summaryrefslogtreecommitdiff
path: root/graphics/ellipse/ellipse.dtx
blob: c22bfe2a58f0aea1b333a2d0df46a46dc8b2d02e (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
% \iffalse meta-comment
%
% Copyright (C) 2015 Daan Leijen
% -------------------------------------------------------
% 
% This file may be distributed and/or modified under the
% conditions of the LaTeX Project Public License, either version 1.2
% of this license or (at your option) any later version.
% The latest version of this license is in:
%
%    http://www.latex-project.org/lppl.txt
%
% and version 1.2 or later is part of all distributions of LaTeX 
% version 1999/12/01 or later.
%
% \fi
%
% \iffalse
%<*driver>
\ProvidesFile{ellipse.dtx}
%</driver>
%<package>\NeedsTeXFormat{LaTeX2e}[1999/12/01]
%<package>\ProvidesPackage{ellipse}
%<*package>
    [2004/11/05 v1.0 .dtx ellipse file]
%</package>
%<package>\RequirePackage{pict2e}
%
%<*driver>
\documentclass{ltxdoc}
\usepackage{graphicx}
\usepackage{ellipse}[2004/11/05]
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{hyperref}
\usepackage{lmodern}
\usepackage{xcolor}
\usepackage{wrapfig}
\newcommand\kk[1]{\iota_{#1}}
\newcommand\sk[1]{\rho_{#1}}
\newcommand\sint[1]{\textit{sint}_{#1}}
\newcommand\cost[1]{\textit{cost}_{#1}}
\newcommand\sign[1]{\pm_{#1}}
\newcommand\csqrt{\textit{csqrt}}
\renewcommand\cos{\textit{cos}}\renewcommand\sin{\textit{sin}}
\renewcommand\arctan{\textit{arctan}}\renewcommand\tan{\textit{tan}}
\renewcommand\max{\textit{max}}\renewcommand\sin{\textit{sin}}
\providecolor{teal}{HTML}{008080}
\providecolor{purple}{HTML}{800080}
\providecolor{navy}{HTML}{000080}
\providecolor{maroon}{HTML}{800000}
\providecolor{floralwhite}{HTML}{FFFAF0}
\providecolor{ivory}{HTML}{FFFFF0}
\providecolor{white}{HTML}{FFFFFF}
\providecolor{transparent}{named}{white}
\providecolor{gainsboro}{HTML}{DCDCDC}
\hypersetup{colorlinks=true,citecolor=navy,linkcolor=navy,urlcolor=navy,filecolor=navy,bookmarksdepth=3,bookmarksopenlevel=1}
\makeatletter
\DeclareRobustCommand*\package[2][]{%
  \def\@tempa{#1}%
  \ifx\@tempa\@empty
    \textsf{#2}%
  \else
    \href{http://mirrors.ctan.org/macros/latex/#1}{\textsf{#2}}%
  \fi
}%
\DeclareRobustCommand*\pkgpicte{\package[contrib/pict2e/pict2e.pdf]{pict2e}}
\newsavebox{\@ebox}
\newcommand*\@unit[1]{\strip@pt\dimexpr#1\relax}%
\newcommand*\ellipbox[1]{%
  \begingroup
  \savebox{\@ebox}{#1}%
  \setlength{\unitlength}{1pt}%
  \hspace*{0.8ex}%
  \begin{picture}(0,0)%
   \put(\@unit{0.5\wd\@ebox},\@unit{0.5\ht\@ebox - 0.5\dp\@ebox}){%
     \ellipse{\@unit{0.8ex + 0.5\wd\@ebox}}{\@unit{0.8ex + 0.5\ht\@ebox}}%
  }%
  \end{picture}%
  \usebox{\@ebox}%
  \hspace{0.8ex}%
  \endgroup%
}
\makeatother
\EnableCrossrefs         
\CodelineIndex
\RecordChanges
\begin{document}
  \setlength\emergencystretch{3em}  
  \DocInput{ellipse.dtx}
  \PrintChanges
  \setcounter{IndexColumns}{2}%
  \PrintIndex
\end{document}
%</driver>
% \fi
%
% \CheckSum{0}
%
% \CharacterTable
%  {Upper-case    \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z
%   Lower-case    \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z
%   Digits        \0\1\2\3\4\5\6\7\8\9
%   Exclamation   \!     Double quote  \"     Hash (number) \#
%   Dollar        \$     Percent       \%     Ampersand     \&
%   Acute accent  \'     Left paren    \(     Right paren   \)
%   Asterisk      \*     Plus          \+     Comma         \,
%   Minus         \-     Point         \.     Solidus       \/
%   Colon         \:     Semicolon     \;     Less than     \<
%   Equals        \=     Greater than  \>     Question mark \?
%   Commercial at \@     Left bracket  \[     Backslash     \\
%   Right bracket \]     Circumflex    \^     Underscore    \_
%   Grave accent  \`     Left brace    \{     Vertical bar  \|
%   Right brace   \}     Tilde         \~}
%
%
% \changes{v1.0}{2015/03/24}{Initial version}
%
% \GetFileInfo{ellipse.dtx}
%
% \DoNotIndex{\newcommand,\renewcommand,\newenvironment,\renewenvironment}
% \DoNotIndex{\providecommand,\def,\edef,\let,\gdef,\xdef,\global,\newtoks}
% \DoNotIndex{\RequirePackage,\DeclareOption,\ProcessOptions,\ExecuteOptions}
% \DoNotIndex{\input,\InputIfFileExists}
% \DoNotIndex{\@ifdefinable,\@ifundefined,\@percentchar}
% \DoNotIndex{\AtBeginDocument,\AtEndOfPackage}
% \DoNotIndex{\PackageError,\PackageWarning,\PackageWarningNoLine,\PackageInfo}
% \DoNotIndex{\MessageBreak,\typeout}
%
% \DoNotIndex{\z@,\z@skip,\p@,\@ne,\tw@,\thr@@,\@iv,\two@fourteen,\strip@pt}
% \DoNotIndex{\the,\if,\else,\or,\fi,\ifnum,\ifdim,\ifcase,\ifodd}
% \DoNotIndex{\advance,\multiply,\divide}
% \DoNotIndex{\@tfor,\do}
% \DoNotIndex{\csname,\endcsname,\begingroup,\endgroup}
% \DoNotIndex{\expandafter,\afterassignment,\noexpand}
% \DoNotIndex{\@ovxx,\@ovyy,\@ovdx,\@ovdy}
% \DoNotIndex{\undefined,\dimexpr,\relax,\space,\protect,\begin}
% \DoNotIndex{\@tempdima,\@tempdimb,\@tempdimc,\@tempdimd,\dimen@,\@tempa}
% \DoNotIndex{\@tempswafalse,\@tempswatrue,\if@tempswa,\iffalse,\ifx,\ignorespaces}
%
% \title{The {\linethickness{0.6pt}\ellipbox{\textsf{ellipse}}} package}
% \author{Daan Leijen \\ \texttt{daan@microsoft.com}}
%
% \maketitle
%  
% \newcommand\pictexample[1]{%
%   \setlength{\unitlength}{10pt}%
%   \raisebox{-30pt}{\begin{picture}(6,8)%
%   \linethickness{0.4pt}\roundjoin%
%   \color{gainsboro}%
%   \put(0,0){\multiput(0,0)(1,0){9}{\line(0,1){6}}%
%    \multiput(0,0)(0,1){7}{\line(1,0){8}}%  
%   }%
%   \put(0,0){\color{gray}%
%     {\vector(1,0){8}}\put(-0.5,6.5){$y$}%
%     {\vector(0,1){6}}\put(8.5,-0.5){$x$}%
%   }%
%   \color{black}%
%   \linethickness{0.8pt}%
%   #1\end{picture}}%
% } 
% 
% \section{Introduction}
%
% \LaTeX{} has many advanced graphics packages now, the most extensive are 
% \href{https://www.ctan.org/pkg/pgf}{\textsf{tikz}}
% and \href{https://www.ctan.org/pkg/pstricks-base}{\textsf{pstricks}}.
% However, these are also large packages that take long to load and
% may not always work on all drivers. The standard \pkgpicte{} package removes many of the previous
% limitations of the `old' \LaTeX{} |picture| environment and makes it a \emph{lean
% and portable} alternative to the more full featured packages. However, even though it can
% draw circles and circle arcs well, it lacks the ability to draw ellipses and elliptical
% arcs. This package adds these functions on top of the standard \pkgpicte{} primitives 
% (i.e. the |\cbezier| command).
%
% \section{Drawing ellipses}
%
% \noindent
% \DescribeMacro\ellipse
% \DescribeMacro{\ellipse*}
% \marg{x-radius}\marg{y-radius}\\
% \strut\\
% These commands draw an ellipse with the specified radi\"\i. The |\ellipse| command draws
% a stroked ellipse with the current |\linethickness| while |\ellipse*| draws a filled
% ellipse with the current |\color|. For example:
%
% \noindent\begin{minipage}{0.7\linewidth}%
% \begin{verbatim}
% \setlength{\unitlength}{10pt}%
% \begin{picture}(6,8)
%   \linethickness{0.8pt}%
%   \put(6,3){\color{teal}\ellipse*{2}{3}}%
%   \put(3,3){\color{blue}\ellipse{3}{2}}%
% \end{picture}
% \end{verbatim}\end{minipage}
% \pictexample{%
%   \put(6,3){\color{teal}\ellipse*{2}{3}}%
%   \put(3,3){\color{blue}\ellipse{3}{2}}%
% }\\[2ex]
%
% \noindent
% \DescribeMacro\earc
% \DescribeMacro{\earc*}
% \oarg{start-angle$\rangle$|,|$\langle$end-angle}\marg{x-radius}\marg{y-radius}\\
% \strut\\
% These commands draw part of an ellipse with the specified radi\"\i. 
% The |\earc| command draws
% a stroked elliptical arc with the current |\linethickness| while |\earc*| draws a filled
% elliptical `pie slice' with the current |\color|. The optional argument specifies a start and 
% end-angle in degrees which must be between $-720$ and $720$ (but can be fractional).
% The endings of the arcs are determined by the \emph{cap} setting: |\buttcap| (default),
% |\roundcap| (add half disc), or |\squarecap| (add half square).
%
% \noindent\begin{minipage}{0.7\linewidth}%
% \begin{verbatim}
% \put(3,3){%
%   \color{blue}\roundcap\earc[135,330]{3}{2}}%
% \put(6,3){%
%   \color{teal}\earc*[-45,90]{2}{3}}%
% \end{verbatim}
% \end{minipage}
% \pictexample{%
%   \put(3,3){\color{blue}\roundcap\earc[135,330]{3}{2}}%
%   \put(6,3){\color{teal}\earc*[-45,90]{2}{3}}%
% }\\[2ex]
%
% \noindent
% \DescribeMacro\elliparc
% \oarg{initial}\marg{center-x}\marg{center-y}\marg{x-rad}\marg{y-rad}\marg{start-angle}\marg{end-angle}\\
%
% \noindent The core elliptical arc routine. These are to be used with path commands, like
% |\lineto|, |\moveto|, |\strokepath|, etc, and can draw an elliptical arc at any center point. 
% The optional argument specifies the initial drawing
% action: the default is $0$ (|\lineto|) which draws a line to the arc starting point,
% the value $1$
% (|\moveto|) just moves to the starting point, and $2$ does nothing as an initial action. 
% If the start angle is larger than the end angle, the arc is drawn clockwise, and otherwise
% anti-clockwise.\\[1ex]
%
% \noindent\begin{minipage}{0.7\linewidth}%
% \begin{verbatim}
% \elliparc[1]{3}{3}{3}{2}{90}{270}%
% \elliparc{5}{3}{2}{2}{-90}{90}%
% \closepath\strokepath
% \color{teal}%
% \moveto(1,3)
% \elliparc{3}{3}{2}{1}{-135}{135}%
% \closepath
% \fillpath
% \end{verbatim}\end{minipage}
% \pictexample{%
%   \elliparc[1]{3}{3}{3}{2}{90}{270}%
%   \elliparc{5}{3}{2}{2}{-90}{90}%
%   \closepath\strokepath
%   \color{teal}%
%   \moveto(1,3)
%   \elliparc{3}{3}{2}{1}{-135}{135}%
%   \closepath
%   \fillpath
% }\\[2ex]
%
% \noindent Note how the two initial arcs are automatically connected by a line
% segment from $(3,1)$ to $(5,1)$ (due to the default optional argument of $0$ that
% uses a |\lineto| command to the starting point of the arc). Similarly, we use
% such initial line segment and a |\closepath| to draw the triangular side of the
% inner ellipse.
%
% \subsection{Rotated ellipses}
%
% There is no direct command to rotate an ellipse but you can use the
% standard |\rotatebox| command from the \package[required/graphics/grfguide.pdf]{graphicx} package. For example:
%
% \noindent\begin{minipage}{0.7\linewidth}%
% \begin{verbatim}
% \put(3,3){%
%   \rotatebox[origin=c]{45}{\ellipse{3}{2}}%
% }%
% \end{verbatim}\end{minipage}
% \pictexample{%
% \put(3,3){\rotatebox[origin=c]{45}{\ellipse{3}{2}}%
% }%
% }\\[2ex]
%
% \subsection{Using the picture environment inline}
%
% The standard \LaTeX{} |picture| environment is nowadays quite
% powerful and convenient. Read the latest \pkgpicte{} documentation
% and ``The unknown \emph{picture} environment'' \cite{picture} for more information. 
% One particularly nice feature is that we
% can create a picture as |\begin{picture}(0,0)| to give it zero
% space. This can be used for example to define an |\ellipbox| command
% like:\\[1ex]
%
% \noindent\begin{minipage}{0.6\linewidth}%
% \begin{verbatim}
% Boxed numbers: 
%   \ellipbox{1}, \ellipbox{123}.  
% \end{verbatim}
% \end{minipage}
% \begin{minipage}{0.38\linewidth}%
% Boxed numbers: \ellipbox{1}, \ellipbox{123}.
% \end{minipage}\\[1ex]
%
% \noindent We also used this command to draw the ellipse in the title
% of this article, and it is defined as:
% \begin{verbatim}
%\newsavebox{\@ebox}
%\newcommand*\@unit[1]{\strip@pt\dimexpr#1\relax}%
%\newcommand*\ellipbox[1]{%
%  \begingroup
%  \savebox{\@ebox}{#1}%
%  \setlength{\unitlength}{1pt}%
%  \hspace*{0.8ex}%
%  \begin{picture}(0,0)%
%   \put(\@unit{0.5\wd\@ebox},\@unit{0.5\ht\@ebox - 0.5\dp\@ebox}){%
%     \ellipse{\@unit{0.8ex + 0.5\wd\@ebox}}{\@unit{0.8ex + 0.5\ht\@ebox}}%
%  }%
%  \end{picture}%
%  \usebox{\@ebox}\hspace{0.25ex}\endgroup}
% \end{verbatim}
% \noindent This is not the best code possible but it hopefully gives you a good
% idea on how to implement your own boxes. Note the use of the |\@unit| macro
% to convert dimensions to units, which is also why we need to set the |\unitlength|
% to |1pt| here.
%
% \StopEventually{}
%
% \begin{thebibliography}{9}
% \raggedright
%
% \bibitem{abst} M.~Abramowitz and I.A.~Stegun: \textit{Handbook of Mathematical Functions},
%  \url{people.math.sfu.ca/~cbm/aands}, 1964
%
% \bibitem{picture} Claudio Beccari: \emph{The unknown \emph{picture} environment},
%   TUGBoat, vol. 33(1), 2012. \url{tug.org/TUGboat/tb33-1/tb103becc-picture.pdf}
%
% \bibitem{ellipse} Luc Maisonobe: \textit{Drawing an elliptical arc using polylines, quadratic or cubic B\'ezier lines.}
%    \url{www.spaceroots.org/documents/ellipse/elliptical-arc.pdf}, 2003
%
% \bibitem{rajan:atan} S.~Rajan, Sichun Wang, R.~Inkol, and A.~Joyal: \textit{Efficient approximations for the arctangent function}. 
%  In Signal Processing Magazine, vol. 23(3), pages 108--111, May 2006
%
% \end{thebibliography}
%
% \clearpage
%
% \newcommand\abs[1]{\lvert #1\rvert}
% \newcommand\rarg[1]{$\langle$\textit{#1}$\rangle$}
% \newcommand\xellipse{\mathcal{E}}
%
% \section{Elliptical arcs as B\'ezier curves}
% \begin{figure}\begin{center}
% \setlength\unitlength{18pt}
% \begin{picture}(12,9)(-1.5,-0.5)%
%   \linethickness{0.4pt}\roundjoin%
%   \iffalse
%    \color{gainsboro}%
%    \put(0,0){%
%      \multiput(-2,-1)(1,0){13}{\line(0,1){9}}%
%      \multiput(-2,-1)(0,1){10}{\line(1,0){12}}%
%    }%
%   \fi
%   \color{black}%
%   \put(0,0){%
%     \put(0,-0.5){\vector(0,1){8}}\put(8.5,-0.5){$x$}%
%     \put(-1.5,0){\vector(1,0){11}}\put(-0.5,6.5){$y$}%
%   }%
%   \put(4,3.3){\color{navy}\earc{5}{3}
%     \put(3,-2){$\xellipse$}%%  
%     \color{teal}%
%     \put(-5.5,0){\line(1,0){11}}%
%     \put(0,-3.5){\line(0,1){8}}%
%     \put(3,0){\vector(-1,0){3}\vector(1,0){2}\raisebox{0.3ex}{$a$}}%
%     \put(0,-1.5){\vector(0,1){1.5}\vector(0,-1){1.5}$\,b$}%
%     \color{maroon}%
%     \put(0,0){\moveto(0,0)\elliparc{0}{0}{5}{3}{30}{120}\closepath\strokepath}%
%     {\linethickness{2pt}\earc[30,120]{5}{3}}%
%     \put(3.7,2.2){$p_1$}\put(-2,3.1){$p_2$}%
%     \put(2.25,2.92){\moveto(0,0)\lineto(1.35,-0.84)\strokepath\circle*{0.1}$\;q_1$}%
%     \put(-0.075,3.22){\moveto(0,0)\lineto(-1.56,-0.39)\strokepath\circle*{0.1}$\,q_2$}%
%     \color{black}
%     \linethickness{0.2pt}\arc[0,30]{1}\arc[0,120]{2}%
%     \put(1,0.2){$\alpha_1$}\put(1,1.8){$\alpha_2$}%
%   }%
%   \color{navy}\put(4,-0.3){$\,c_x$}\put(0,2.9){$\,c_y$}%
%   %
% \end{picture}\end{center}
% \caption{Approximating an elliptical arc with a cubic B\'ezier curve. The center of the ellipse
%  is at $(c_x,c_y)$ with a horizontal radius of $a$ and a vertical one $b$. The elliptical arc goes
%  from $\alpha_1$ to $\alpha_2$ and is approximated with a thick red cubic B\'ezier curve. The curve
%  starts at $p_1$ and ends in $p_2$ with two control points $q_1$ and $q_2$. The curve was drawn
%  using the command \texttt{\textbackslash{}elliparc\{4\}\{3.3\}\{5\}\{3\}\{30\}\{120\}}.}
% \label{fig:ellipse}
% \end{figure}
% \begin{figure}
% \setlength\unitlength{20pt}\begin{center}
% \begin{picture}(12,8)(-1.5,-0.5)%
%   \linethickness{0.4pt}\roundjoin%
%   \iffalse
%   \put(0,0){\color{gainsboro}%
%     \multiput(-2,-1)(1,0){13}{\line(0,1){8}}%
%     \multiput(-2,-1)(0,1){9}{\line(1,0){12}}%
%   }%
%   \fi
%   \put(4,3){\color{navy}\earc{5}{3}\put(-4,1){$\xellipse$}\color{teal}%
%     \put(-5.5,0){\line(1,0){11}}%
%     \put(0,-3.5){\line(0,1){7}}%
%     \put(-2,0){\vector(-1,0){3}\vector(1,0){2}\raisebox{0.3ex}{$a$}}%
%     \put(0,-1.5){\vector(0,1){1.5}\vector(0,-1){1.5}$\,b$}%
%     \color{black}%
%     \put(0,0){\circle{6}}%
%     \put(0,0){\vector(3.6,2.08){3.6}\vector(2.16,2.08){2.16}}%
%     \put(0,0){\color{maroon}%
%      \put(3.6,0){\line(0,1){2.08}\circle*{0.1}}%
%      \put(3.2,-0.5){$a\cos(t_1)$}%
%      \put(2.16,0){\line(0,1){2.08}\circle*{0.1}}%
%      \put(1.3,-0.5){$\cos(t_1)$}%
%     }%
%     \linethickness{0.2pt}\arc[0,30]{1}\arc[0,43.9]{2}%
%     \put(1,0.2){$\alpha_1$}\put(1.65,1.25){$t_1$}%
%   }%
%   %
% \end{picture}\end{center}
% \caption{The relation between the parametric angle $t_1$ and the angle $\alpha_1$ to the point on the ellipse.
%   All points on the ellipse are defined by the parametric equation $\xellipse(t) = (c_x + a\cdot\cos(t), c_y + b\cdot\sin(t))$}
% \label{fig:parametric}
% \end{figure}
% %
% %
% Drawing an ellipse or part of an ellipse (\emph{elliptical arc}) using B\'ezier
% curves requires some math to determine the right control points of the B\'ezier curve.
% Figure~\ref{fig:ellipse} establishes some notation. We do not consider rotated
% ellipses here and always use $a$ for the $x$-radius and $b$ for the $y$-radius.
% We are interested in finding the B\'ezier curve between the $\alpha_1$ and
% $\alpha_2$ angles, which implies finding the starting 
% point $p_1$, the end point $p_2$ and the control points $q_1$ and $q_2$. 
%
% Each point on an ellipse is determined by the following parametric equation:
% \[ \xellipse(t) = (c_x + a\cdot\cos(t), c_y + b\cdot\sin(t)) \]
% where $t$ is the
% parametric angle. The parametric angle $t$ is just a property of the ellipse and has no
% `real' counterpart. Figure~\ref{fig:parametric} gives some helpful intuition how
% the $\alpha$ angles and $t$ angles are related: we can imagine drawing a unit circle inside
% an ellipse where for every $t$ angle on the unit circle we have a corresponding point
% and angle $\alpha$ on the ellipse. From the definition of $\xellipse$ it 
% is straightforward to derive a parametric angle $t_i$ for some $\alpha_i$: 
% \[ t_i = \arctan_2(\frac{\sin(\alpha_i)}{b},\frac{\cos(\alpha_i)}{a}) \]
%
% \noindent Given this relation, the start and end points of our curve are simply:
% \begin{align*}
% p_1 &= \xellipse(t_1)\\
% p_2 &= \xellipse(t_2)
% \end{align*}
%
% \noindent To be able to calculate optimal control points $q$ we need to also determine the 
% tangent of each point on the ellipse, which is given by the derivative of $\xellipse$:
% \[ \xellipse'(t) = (-a\cdot\sin(t), b\cdot\cos(t)) \]
% 
% \noindent 
% The derivation of the optimal B\'ezier control points for an ellipse is quite involved, 
% see~\cite{ellipse} for a nice overview. 
% For a quadratic B\'ezier curve, it turns out the optimal control points are determined as:
% \begin{align*}
% q_1 &= p_1 + \tan(\frac{t_2 - t_1}{2})\cdot\xellipse'(t_1)\\
%     &= p_2 - \tan(\frac{t_2 - t_1}{2})\cdot\xellipse'(t_2)
% \end{align*}
% %
% \noindent while for a cubic B\'ezier curve, one solution for optimal control points is:
% %
% \begin{align*}
% q_1 &= p_1 + \kappa\cdot\xellipse'(t_1)\\
% q_2 &= p_2 - \kappa\cdot\xellipse'(t_2)\\
% \kappa &= \sin(t_2 - t_1)\frac{\sqrt{4 + 3\tan^2(\frac{t_2 - t_1}{2})}-1}{3}
% \end{align*}
% \noindent We will use cubic bezier curves since they look best. However, a na\"\i{}ve implementation
% may be too expensive in \LaTeX: if we count the expensive operations, we need about 11 $\cos$/$\sin$ 
% operations, plus a $\sqrt{}$ and 2 $\arctan$ operations.
% %
% \subsection{Optimizing elliptic arc equations} 
% Fortunately, we can improve upon this. First we note:
% \begin{flalign*}
% t_i &= \arctan_2(\frac{\sin(\alpha_i)}{b},\frac{\cos(\alpha_i)}{a}) & \hfill\\ 
%     &= \arctan(\frac{a}{b}\tan(\alpha_i)) \\
%     &= \arctan(\kk{i}) & \mbox{(introducing $\kk{i}$ for $\frac{a}{b}\tan(\alpha_i)$)}
% \end{flalign*}
% \noindent where we write $\kk{i}$ for $\frac{a}{b}\tan(\alpha_i)$. Now, 
% \begin{flalign*}
% \cost{i} &= \cos(t_i)\\
%           &= \cos(\arctan(\kk{i})) & \mbox{(geometry and pythagorean theorem)}\\
%           &= \sign{i}\frac{1}{\sqrt{1 + \kk{i}^2}} \\
% \mbox{with}\\
% \sign{i} &= \textsf{if}\; \cos(\alpha_i) < 0\; \textsf{then}\; -\; \textsf{else}\; + 
% \end{flalign*}
% Later we will see how we can efficiently calculate the square root term, but first 
% do the same derivation for the $\sin$ function:
% \begin{flalign*}
% \sint{i}&= \sin(t_i) \\
%         &= \sin(\arctan(\frac{a}{b}\tan(\alpha_i))) & \\
%           &= \sin(\arctan(\kk{i})) \\
%           &= \sign{i}\frac{\kk{i}}{\sqrt{1 + \kk{i}^2}}  
% \end{flalign*}
% \noindent Note that the interaction between the $\sin$ and $\kk{i}$ term (whose sign is determined by $\tan(\alpha_i)$) allows us to reuse the sign function used for $\cost{i}$. 
% %
% Using the previous equalities we can restate the parametric equations in terms of $\sint{i}$
% and $\cost{i}$:
% \begin{flalign*}
% \xellipse_i &= (c_x + a\cdot\cost{i}), c_y + b\cdot\sint{i}) \\
% \xellipse'_i &= (-a\cdot\sint{i}, b\cdot\cost{i})
% \end{flalign*}
% %
% \noindent This takes care of $p_1$ and $p_2$. The control points $q$ still need $\sin(t_2 - t_1)$ and $\tan(\frac{t_2 - t_1}{2})$.
% The halving rule on $\tan$ gives us:
% \[
% \tan(\frac{t_2 - t_1}{2}) = \frac{1 - \cos(t_2 - t_1)}{\sin(t_2 - t_1)} \mbox{\quad(\cite[page 71, 4.3.20]{abst})}
% \]
% \noindent So that leaves $\sin(t_2 - t_1)$ and $\cos(t_2 - t_1)$. Using the addition laws it follows:
% \begin{flalign*}
% \sin(t_2 - t_1) = \sint2\cost1 - \cost2\sint1 & \mbox{\quad(\cite[page 72, 4.3.16]{abst})}\\
% \cos(t_2 - t_1) = \cost2\cost1 + \sint2\sint1 & \mbox{\quad(\cite[page 72, 4.3.17]{abst})}
% \end{flalign*}
%
% \subsection{Circular square roots}
% \noindent Now, we only need two $\tan$ operations to calculate the initial $\kk{1}$ and $\kk{2}$ terms but we still have
% three square roots: $\sqrt{1 + \kk{i}^2}$ and $\sqrt{4 + 3\tan^2(\frac{t_2 - t_1}{2})}$. 
% Fortunately, both have the form $\sqrt{x^2 + y^2}$. For this form, we can make a very good 
% \begin{wrapfigure}[10]{o}{0.3\textwidth}
% \setlength\unitlength{3pt}
% \begin{picture}(30,30)(-15,-15)%
%   \linethickness{0.4pt}\roundjoin%
%   \iffalse
%   \put(0,0){\color{gainsboro}%
%     \multiput(-15,-15)(1,0){31}{\line(0,1){30}}%
%     \multiput(-15,-15)(0,1){31}{\line(1,0){30}}%
%   }%
%   \fi
%   \color{gainsboro}%
%   \put(0,0){\put(0,-15){\line(0,1){30}}%
%     \put(-15,0){\line(1,0){30}}%
%   }%
%   \color{black}%
%   \put(0,0){\circle{20}\put(-9,3){$\,x^2 + y^2$}%
%     \color{maroon}\moveto(14.14,0)\lineto(0,14.14)\lineto(-14.14,0)\lineto(0,-14.14)
%     \closepath\strokepath
%     \put(3,13){$\frac{1}{\sqrt{2}}(x+y)$}%
%     \color{navy}\moveto(10,10)\lineto(-10,10)\lineto(-10,-10)\lineto(10,-10)
%     \closepath\strokepath
%     \put(6,-12){$\,\max(x,y)$}%
%   }
%   %
% \end{picture}\iffalse\caption{Estimating an initial value for $\sqrt{x^2+y^2}$}\fi
% \end{wrapfigure}
% initial guess for the square root, since this is the parametric equation for a circle.
% The two good initial guesses form a `square' and `diamond' around this circle, namely
% $\max(\abs{x},\abs{y})$ and $\frac{1}{\sqrt{2}}\abs{x+y}$. Each one can be superior depending if $x$ and $y$
% are close or not, but it can be shown that the best choice is always the largest of these.
% Using this guess as an initial seed, we can do a standard Newton-Raphson iteration to 
% find a the square root where we only need 2 or 3 steps to achieve the desired precision. 
% Let's define a `circular square root' function $\csqrt$ such that
% $\csqrt(x,y) \approx \sqrt{x^2 + y^2}$ as:
% \begin{align*}
% \csqrt(x,y) = &\textsf{let} & sqr & = x^2 + y^2 & \hfill \\
%             &               & x_0 & = \max(\abs{x},\abs{y},\frac{1}{\sqrt{2}}\abs{x+y}) & \\
%             &               & x_1 & = (x_0 + \frac{sqr}{x_0})/2 & \hspace{\textwidth}\\
%             &               & x_2 & = (x_1 + \frac{sqr}{x_1})/2 &\\
%             &  \textsf{in} & x_2 & 
% \end{align*}
% 
% \subsection{The optimized elliptical B\'ezier equations}
% \noindent Taking it all together, we get the following equations for a cubic B\'ezier curve approximation of an elliptical arc,
% where we assume as input the center point $(c_x,c_y)$, the $x$- and $y$-radius $(a,b)$, and a start and end angle
% $\alpha_1$ and $\alpha_2$. % It is assumed that $\alpha_1 \ne \alpha_2$ and $a \ge 0, b \ge 0$. Of course,
% with bezier curves one should build a full ellipse of parts where for each part $\rvert\alpha_1 - \alpha_2\lvert \le 90$.
% Given these parameters, the start and end point $p_1$ and $p_2$, and the control points $q_1$ and $q_2$ are
% defined as:
% \begin{flalign*}
% p_1 &= \xellipse_1 & \\
% p_2 &= \xellipse_2\\
% q_1 &= p_1 + \kappa\cdot\xellipse'_1\\
% q_2 &= p_2 - \kappa\cdot\xellipse'_2\\
% \xellipse_i &= (c_x + a\cdot\cost{i}, c_y + b\cdot\sint{i})\\
% \xellipse'_i &= (-a\cdot\sint{i}, b\cdot\cost{i})
% \end{flalign*}
% The $\cost{i}$ and $\sint{i}$ are calculated as:
% \begin{flalign*}
% \sint{i} & = \sign{i}\frac{\kk{i}}{\sk{i}} & \\
% \cost{i} & = \sign{i}\frac{1}{\sk{i}} 
% \end{flalign*}
% with
% \begin{flalign*}
% \kk{i} &= \frac{a}{b}\tan(\alpha_i) & \\
% \sk{i} &= \csqrt(1,\kk{i}) \;\;(\approx \sqrt{1 + \kk{i}^2}) \\
% \sign{i} &= \textsf{if}\; \cos(\alpha_i) < 0\; \textsf{then}\; -\; \textsf{else}\; + 
% \end{flalign*}
% And finally, the $\kappa$ term can be defined as:
% \begin{flalign*}
% \kappa    &= \sint{21}\frac{\kappa_{sqrt}-1}{3} & 
% \end{flalign*}
% with
% \begin{flalign*}
% \sint{21} & = \sint2\cost1 - \cost2\sint1 \;\;(=\sin(t_2 - t_1)) & \\
% \cost{21} & = \cost2\cost1 + \sint2\sint1 \;\;(=\cos(t_2 - t_1))\\
% \kappa_{tan} &= \frac{1 - \cost{21}}{\sint{21}} \;\;\mbox{(note: divides by zero if $\alpha_1 = \alpha_2$)}\\
% \kappa_{sqrt} &= \csqrt(\sqrt{4},\sqrt{3}\cdot\kappa_{tan}) \;\;(\approx \sqrt{4 + 3\kappa_{tan}^2})
% \end{flalign*}
%
%\section{Implementation}
%
% Generally, we use e-\TeX{} division to divide dimensions, where we 
% divide \rarg{dim$_1$} by \rarg{dim$_2$} using:
% |\dimexpr 1pt * |\rarg{dim$_1$}|/|\rarg{dim$_2$}|\relax| 
% since it keeps a 64-bit intermediate result for such `scaling' expressions. 
% Note that both \rarg{dim}
% expressions occur in an integer context and \TeX{} will convert
% them to numbers automatically (i.e. in |sp| units).
%

% \subsection{Generic math and trigonometry routines}
%
% \begin{macro}{\pIIe@csedef}
% \marg{csname}\textit{pattern}\marg{body}\\
% Define a macro by a \textit{csname}. Just like the |\csedef| function 
% from  \textsf{etoolbox} package 
%    \begin{macrocode}
\providecommand*\pIIe@csedef[1]{\expandafter\edef\csname #1\endcsname}
%    \end{macrocode}
% \end{macro}
%
%\begin{macro}{\pIIe@ellip@csqrt}
% \marg{dimen$_x$}\marg{dimen$_y$}\marg{dimreg$_{res}$}\\
% Calculates $res \approx \sqrt{x^2 + y^2}$ and caches previous results
% for efficiency.
%
% Overwrites |\@ovxx|,|\@ovyy|,|\@ovdx|,|\@ovdy|,|\@tempa|, and |\dimen@|.
%    \begin{macrocode}
\newcommand*\pIIe@ellip@csqrt[3]{%
  \@ovxx=#1\relax
  \ifdim\@ovxx<\z@\@ovxx-\@ovxx\fi
  \@ovyy=#2\relax
  \ifdim\@ovyy<\z@\@ovyy-\@ovyy\fi
  \edef\pIIe@csname{@csqrt(\number\@ovxx,\number\@ovyy)}%
  \expandafter\ifx\csname\pIIe@csname\endcsname\relax
    \pIIe@ellip@csqrt@%    
    \pIIe@csedef{\pIIe@csname}{\the\dimen@}%
    #3\dimen@
  \else
    #3\dimexpr\csname\pIIe@csname\endcsname\relax
  \fi
}
%    \end{macrocode}
%\end{macro}
%\begin{macro}{\pIIe@ellip@csqrt@}
% Internal routine: calculates |\dimen@| $\approx \sqrt{x^2 + y^2}$.
% where $x \ge 0$ and $y \ge 0$, and |\@ovxx| $=x$ and |\@ovyy| $=y$.
% 
% Overwrites |\@ovdx|,|\@ovdy|, and |\@tempa|.
%    \begin{macrocode}
\newcommand*\pIIe@ellip@csqrt@{%
%    \end{macrocode}
% First determine $\max(x,y,\frac{1}{\sqrt{2}}(x+y))$ in |\dimen@|.
% Put the sum $x+y$ in |\@ovdx|.
%    \begin{macrocode}  
  \@ovdx\@ovxx
  \advance\@ovdx by \@ovyy
%    \end{macrocode}
% Put initial guess in |\dimen@| $=\max(\abs{x},\abs{y},\frac{1}{\sqrt{2}}(x+y))$.
%    \begin{macrocode}  
  \dimen@0.7071067\@ovdx
  \ifdim\dimen@<\@ovyy\dimen@\@ovyy\fi
  \ifdim\dimen@<\@ovxx\dimen@\@ovxx\fi
%    \end{macrocode}
% To prevent overflowing \TeX{} dimensions we only do
% a further Newton-Raphson approximation if the sum $x+y$ is less than 128pt.
% Otherwise, for our application, the initial guess is still very precise since $x \ll y$ in that case.
%    \begin{macrocode}  
  \ifdim\@ovdx<128\p@    
%    \end{macrocode}
% Set |\@ovxx| to $x^2 + y^2$
%    \begin{macrocode}  
    \edef\@tempa{\strip@pt\@ovxx}%
    \@ovxx\@tempa\@ovxx
    \edef\@tempa{\strip@pt\@ovyy}%
    \@ovyy\@tempa\@ovyy
    \advance\@ovxx by \@ovyy
%    \end{macrocode}
% Do two steps of Newton-Raphson (should we do three?)
%    \begin{macrocode}    
    \advance\dimen@ by \dimexpr1pt * \@ovxx/\dimen@\relax
    \divide\dimen@ by 2%
    \advance\dimen@ by \dimexpr1pt * \@ovxx/\dimen@\relax
    \divide\dimen@ by 2%
  \fi
%    \end{macrocode}
% Result is |\dimen@|.
%    \begin{macrocode}    
}
%    \end{macrocode}
% \end{macro}
%


% \begin{macro}{\pIIe@atan@}
%
% Approximate the $\arctan$ using 
% \[x\cdot\frac{\pi}{4} - x \cdot (\abs{x} - 1) \cdot (0.2447 + 0.0663\cdot\abs{x})\]
% This approximation was described by Rajan et al.~\cite{rajan:atan}.
%
% The \cmd{pIIe@atan@} computes the arctan of |\dimen@| which must be between $-1$ and $1$, and stores it in |\dimen@| again.
% Overwrites |\@tempdim|(|a|,|b|,|c|,|d|),|\@tempa|, and |\dimen@|.
%    \begin{macrocode}
\newcommand*\pIIe@atan@{%
%    \end{macrocode}
% |\dimen@| contains $x$.
%    \begin{macrocode}  
  \@tempdima\dimen@
%    \end{macrocode}
% Set |\@dimtmpb| to $\abs{x}$
%    \begin{macrocode}  
  \@tempdimb\@tempdima
  \ifdim\@tempdimb<\z@\@tempdimb-\@tempdimb\fi
  \dimen@0.0663\@tempdimb
  \advance\dimen@ 0.2447pt\relax
  \advance\@tempdimb -1pt\relax
  \edef\@tempa{\strip@pt\@tempdimb}%
  \dimen@\@tempa\dimen@
  \edef\@tempa{\strip@pt\@tempdima}%
  \dimen@\@tempa\dimen@
  \dimen@-\dimen@
%    \end{macrocode}
% Add $x\cdot\frac{\pi}{4}$ ($\approx 0.7853\cdot x$).
%    \begin{macrocode}   
  \advance\dimen@ 0.7853\@tempdima
}
%    \end{macrocode}
% \end{macro}

% \begin{macro}{\pIIe@atantwo}
% \marg{dimen$_y$}\marg{dimen$_x$}\marg{dimreg$_{res}$}\\
% Calculate \rarg{res} $= \arctan_2(y,x)$ and caches the result for later use.
%
% Overwrites |\@tempdim|(|a|,|b|,|c|,|d|),|\@tempa|, and |\dimen@|.
% Both $y$ and $x$ must be dimensions.
%    \begin{macrocode}
\newcommand*\pIIe@atantwo[3]{%
  \edef\pIIe@csname{@atan2(\number\dimexpr#1\relax,\number\dimexpr#2\relax)}%
  \expandafter\ifx\csname\pIIe@csname\endcsname\relax
    \pIIe@atantwo@{#1}{#2}{#3}%    
    \pIIe@csedef{\pIIe@csname}{\the\dimexpr#3\relax}%
  \else
    #3\dimexpr\csname\pIIe@csname\endcsname\relax
  \fi
}
%    \end{macrocode}
% \end{macro}

% \begin{macro}{\pIIe@atantwo@}
% \marg{dimen$_y$}\marg{dimen$_x$}\marg{dimreg$_{res}$}\\
% Calculate \rarg{res} $= \arctan_2(y,x)$.
% Overwrites |\@tempdim|(|a|,|b|,|c|,|d|),|\@tempa|, and |\dimen@|.
% Both $y$ and $x$ must be dimensions.
%    \begin{macrocode}
\newcommand*\pIIe@atantwo@[3]{%
  \@tempdima\dimexpr#2\relax
  \@tempdimb\dimexpr#1\relax
%    \end{macrocode}
% Handle extremes
%    \begin{macrocode}  
  \ifdim\@tempdima=\z@\relax
    \ifdim\@tempdimb>\z@\relax\dimen@90\p@
    \else\ifdim\@tempdimb<\z@\relax\dimen@-90\p@
    \else\dimen@0\p@
    \fi\fi
  \else
%    \end{macrocode}
% Save angle adjustment term in |\@tempdimd|.
%    \begin{macrocode}  
    \@tempdimd\z@
    \ifdim\@tempdima<\z@\relax
      \ifdim\@tempdimb<\z@\relax\@tempdimd-180\p@
      \else\@tempdimd180\p@
      \fi
    \fi
%    \end{macrocode}
% Divide $\frac{y}{x}$ and check if $-1 \le \frac{y}{x} \le 1$.
%    \begin{macrocode}  
    \dimen@\dimexpr1pt * \@tempdimb/\@tempdima\relax
    \@tempdimc\dimen@
    \ifdim\@tempdimc<\z@\relax\@tempdimc-\@tempdimc\fi
    \ifdim\@tempdimc>\p@\relax
%    \end{macrocode}
% Use the equality $\arctan(x) = \pm\frac{1}{2}\pi - \arctan(\frac{1}{x})$ 
% to stay within the valid domain of |\pIIe@atan@|. The sign $\pm$ is
% positive when $x \ge 0$ and negative otherwise.
%    \begin{macrocode}
      \dimen@\dimexpr1pt * \@tempdima/\@tempdimb\relax
      \ifdim\dimen@<\z@\relax\def\@tempsign{-}\else\def\@tempsign{}\fi
      \pIIe@atan@
      \dimen@-\dimen@
      \advance\dimen@ by \@tempsign1.5707pt\relax
    \else
      \pIIe@atan@
    \fi
%    \end{macrocode}
% And convert back to degrees ($\frac{180}{\pi} \approx 57.29578$)
%    \begin{macrocode}
    \dimen@57.29578\dimen@
%    \end{macrocode}
% Apply angle adjustment 
%    \begin{macrocode}    
    \advance\dimen@ by \@tempdimd
  \fi
  #3\dimen@%
}
%    \end{macrocode}
% \end{macro}

% \subsection{Sub routines for drawing an elliptical arc}
%
%
% \begin{macro}{\pIIe@noneto}
% \marg{dimen$_x$}\marg{dimen$_y$}\\
% Ignores its arguments. Used as a no-op instead of |\pIIe@lineto| or |pIIe@moveto|.
%    \begin{macrocode}
\newcommand*\pIIe@noneto[2]{}
%    \end{macrocode}
% \end{macro}
%
%\begin{macro}{\pIIe@ellip@sincost@}
% \marg{$\alpha_i$}\marg{$i$ = |one| or |two|}\\
% Calculate $\sint{i}$ and $\cost{i}$ into the |\@ellip|(|sin|/|cos|)$i$.
% Assumes |\@ellipratio| $=\frac{a}{b}$.
%    \begin{macrocode}
\newcommand*\pIIe@ellip@sincost@[2]{%
%    \end{macrocode}
% Put the $\sin(\alpha_i)$ and $\cos(\alpha_i)$ into |\@tempdima| and |\@tempdimb|.
%    \begin{macrocode}    
  \CalculateSin{#1}%
  \CalculateCos{#1}% 
  \@tempdima\UseSin{#1}\p@   
  \@tempdimb\UseCos{#1}\p@
%    \end{macrocode}
% Check for extremes where $\tan = \pm\infty$.
%    \begin{macrocode}    
  \ifdim\@tempdima=\p@\relax
    \pIIe@csedef{@ellipsin#2}{1}%
    \pIIe@csedef{@ellipcos#2}{0}%
  \else\ifdim\@tempdima=-\p@\relax
    \pIIe@csedef{@ellipsin#2}{-1}%
    \pIIe@csedef{@ellipcos#2}{0}%
  \else
%    \end{macrocode}
% Calculate $\kk{i}$ in |\@tempdimc| and $\sqrt{1 + \kk{i}^2}$ in |\@tempdimd|, 
% and derive $\sint{i}$ and $\cost{i}$.
%    \begin{macrocode}    
    \@tempdimc\@ellipratio\dimexpr1pt * \@tempdima/\@tempdimb\relax
    %\typeout{ i#2=\the\@tempdimc, sin(#1)=\the\@tempdima}%
    \pIIe@ellip@csqrt{\p@}{\@tempdimc}\@tempdimd
    \ifdim\@tempdimb<\z@\relax\@tempdimd-\@tempdimd\fi
    \pIIe@csedef{@ellipsin#2}{\strip@pt\dimexpr1pt * \@tempdimc/\@tempdimd\relax}%
    \pIIe@csedef{@ellipcos#2}{\strip@pt\dimexpr1pt * \p@/\@tempdimd\relax}%
  \fi\fi
}
%    \end{macrocode}
% \end{macro}

%\begin{macro}{\pIIe@ellip@sincost}
% \marg{$\alpha_1$}\marg{$\alpha_2$}\\
% Calculate $\sint{i}$ and $\cost{i}$ into the |\@ellip|(|sin|/|cos|)(|one|/|two|).
% Assumes |\@ovro|$=a$ and |\@ovri|$=b$ with $b \ne 0$.
%    \begin{macrocode}
\newcommand*\pIIe@ellip@sincost[2]{%
%    \end{macrocode}
% Set |\@ellipratio| to the ratio $\frac{a}{b}$.
%    \begin{macrocode}    
  %\typeout{ calc sin cos: angles (#1,#2), radii: (\the\@ovro,\the\@ovri)}%
  \edef\@ellipratio{\strip@pt\dimexpr1pt * \@ovro/\@ovri\relax}%
%    \end{macrocode}
% And calculate $\sint{i}$ and $\cost{i}$
%    \begin{macrocode} 
  \pIIe@ellip@sincost@{#1}{one}%
  \pIIe@ellip@sincost@{#2}{two}%
  %\typeout{ sincos(a=#1)=(\@ellipsinone,\@ellipcosone), sincos(a=#2)=(\@ellipsintwo,\@ellipcostwo), }%
}
%    \end{macrocode}
% \end{macro}
%
%
%\begin{macro}{\pIIe@omega}
% \marg{$i$ = |one| or |two|}\\
% Calculates $\xellipse_i$ into |\@tempdima| and |\@tempdimb|.
% Assumes |\@ovro|$=a$ and |\@ovri|$=b$.
%    \begin{macrocode}
\newcommand*\pIIe@omega[3]{%
  \@tempdima\csname @ellipcos#3\endcsname\@ovro
  \advance\@tempdima by #1\relax
  \@tempdimb\csname @ellipsin#3\endcsname\@ovri
  \advance\@tempdimb by #2\relax
}
%    \end{macrocode}
% \end{macro}

%\begin{macro}{\pIIe@omegai}
% \marg{$i$ = |one| or |two|}\\
% Calculates $\xellipse'_i$ into |\@tempdimc| and |\@tempdimd|.
% Assumes |\@ovro|$=a$ and |\@ovri|$=b$.
%    \begin{macrocode}
\newcommand*\pIIe@omegai[1]{%
  \@tempdimc\csname @ellipsin#1\endcsname\@ovro
  \@tempdimc-\@tempdimc
  \@tempdimd\csname @ellipcos#1\endcsname\@ovri
}
%    \end{macrocode}
% \end{macro}

%\begin{macro}{\pIIe@ellip@kappa}
% Calculates $\kappa$, expects |\@ellip|(|sin|/|cos|)(|one|/|two|) to be defined.
%    \begin{macrocode}
\newcommand*\pIIe@ellip@kappa{%
%    \end{macrocode}
% Calculate $\sint{21}$ and $\cost{21}$ in |\@tempdima| and |\@tempdimb|.
%    \begin{macrocode}  
  \@ovyy\@ellipsinone\p@
  \@ovxx\@ellipcosone\p@
  \@tempdima\@ellipcostwo\@ovyy
  \@tempdima-\@tempdima
  \advance\@tempdima by \@ellipsintwo\@ovxx
  \@tempdimb\@ellipcostwo\@ovxx
  \advance\@tempdimb by \@ellipsintwo\@ovyy
%    \end{macrocode}
% First test if $\sint{21} = 0$ to prevent division by zero. In that
% case, it must have been that $\alpha_1 = \alpha_2$ and we set $\kappa$ to zero
% so it the control points become equal to the start and end point.
%    \begin{macrocode}  
  \ifdim\@tempdima=\z@\relax
    \edef\@ellipkappa{0}%
  \else
%    \end{macrocode}
% Calculate $\kappa_{tan}$ in |\dimen@|
%    \begin{macrocode}  
    \dimen@\dimexpr1pt - \@tempdimb\relax
    \dimen@\dimexpr1pt * \dimen@/\@tempdima\relax
%    \end{macrocode}
% Calculate $\kappa_{sqrt}$ in |\dimen@|
%    \begin{macrocode}  
    \pIIe@ellip@csqrt{2\p@}{1.73205\dimen@}{\dimen@}%
%    \end{macrocode}
% Calculate $\kappa$ in |\dimen@|
%    \begin{macrocode}  
    \advance\dimen@ by -\p@
    \divide\dimen@ by 3%
    \edef\@tempa{\strip@pt\@tempdima}%
    \dimen@\@tempa\dimen@
    \edef\@ellipkappa{\strip@pt\dimen@}%
  \fi
  %\typeout{ calculated kappa: \@ellipkappa}%
}
%    \end{macrocode}
% \end{macro}

% \subsection{Core routines for drawing elliptical arcs}
%
%\begin{macro}{\pIIe@elliparc@}
% \oarg{start}\marg{$c_x$}\marg{$c_y$}\marg{$\alpha_1$}\marg{$\alpha_2$}\\
% Assumes that the radii are set as |\@ovro|$=a$ and |\@ovri|$=b$.
% This is the main routine for drawing an elliptic arc, where $\abs{\alpha_2 - \alpha_1}\,\le 90$.
%    \begin{macrocode}
\newcommand*\pIIe@elliparc@[5]{%
  %\typeout{elliparc: #1, center: (#2, #3), radius (\the\@ovro, \the\@ovri),angle (#4, #5)}%
%    \end{macrocode}
% Define initial action: 0 (lineto), 1(moveto), or 2 (nothing)
%    \begin{macrocode}    
  \ifcase #1\relax
      \let\@ellip@startto\pIIe@lineto
  \or \let\@ellip@startto\pIIe@moveto
  \or \let\@ellip@startto\pIIe@noneto%
  \else\PackageWarning{ellipse}{Illegal initial action in \protect\elliparc: %
          must be one of 0 (lineto), 1 (moveto) or 2 (do nothing) but I got: #1}%
  \fi
%    \end{macrocode}
% Perform just the start action if the radii are zero
%    \begin{macrocode}    
  \ifdim\@ovro=\z@\relax\@ovri\z@\fi
  \ifdim\@ovri=\z@\relax
    \@ellip@startto{#2}{#3}%
  \else
%    \end{macrocode}
% Calculate $\sint{i}$ and $\cost{i}$ first into the |\@ellip|(|sin|/|cos|)(|one|/|two|) registers.
%    \begin{macrocode}    
    \pIIe@ellip@sincost{#4}{#5}%
%    \end{macrocode}
% And draw..
%    \begin{macrocode}
    \pIIe@elliparc@draw{#2}{#3}%
  \fi
}
%    \end{macrocode}
% \end{macro}
%
%\begin{macro}{\pIIe@elliparc@t}
% \oarg{start}\marg{$c_x$}\marg{$c_y$}\marg{$t_1$}\marg{$t_2$}\\
% Assumes that the radii are set as |\@ovro|$=a$ and |\@ovri|$=b$.
% Moreover, this routine take $t_1$ and $t_2$ as the angles of the ellipse equation (instead of real angles $\alpha_i$).
% This routine is mainly for other libraries that may already have computed the $t$ angles
% and need a bit more efficiency.
%    \begin{macrocode}
\newcommand*\pIIe@elliparc@t[5]{%
%    \end{macrocode}
% Define initial action: 0 (lineto), 1(moveto), or 2 (nothing)
%    \begin{macrocode}    
  \ifcase #1\relax
      \let\@ellip@startto\pIIe@lineto
  \or \let\@ellip@startto\pIIe@moveto
  \or \let\@ellip@startto\pIIe@noneto%
  \else\PackageWarning{ellipse}{Illegal initial action in \protect\elliparc: %
          must be one of 0 (lineto), 1 (moveto) or 2 (do nothing) but I got: #1}%
  \fi
%    \end{macrocode}
% Perform just the start action if the radii are zero
%    \begin{macrocode}    
  \ifdim\@ovro=\z@\relax\@ovri\z@\fi
  \ifdim\@ovri=\z@\relax
    \@ellip@startto{#2}{#3}%
  \else
%    \end{macrocode}
% Calculate $\sint{i}$ and $\cost{i}$ first into the |\@ellip|(|sin|/|cos|)(|one|/|two|) registers.
%    \begin{macrocode}    
    \CalculateSin{#4}\CalculateCos{#4}%
    \edef\@ellipsinone{\UseSin{#4}}%
    \edef\@ellipcosone{\UseCos{#4}}%
    \CalculateSin{#5}\CalculateCos{#5}%
    \edef\@ellipsintwo{\UseSin{#5}}%
    \edef\@ellipcostwo{\UseCos{#5}}%
%    \end{macrocode}
% And draw..
%    \begin{macrocode}
    \pIIe@elliparc@draw{#2}{#3}%
  \fi
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{pIIe@elliparc@draw}
% \marg{$c_x$}\marg{$c_y$}\\
% Expects $a =$|\@ovro|, $b$ = |\@ovri|, and  |\@ellip|(|sin|/|cos|)(|one|/|two|) defined.
% |\@ellipstarto| should contain the initial drawing action and is called with an initial
% $x$ and $y$ coordinate (usually equal to |\pIIe@lineto|,|\pIIe@moveto|, or |pIIe@noneto|).
%    \begin{macrocode}
\newcommand*\pIIe@elliparc@draw[2]{%
% Calculate $\kappa$.
%    \begin{macrocode}    
  \pIIe@ellip@kappa%
%    \end{macrocode}
% Now we are ready to compute the control points. First $p_1$.
%    \begin{macrocode}
  \pIIe@omega{#1}{#2}{one}%
  %\typeout{ point one: (\the\@tempdima,\the\@tempdimb)}%
%    \end{macrocode}
% The coordinates are added to the path if and how necessary:
%    \begin{macrocode}
  \@ellip@startto\@tempdima\@tempdimb
%    \end{macrocode}
% Add control point $q_1$
%    \begin{macrocode}
  \pIIe@omegai{one}%
  \advance\@tempdima by \@ellipkappa\@tempdimc
  \advance\@tempdimb by \@ellipkappa\@tempdimd
  \pIIe@add@nums\@tempdima\@tempdimb
  %\typeout{ control one: (\the\@tempdima,\the\@tempdimb)}%
%    \end{macrocode}
% Calculate $p_2$
%    \begin{macrocode}
  \pIIe@omega{#1}{#2}{two}%
%    \end{macrocode}
% Add control point $q_1$
%    \begin{macrocode}
  \pIIe@omegai{two}%
  \@tempdimc\@ellipkappa\@tempdimc
  \@tempdimd\@ellipkappa\@tempdimd
  \@tempdimc-\@tempdimc
  \@tempdimd-\@tempdimd
  \advance\@tempdimc by \@tempdima
  \advance\@tempdimd by \@tempdimb
  \pIIe@add@nums\@tempdimc\@tempdimd
  %\typeout{ control two: (\the\@tempdimc,\the\@tempdimd)}%
%    \end{macrocode}
% And finally add $p_2$ to the path
%    \begin{macrocode}
  \pIIe@add@CP\@tempdima\@tempdimb
  %\typeout{ point two: (\the\@tempdima,\the\@tempdimb)}%
  \pIIe@addtoGraph\pIIe@curveto@op
}
%    \end{macrocode}
% \end{macro}
%
%
% \subsection{Normalizing elliptical arcs}
%
% \begin{macro}{pIIe@elliparc}
% \begin{macro}{pIIe@@elliparc}
% \oarg{start}\marg{$c_x$}\marg{$c_y$}\marg{a}\marg{b}\marg{$\alpha_1$}\marg{$\alpha_2$}\\
% \strut\\
% These two macros check the arguments and normalize the angles.
%    \begin{macrocode}
\newcommand*\pIIe@elliparc[7][0]{%
%    \end{macrocode}
% Store the radii in registers, where |\@ovro|$=a$ and |\@ovri|$=b$.
%    \begin{macrocode}  
  \@ovro #4\relax
  \@ovri #5\relax
  \iffalse%dim\@ovro=\@ovri
%    \end{macrocode}
% Call the circular arc routine if the x- and y-radius are equal
%    \begin{macrocode}  
    \pIIe@arc[#1]{#2}{#3}{#4}{#6}{#7}
  \else
%    \end{macrocode}
% Normalize angles such that the arc angle $\abs{\alpha_2 - \alpha_1}\,\le 720$.
% Store the arc angle in |\@arclen|.
%    \begin{macrocode}  
    \ifdim \@ovro<\z@ \pIIe@badcircarg\else
      \ifdim \@ovri<\z@ \pIIe@badcircarg\else
        \@arclen #7\p@ \advance\@arclen -#6\p@
        \ifdim \@arclen<\z@ \def\@tempsign{-}\else\def\@tempsign{}\fi
        \ifdim \@tempsign\@arclen>720\p@
          \PackageWarning {ellipse}{The arc angle is reduced to -720..720}%
          \@whiledim \@tempsign\@arclen>720\p@ \do {\advance\@arclen-\@tempsign360\p@}%
          \@tempdima #6\p@ \advance\@tempdima \@arclen
          \edef\@angleend{\strip@pt\@tempdima}%
          \pIIe@@elliparc{#1}{#2}{#3}{#6}{\@angleend}%
        \else
          \pIIe@@elliparc{#1}{#2}{#3}{#6}{#7}%
        \fi
      \fi
    \fi
  \fi
}
%    \end{macrocode}
% |\pIIe@@elliparc| divides the total angle in parts of at most $90$ degrees.
% Assumes |\@ovro|$=a$ and |\@ovri|$=b$, and |\@arclen| the arc angle, with |\@tempsign| 
% sign of the arc angle.
%    \begin{macrocode}  
\newcommand*\pIIe@@elliparc[5]{%
  \begingroup
  \ifdim \@tempsign\@arclen>90\p@
%    \end{macrocode}
% If the arc angle is too large, the arc is recursively
% divided into 2 parts until the arc angle is at most 90~degrees.
%    \begin{macrocode}
    \divide\@arclen 2%
    \@tempdima #4\p@\advance\@tempdima by \@arclen
    \edef\@anglemid{\strip@pt\@tempdima}%
    \def\@tempa{\pIIe@@elliparc{#1}{#2}{#3}{#4}}%
    \expandafter\@tempa\expandafter{\@anglemid}%
    \def\@tempa{\pIIe@@elliparc{2}{#2}{#3}}%
    \expandafter\@tempa\expandafter{\@anglemid}{#5}%
  \else
%    \end{macrocode}
% The arc angle is smaller than 90 degrees.
%    \begin{macrocode}
    \pIIe@elliparc@{#1}{#2}{#3}{#4}{#5}%
  \fi
  \endgroup
}%
%    \end{macrocode}
% \end{macro}
%\end{macro}

% \subsection{Drawing elliptical arcs}
%
%\begin{macro}{\elliparc}
%\begin{macro}{\pIIeelliparc}
% \oarg{start}\marg{center-x}\marg{center-y}\marg{radius-x}\marg{radius-y}\marg{start-angle}\marg{end-angle}\\
% \strut\\
% The main elliptical arc drawing routine. We start with |\pIIeelliparc| to avoid conflicts with
% other packages.
%    \begin{macrocode}
\newcommand*\pIIeelliparc[7][0]{%
  \@killglue
  \pIIe@elliparc[#1]{#2\unitlength}{#3\unitlength}{#4\unitlength}{#5\unitlength}{#6}{#7}%
  \ignorespaces%
}
\ifx\undefined\elliparc\else
  \PackageWarning{ellipse}{\protect\elliparc\space is redefined}%
\fi
\let\elliparc\pIIeelliparc
%    \end{macrocode}
% \end{macro}
% \end{macro}

% \begin{macro}{\earc}
% \begin{macro}{\earc*}
% |[|\rarg{$\alpha_0$}|,|\rarg{$\alpha_1$}|]|\marg{radius-x}\marg{radius-y}\\
% \strut\\
% The \cmd{\earc} command generalizes the standard \cmd{\arc} with both
% a $x$- and $y$-radius. The |\earc*| version draws a filled elliptical arc while
% |\earc| only strokes the elliptical arc. Both take an optional comma separated pair of 
% angles which specify the initial and final angle ($0$ and $360$ by default). 
% We start with \cmd{\pIIeearc} to avoid conflicts with otherpackages.
%    \begin{macrocode}
\newcommand*\pIIeearc
  {\@ifstar{\@tempswatrue\pIIe@earc@}{\@tempswafalse\pIIe@earc@}}
\newcommand*\pIIe@earc@[3][0,360]{\pIIe@earc@@(#1){#2}{#3}}
\def\pIIe@earc@@(#1,#2)#3#4{%
  \if@tempswa
    \pIIe@moveto\z@\z@
    \pIIe@elliparc{\z@}{\z@}{#3\unitlength}{#4\unitlength}{#1}{#2}%
    \pIIe@closepath\pIIe@fillGraph
  \else
    \pIIe@elliparc[1]{\z@}{\z@}{#3\unitlength}{#4\unitlength}{#1}{#2}%
    \pIIe@strokeGraph
  \fi}
\ifx\undefined\earc\else
  \PackageWarning{ellipse}{\protect\earc\space is redefined}%
\fi
\let\earc\pIIeearc
%    \end{macrocode}
% \end{macro}
% \end{macro}

% \begin{macro}{\ellipse}
% \begin{macro}{\ellipse*}
% \marg{radius-x}\marg{radius-y}\\
% \strut\\
% The \cmd{\ellipse} draws an ellipse with the specified $x$- and $y$-radius. 
% The |\ellipse*| version draws a filled ellipse. 
% We start with \cmd{\pIIeellipse} to avoid conflicts with other packages.
% The implementation redirects immediately to |earc| which generalized this command.
%    \begin{macrocode}
\newcommand*\pIIeellipse
  {\@ifstar{\@tempswatrue\pIIe@earc@}{\@tempswafalse\pIIe@earc@}}
\let\ellipse\pIIeellipse
%    \end{macrocode}
% \end{macro}
% \end{macro}

%
% \Finale
\endinput