unit@ratio@reference
		\else
			\pgfmathmultiply@{\pgfplots@loc@TMPa}{\pgfplots@unit@ratio@y}%
			\let\pgfplots@loc@TMPa=\pgfmathresult
		\fi
		%
		\pgfmathmultiply@{\pgfplots@loc@TMPa}{\pgfplots@unit@ratio@x}%
		\let\pgfplots@unit@ratio@x=\pgfmathresult
		%
		\pgfmathmultiply@{\pgfplots@loc@TMPa}{\pgfplots@unit@ratio@y}%
		\let\pgfplots@unit@ratio@y=\pgfmathresult
		%
		\def\pgfplots@unit@ratio@z{<unused>}%
	\fi
	%
}%


% Computes a new unit vector E_#1 for direction #1 such that
% ||E_#1|| = #2 * ||e_reference||.
% Here, #2 is a scaling factor and e_reference is a reference axis.
% The reference axis is stored in
% \pgfplots@apply@unit@ratio@reference, the macro contains one of
% {x,y,z}.
%
% The data limits for '#1' will be enlarged as well (for 'unit rescale
% keep size').
%
% #1 is the axis which should be scaled (i.e. #1 in {x,y,z}).
% It is allowed if #1 = \pgfplots@apply@unit@ratio@reference. In this
% case, you can provide a scale '#2' to rescale the axis.
%
% #2 is a desired scale, relative to
% \pgfplots@apply@unit@ratio@reference. #2 should be a number without
% unit.
%
% The parameter \pgfplots@apply@unit@ratio@reference is also one of
% {x,y,z}.
%
\def\pgfplots@rescale@unit@vector@reltoreference#1#2{%
	\def\pgfplots@loc@TMPa{0}%
	\if#1\pgfplots@apply@unit@ratio@reference
		\pgfplotsmath@ifapproxequal@dim{#2pt}{1pt}{0.0002pt}{%
		}{%
			\def\pgfplots@loc@TMPa{1}%
		}%
	\else
		\def\pgfplots@loc@TMPa{1}%
	\fi
	\if1\csname b@pgfplots@unitvec@is@zero@#1\endcsname
		\def\pgfplots@loc@TMPa{0}%
	\fi
	\if1\pgfplots@loc@TMPa
		%
		\pgfplots@getscale@unit@vector@reltoreference{#1}{#2}%
		\global\let\pgfplots@glob@TMPa=\pgfmathresult
		%
%\message{Rescaling '#1' by \pgfplots@glob@TMPa.^^J}%
		%
		\pgfmathdivide@{\csname pgfplots@target@unit@scale@inv@#1\endcsname}{\pgfplots@glob@TMPa}%
		\expandafter\let\csname pgfplots@target@unit@scale@inv@#1\endcsname=\pgfmathresult
		%
		\pgfmathreciprocal@\pgfplots@glob@TMPa
		\expandafter\let\csname pgfplots@target@limitrescale@#1@\endcsname=\pgfmathresult
		%
	\fi
}

% Updates the #1 axis limits such that the axis' dimensions
% stay the same after scaling the #1 unit vector by a scale 's'.
%
% PRECONDITION:
%   - the #1 unit vector has been rescaled by a factor s.
%   For example, e_xnew := e_x * 0.5 .
% 
% POSTCONDITION:
%   - the axis limits are enlarged by a factor 1/s such that 
%   1/s (#1max - #1min) * e_xnew = (#1max- #1min) * e_x.
%
% In other words, the unit vector rescale is componensated by
% modifying the axis limits: we want to add an absolute component 'd'
% to the range:
% 1/s (xmax - xmin ) = xmax - xmin +d 
% =>
% d = (1/s - 1) * (xmax - xmin)
%
% The only remaining thing to do is to distribute 'd' to 'xmax' and
% 'xmin'. Typically, 50% to each will be fine, I guess...
% 
% #1: either x, y or z. It denotes the direction which has been
% modified.
% #2: the INVERSE of the scaling factor, #2 = 1/s .
%
\def\pgfplots@apply@unit@vector@rescale@keep@size#1#2{%
	\ifdim#2pt=1pt
	\else
	\if0\pgfplots@unit@vector@rescale@keep@size
		% unit rescale keep size=false : do nothing. Ignore the
		% scaling request.
	\else
		% unit rescale keep size=true|unless limits declared
		%
%\message{'unit rescale keep size': Resizing data range for #1 by #2: from \csname pgfplots@#1min\endcsname:\csname pgfplots@#1max\endcsname\ to}%
		\pgfmathsubtract@{\csname pgfplots@#1max\endcsname}{\csname pgfplots@#1min\endcsname}%
		\begingroup
			\pgf@xa=\pgfmathresult pt
			\pgfmathsubtract@{#2}{1.0}%
			\pgf@xa=\pgfmathresult \pgf@xa% this is 'd'
			%
			% \pgfplots@glob@TMPb : will be subtracted from #1min
			% \pgfplots@glob@TMPc : will be added to #1max
			\pgfplots@if{pgfplots@autocompute@#1min}{%
				\pgfplots@if{pgfplots@autocompute@#1max}{%
					\pgf@xa=0.5 \pgf@xa
					\xdef\pgfplots@glob@TMPb{\pgf@sys@tonumber{\pgf@xa}}%
					\xdef\pgfplots@glob@TMPc{\pgfplots@glob@TMPb}%
				}{%
					\xdef\pgfplots@glob@TMPb{\pgf@sys@tonumber{\pgf@xa}}%
					\xdef\pgfplots@glob@TMPc{0.0}%
				}%
			}{%
				\pgfplots@if{pgfplots@autocompute@#1max}{%
					\xdef\pgfplots@glob@TMPb{0.0}%
					\xdef\pgfplots@glob@TMPc{\pgf@sys@tonumber{\pgf@xa}}%
				}{%
					\if1\pgfplots@unit@vector@rescale@keep@size
						% unit rescale keep size=true : FORCE
						% enlargement!
						\pgf@xa=0.5 \pgf@xa
						\xdef\pgfplots@glob@TMPb{\pgf@sys@tonumber{\pgf@xa}}%
						\xdef\pgfplots@glob@TMPc{\pgfplots@glob@TMPb}%
					\else
						% unit rescale keep size=unless limits declared: 
						% do not scale - all limits are declared
						% explicitly
						\xdef\pgfplots@glob@TMPb{0.0}%
						\xdef\pgfplots@glob@TMPc{0.0}%
					\fi
				}%
			}%
		\endgroup
		\pgfmathsubtract@{\csname pgfplots@#1min\endcsname}{\pgfplots@glob@TMPb}%
		\expandafter\global\expandafter\let\csname pgfplots@#1min\endcsname=\pgfmathresult
		\pgfmathadd@{\csname pgfplots@#1max\endcsname}{\pgfplots@glob@TMPc}%
		\expandafter\global\expandafter\let\csname pgfplots@#1max\endcsname=\pgfmathresult
%\message{\csname pgfplots@#1min\endcsname:\csname pgfplots@#1max\endcsname. [- \pgfplots@glob@TMPb; + \pgfplots@glob@TMPc]^^J}%
		%
		% Update auxiliary data members:
		\pgfplots@visphase@notify@changeofcanvaslimits{#1}%
	\fi
	\fi
}%

% #1: an axis which should be scaled
% #2: the desired final ratio  ||e_#1||/||e_ref||
\def\pgfplots@getscale@unit@vector@reltoreference#1#2{%
	% 
	% If the datascaling transformation is active (which is almost
	% everytime the case here), we have a transformation
	% T^{-1}(x)= 10^scale * x
	% with different scales for every axis.
	%
	% If the datascaling transformation is NOT active, scale is 0
	% and T^{-1} = Identity.
	%
	% Note that the datascaling transformation also has
	% translations (shifts). These are not important here.
	%
	% Goal:
	% compute E_#1 such that
	%   #2* || T^{-1} e_ref || = || T^{-1} E_#1 ||
	% where T^{-1} is the data scaling transformation and e_ref the
	% reference unit vector. Keep in mind that there are
	% *different* data scaling transformations for each axis.
	%
	% We are given e_ref and e_#1 and the desired aspect ratio
	% between e_ref and E_#1, which is available as #2.
	%
	% So: T^{-1} E_#1 :=  s* T^{-1} e_#1 where 
	%  s = #2 * ||T^{-1} e_ref|| / || T^{-1} e_#1 || 
	%    = |10^{scale_ref}| / |10^{scale_#1}| * #2 * || e_ref|| / ||e_#1||.
	% 
	% Then, E_#1 = T ( T^{-1} E_#1 ) = s * e_#1.
	%
	% -> compute 's'! 
	%
	% Part 1: compute
	% #2 * ||e_ref|| / ||e_#1||.
	%
	\def\pgfplots@loc@TMPa{1}%
	\if1\csname b@pgfplots@unitvec@is@zero@#1\endcsname
		\def\pgfplots@loc@TMPa{0}%
	\else
		\if1\csname b@pgfplots@unitvec@is@zero@\pgfplots@apply@unit@ratio@reference\endcsname
			\def\pgfplots@loc@TMPa{0}%
		\fi
	\fi
	\if0\pgfplots@loc@TMPa
		\def\pgfmathresult{16001}%
	\else
		% note that  x^{-1} / y^{-1}  == ( x/y )^{-1} == y/x .
		% consequently, we can use our @inv@[xyz] values here:
		\pgfmathdivide@
			{\csname pgfplots@target@unit@scale@inv@#1\endcsname}%
			{\csname pgfplots@target@unit@scale@inv@\pgfplots@apply@unit@ratio@reference\endcsname}
		\pgfmathmultiply@
			{\pgfmathresult}%
			{#2}%
		\global\let\pgfplots@glob@TMPa=\pgfmathresult
		% 
		% also compute 1/s, required as temporary value:
		%\pgfmathmultiply@
		%	{\csname pgfplots@\pgfplots@apply@unit@ratio@reference @inverseveclength\endcsname}
		%	{\csname pgfplots@target@unit@scale@#1\endcsname}%
		%\ifdim#2pt=1pt
		%\else
		%	\pgfmathdivide@{\pgfmathresult}{#2}%
		%\fi
		%\global\let\pgfplots@glob@TMPb=\pgfmathresult
		%
		% Part 2: handle data scaling trafo scales:
		\begingroup
			\def\pgfplots@tmp@exponentref{0}%
			\def\pgfplots@tmp@exponentK{0}%
			\pgfplots@if{pgfplots@apply@datatrafo@\pgfplots@apply@unit@ratio@reference }{%
				\pgfplots@letcsname{pgfplots@tmp@exponentref}={pgfplots@data@scale@trafo@EXPONENT@\pgfplots@apply@unit@ratio@reference }%
			}{}%
			\pgfplots@if{pgfplots@apply@datatrafo@#1}{%
				\pgfplots@letcsname{pgfplots@tmp@exponentK}={pgfplots@data@scale@trafo@EXPONENT@#1}%
			}{}%
			\c@pgf@counta=\pgfplots@tmp@exponentref\relax
			\advance\c@pgf@counta by-\pgfplots@tmp@exponentK\relax
			\ifnum\c@pgf@counta=0
			\else
				\pgfplotsmathmultiplypowten@{\pgfplots@glob@TMPa}{\c@pgf@counta}%
				\global\let\pgfplots@glob@TMPa=\pgfmathresult
			%	\pgfplotsmathmultiplypowten@{\pgfplots@glob@TMPb}{-\c@pgf@counta}%
			%	\global\let\pgfplots@glob@TMPb=\pgfmathresult
			\fi
			\xdef\pgfplots@glob@TMPc{\the\c@pgf@counta}%
		\endgroup
		\let\pgfmathresult=\pgfplots@glob@TMPa
	\fi
%\message{\string\pgfplots@getscale@unit@vector@reltoreference{#1}{#2} (reference \pgfplots@apply@unit@ratio@reference) = \pgfmathresult.^^J}%
}

% helper for \pgfplots@check@and@apply@datatrafo@for.
\def\pgfplots@compute@number@order@for@trafo@isdimen#1\tocount#2{%
	\edef\pgfplots@loc@TMPa{\pgf@sys@tonumber{#1}}%
	\pgfmathfloatparsenumber{\pgfplots@loc@TMPa}%
	\expandafter\pgfmathfloat@decompose@E\pgfmathresult\relax#2
	\advance#2 by1\relax
}

% helper for \pgfplots@check@and@apply@datatrafo@for.
% 
\def\pgfplots@compute@number@order@for@trafo@isfloat#1\tocount#2{%
	\pgfmathfloatparsenumber{#1}%
	\expandafter\pgfmathfloat@decompose@E\pgfmathresult\relax#2\relax
	\advance#2 by1
}

\def\pgfplots@if@is@float@zero#1#2#3{%
	\pgfmathfloatparsenumber{#1}%
	\pgfmathfloatgetflags{\pgfmathresult}{\c@pgf@countd}%
	\ifnum\c@pgf@countd=0 %
		% ah - it *is* 0.0:
		#2%
	\else
		#3%
	\fi
}

% Initialises the data scale transformation such that it is optimal
% for direction #1 (using its axis limits and the target scaling size).
%
% Note that it will not be applied in any way; and it may still be
% modified.
%
% PRECONDITION:
%   - all axis limits are available in float representation
%   - \pgfplots@set@default@size@options has been called before
% POSTCONDITION:
%   - the scaling transformation is set up,
\def\pgfplots@set@optimal@datatrafo@for@#1{%
	\pgfplots@if{pgfplots@apply@datatrafo@#1}{%
		% initialise data scale transformation 
		%   T(x) = 10^{q-m} * x
		%
		\ifpgfplots@disabledatascaling
			% this here is a waste of time, because the NO-OP trafo
			% will be applied to all coordinates. One could really
			% safe a lot of CPU time when disabledatascaling is enabled...
			% but it requires so much extra cases; I really don't want
			% that!
			\gdef\pgfplots@glob@TMPa{0}%
			\gdef\pgfplots@glob@TMPb{0}%
		\else
			\begingroup
				\let\data@max@order=\c@pgf@counta
				\let\data@cur@order=\c@pgf@countb
				\let\data@dimen=\pgf@xa
				\let\data@tmp=\pgf@xb
				\let\data@dimen@order=\c@pgf@countc
				\let\data@EXPONENT=\c@pgf@countd
				\expandafter\let\expandafter\pgfplots@display@min@float\csname pgfplots@#1min\endcsname
				\expandafter\let\expandafter\pgfplots@display@max@float\csname pgfplots@#1max\endcsname
				\expandafter\let\expandafter\pgfplots@data@min@float\csname pgfplots@data@#1min\endcsname
				\expandafter\let\expandafter\pgfplots@data@max@float\csname pgfplots@data@#1max\endcsname
				\ifpgfplots@autocompute@all@limits
				\else
					\pgfplotscoordmath{#1}{max}{\pgfplots@display@max@float}{\pgfplots@data@max@float}%
					\let\pgfplots@data@max@float=\pgfmathresult
					\pgfplotscoordmath{#1}{min}{\pgfplots@display@min@float}{\pgfplots@data@min@float}%
					\let\pgfplots@data@min@float=\pgfmathresult
				\fi
				%
%\message{minmax = [\pgfplots@data@min@float,\pgfplots@data@max@float]^^J}%
				% Step 1: compute 'm', the data order
				\pgfplots@if@is@float@zero{\pgfplots@data@min@float}{%
					\pgfplots@if@is@float@zero{\pgfplots@data@max@float}{%
						\data@max@order=1 % both are zero. ok.
						% Note that this is '1' due to backwards
						% compatibility.
					}{%
						% one of them is zero. Take the other one!
						\pgfplots@compute@number@order@for@trafo@isfloat
							\pgfplots@data@max@float
							\tocount\data@cur@order
						\data@max@order=\data@cur@order
					}%
				}{%
					\pgfplots@if@is@float@zero{\pgfplots@data@max@float}{%
						% one of them is zero. Take the other one!
						\pgfplots@compute@number@order@for@trafo@isfloat
							\pgfplots@data@min@float
							\tocount\data@cur@order
						\data@max@order=\data@cur@order
					}{%
						% none of them is zero. Compute MAX:
						\pgfplots@compute@number@order@for@trafo@isfloat
							\pgfplots@data@min@float
							\tocount\data@cur@order
						\data@max@order=\data@cur@order
						\pgfplots@compute@number@order@for@trafo@isfloat
							\pgfplots@data@max@float
							\tocount\data@cur@order
						\ifnum\data@cur@order>\data@max@order
							\data@max@order=\data@cur@order
						\fi
					}%
				}%
				%
				%
				%
				% Step 2: compute 'q', the #1-size of the axis.
				%\expandafter\ifx\csname pgfplots@#1\endcsname\pgfutil@empty
					% We have 'width' or 'height' (I always have them).
					%
					% Use the order of these parameters.
					\def\pgfplots@loc@TMPa{#1}%
					\def\pgfplots@loc@TMPb{x}%
					\ifx\pgfplots@loc@TMPa\pgfplots@loc@TMPb
						\data@dimen=\pgfplots@width\relax
					\else
						\if1\pgfplots@compat@scaling@zunitfix@enable
							\data@dimen=\pgfplots@height\relax
						\else
							% this code here belongs to versions up to
							% 1.3.1.
							% It is now deprecated and produces small
							% pixel differences.
							\def\pgfplots@loc@TMPb{y}%
							\ifx\pgfplots@loc@TMPa\pgfplots@loc@TMPb
								\data@dimen=\pgfplots@height\relax
							\else
								\data@dimen=42pt % this is actually different from 1.3.1: there, it was UNDEFINED.
							\fi
						\fi
					\fi
					\pgfplots@compute@number@order@for@trafo@isdimen
						\data@dimen
						\tocount\data@dimen@order
					% This here is to avoid inaccuracies in the final
					% axis rectangle size, see \pgfplots@initsizes:
					%\advance\data@dimen@order by-1
				%\else
					% FIXME:
					% we have either the 'x=1cm' or 'y=1cm' option!
					% How should I initialise the trafo!?
				%	\data@dimen@order=3
				%\fi
				%
%\message{Direction #1: data max order=\the\data@max@order;  data dimen order=\the\data@dimen@order. ^^J}%
				\data@EXPONENT=\data@dimen@order
				\advance\data@EXPONENT by-\data@max@order
				% Now, I introduce a loop which shall avoid cancellation of
				% significant digits.
				%
				% Harmless Example: 
				%  if we have data shift = -3 and 
				%  max = 2e6, min = 1e6, then max-min = 1e6; T(max)-T(min) = 1e3 which is ok.
				%  In this case, the loop won't change anything.
				%
				% Critical Example:
				%  if we have data shift = -3 and
				%  max = 1980, min = 1930 then 
				%    T(max) = 1.98 and T(min) = 1.93
				%  and thus T(max)-T(min) = 0.05 . 
				%  Considering that this is the axis range
				%  in which tick labels and plot points need to be computed, we
				%  only have two or three digits left! That happens because the
				%  prefix '19' is common and is cancelled in the subtraction.
				%  Idea: while T(max)-T(min) < O(10^2) -> increase shift by +1
				%  (and make sure that T(max) < MAX_VALID_TEX_NUMBER).
				%
				\def\pgfplotscoordmathnotifydatascalesetfor##1{}% disable temporarily. We are just testing it.
				\pgfplots@loop@CONTINUEtrue
				\pgfutil@loop
				\pgfplotscoordmath{#1}{datascaletrafo set params}{\the\data@EXPONENT}{0}%
				\pgfplotscoordmath{#1}{datascaletrafo}{\pgfplots@data@min@float}%
				\let\pgfplots@min@fixed=\pgfmathresult
				\ifpgfplots@loop@CONTINUE
					\pgfplotscoordmath{#1}{datascaletrafo}{\pgfplots@data@max@float}%
					\let\pgfplots@max@fixed=\pgfmathresult
					\data@tmp=\pgfplots@max@fixed pt
%\message{Current trafo EXPONENT for #1 direction: \the\data@EXPONENT; original #1 data limits: [\pgfplots@data@min@float:\pgfplots@data@max@float]; current transformed #1 limits: [\pgfplots@min@fixed:\pgfplots@max@fixed]; cancellation check max-min running...^^J}%
					\ifdim\data@tmp<0pt
						% I need absolute values here:
						\multiply\data@tmp by-1\relax
					\fi
					\pgfmathsubtract@{\pgfplots@max@fixed}{\pgfplots@min@fixed}%
					\data@dimen=\pgfmathresult pt
					\pgfplots@loop@CONTINUEfalse
					\ifdim\data@tmp<1500pt % a multiplication with '10' results in max = 15000 which is the upper limit.
						\ifdim\data@dimen<100pt % I guess if max-min = O(100), we have quite good accuracy
							\ifdim\data@dimen<0.0001pt
							\else
								\advance\data@EXPONENT by1
								\pgfplots@loop@CONTINUEtrue
							\fi
						\fi
					\fi
					%--------------------------------------------------
					% \ifdim\data@dimen>1200pt% FIXME : is this here ok!? CHECK IT!
					% 	\ifdim\data@dimen>7999pt
					% 		\advance\data@EXPONENT by-2
					% 	\else
					% 		\advance\data@EXPONENT by-1
					% 	\fi
					% 	\pgfplots@loop@CONTINUEfalse
					% \fi
					%-------------------------------------------------- 
				\pgfutil@repeat
				\xdef\pgfplots@glob@TMPa{\the\data@EXPONENT}%
				\xdef\pgfplots@glob@TMPb{\pgfplots@min@fixed}%
			\endgroup
		\fi
		% COMPLETE INITIALISATION:
%\message{Initialising the data scale transformation in direction #1 to 10^\pgfplots@glob@TMPa*#1 - \pgfplots@glob@TMPb...^^J}%
		\pgfplotscoordmath{#1}{datascaletrafo set params}{\pgfplots@glob@TMPa}{\pgfplots@glob@TMPb}%
	}{%
		% case apply trafo == false:
		\pgfplotscoordmath{#1}{datascaletrafo set params}{0}{0}%
	}%
}


\def\pgfplots@set@optimal@datatrafos@allaxes{%
	\pgfplots@letcsname pgfplots@xmin@unscaled@as@float={pgfplots@xmin}%
	\pgfplots@letcsname pgfplots@xmax@unscaled@as@float={pgfplots@xmax}%
	%
	\pgfplots@letcsname pgfplots@ymin@unscaled@as@float={pgfplots@ymin}%
	\pgfplots@letcsname pgfplots@ymax@unscaled@as@float={pgfplots@ymax}%
	%
	\pgfplots@letcsname pgfplots@zmin@unscaled@as@float={pgfplots@zmin}%
	\pgfplots@letcsname pgfplots@zmax@unscaled@as@float={pgfplots@zmax}%
	%
	\pgfplots@ifneeds@one@uniform@datascale{%
		% Ah - we have to ensure that there is ONE common scale for
		% each unit (x, y, and z have the same).
		%
		% In this case, we need to choose one of the transformations
		% and apply it to all axes -- such that each axis gets the
		% same scale.
		%
		% this mode is used for axis equal and its variants.
		%
		% The strategy to fix the transformation is as follows:
		% 1. we assume that axis limits will be enlarged in order to
		% satisfy 'scale uniformly'. 
		% 2. we assume that the LARGEST axis limit dominates the
		% others.
		% 3. if one of the axes does not have datascaling (i.e. is
		% log scale), we disable all other datascalings.
		%
		% Consequently, we search for the axis with the largest limit
		% - and copy its data scaling to all other axes. If one of the
		% axes is log, that one overrules it and all data scaling
		% effects are disabled..
		\ifpgfplots@disabledatascaling
			\def\pgfplots@loc@TMPd##1{%
				\pgfplotscoordmath{##1}{datascaletrafo set params}{0}{0}%
			}%
		\else
			\begingroup
			\let\pgfplots@data@scale@trafo@EXPONENT@common=\pgfutil@empty
			\ifpgfplots@disabledatascaling
				\def\pgfplots@data@scale@trafo@EXPONENT@common{{0}{0}}% disable scaling!
			\fi
			\def\pgfplots@data@scale@trafo@EXPONENT@common@arg{-}% this should not match anything in this context.
			\pgfplots@if{pgfplots@apply@datatrafo@x}{%
			}{%
				\def\pgfplots@data@scale@trafo@EXPONENT@common{{0}{0}}% disable scaling!
			}%
			\pgfplots@if{pgfplots@apply@datatrafo@y}{%
			}{%
				\def\pgfplots@data@scale@trafo@EXPONENT@common{{0}{0}}% disable scaling!
			}%
			\ifpgfplots@threedim
				\pgfplots@if{pgfplots@apply@datatrafo@z}{%
				}{%
					\def\pgfplots@data@scale@trafo@EXPONENT@common{{0}{0}}% disable scaling!
				}%
			\fi
			\ifx\pgfplots@data@scale@trafo@EXPONENT@common\pgfutil@empty
				% ah - we still need to compute one. ok, search for the
				% largest limit.
				%
				\pgfplots@get@axis@with@largest@limits
				\let\pgfplots@data@scale@trafo@EXPONENT@common@arg=\pgfplotsretval
				%
				% ok, compute data scaling transformation for the target axis:
				\expandafter\pgfplots@set@optimal@datatrafo@for@\pgfplots@data@scale@trafo@EXPONENT@common@arg%
				%
				\pgfplotscoordmath{\pgfplots@data@scale@trafo@EXPONENT@common@arg}{datascaletrafo get params}%
				\let\pgfplots@data@scale@trafo@EXPONENT@common=\pgfmathresult%
			\else
				% hm. early-out - we already have the scaling trafo.
				% return it.
			\fi
			\global\let\pgfplots@glob@TMPa=\pgfplots@data@scale@trafo@EXPONENT@common
			\global\let\pgfplots@glob@TMPb=\pgfplots@data@scale@trafo@EXPONENT@common@arg
			\endgroup
			%
			\xdef\pgfplots@glob@TMPc{\expandafter\pgfutil@firstoftwo\pgfplots@glob@TMPa}%
			%
			\def\pgfplots@loc@TMPd##1{%
				\if ##1\pgfplots@glob@TMPb
					% we need to set the scaling trafo for the target direction
					% (was lost after \endgroup)
					\def\pgfplots@loc@TMPa{\pgfplotscoordmath{##1}{datascaletrafo set params}}%
					\expandafter\pgfplots@loc@TMPa\pgfplots@glob@TMPa%
				\else
					\pgfplotscoordmath{##1}{datascaletrafo set params}{\pgfplots@glob@TMPc}{0}%
					\pgfplotscoordmath{##1}{datascaletrafo}{\csname pgfplots@##1min\endcsname}%
					\pgfplotscoordmath{##1}{datascaletrafo set params}{\pgfplots@glob@TMPc}{\pgfmathresult}%
				\fi
			}%
		\fi
		\pgfplots@loc@TMPd x%
		\pgfplots@loc@TMPd y%
		\ifpgfplots@threedim
			\pgfplots@loc@TMPd z%
		\fi
	}{%
		% optimize individually:
		\pgfplots@set@optimal@datatrafo@for@ x%
		\pgfplots@set@optimal@datatrafo@for@ y%
		\ifpgfplots@threedim
			\pgfplots@set@optimal@datatrafo@for@ z%
		\fi
	}%
	%
}%

% Defines \pgfplotsretval to be one of x, y, or z, such that the
% return value indicates the axis with largest untransformed axis
% limits.
\def\pgfplots@get@axis@with@largest@limits{%
	\begingroup
	\let\pgfplotsretval@extreme=\pgfutil@empty
	\let\pgfplotsretval@extreme@arg=\pgfutil@empty
	\def\pgfplots@@##1{%
		% compute axis range for axis ##1 ...
		\pgfplotscoordmath{default}{parsenumber}{\csname pgfplots@##1min\endcsname}%
		\let\pgfplots@loc@TMPa=\pgfmathresult
		\pgfplotscoordmath{default}{parsenumber}{\csname pgfplots@##1max\endcsname}%
		\pgfplotscoordmath{default}{op}{subtract}{{\pgfmathresult}{\pgfplots@loc@TMPa}}%
		% ... ok, it is in \pgfmathresult.
		\let\candidate=\pgfmathresult
		\ifx\pgfplotsretval@extreme@arg\pgfutil@empty
			% ah: no extreme value so far. use ours.
			\def\pgfplotsretval@extreme@arg{##1}%
			\let\pgfplotsretval@extreme=\candidate
		\else
			\pgfplotscoordmath{default}{if less than}{\pgfplotsretval@extreme}{\candidate}{%
				% update extreme value:
				\def\pgfplotsretval@extreme@arg{##1}%
				\let\pgfplotsretval@extreme=\candidate
			}{%
			}%
		\fi
	}%
	\pgfplots@@ x%
	\pgfplots@@ y%
	\ifpgfplots@threedim
		\pgfplots@@ z%
	\fi
	\let\pgfplotsretval=\pgfplotsretval@extreme@arg
	\pgfmath@smuggleone\pgfplotsretval
	\endgroup
}%

% Initialises the data scale transformation and applies it to any
% user specified options.
%
% PRECONDITION:
%   - all axis limits are available in float representation
%   - \pgfplots@set@default@size@options has been called before
%   - the scaling transformation for direction x is set up
%   (\pgfplots@set@optimal@datatrafo@for@),
% POSTCONDITION:
%   - all axis limits are transformed, but no other axis inputs.
%
% Unit vectors and other axis input parameters will be scaled later.
%
% @see \pgfplots@check@and@apply@datatrafo@for
\def\pgfplots@apply@datatrafo@to@axis@limits#1{%
	\pgfplots@if{pgfplots@apply@datatrafo@#1}{%
		% Transform axis limits:
%\message{#1- display limits BEFORE data transformation: [\csname pgfplots@#1min\endcsname:\csname pgfplots@#1max\endcsname]^^J}%
		\pgfplotscoordmath{#1}{datascaletrafo}{\csname pgfplots@#1min\endcsname}%
		\expandafter\global\expandafter\let\csname pgfplots@#1min\endcsname=\pgfmathresult
		%
		\pgfplotscoordmath{#1}{datascaletrafo}{\csname pgfplots@#1max\endcsname}%
		\expandafter\global\expandafter\let\csname pgfplots@#1max\endcsname=\pgfmathresult
%\message{#1- display limits after data transformation: [\csname pgfplots@#1min\endcsname:\csname pgfplots@#1max\endcsname]^^J}%
	}{%
		% case apply trafo == false:
		\expandafter\let\csname pgfplots@#1min@unscaled@as@float\endcsname=\pgfutil@empty
		\expandafter\let\csname pgfplots@#1max@unscaled@as@float\endcsname=\pgfutil@empty
	}%
}