summaryrefslogtreecommitdiff
path: root/Master/texmf-dist/source/latex/l3experimental/l3draw/l3draw-points.dtx
blob: 69ca8b6ae73735e531dfa50ccf2b000fa4a6f82f (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
% \iffalse meta-comment
%
%% File: l3draw-points.dtx Copyright(C) 2018 The LaTeX3 Project
%
% It may be distributed and/or modified under the conditions of the
% LaTeX Project Public License (LPPL), either version 1.3c of this
% license or (at your option) any later version.  The latest version
% of this license is in the file
%
%    http://www.latex-project.org/lppl.txt
%
% This file is part of the "l3experimental bundle" (The Work in LPPL)
% and all files in that bundle must be distributed together.
%
% -----------------------------------------------------------------------
%
% The development version of the bundle can be found at
%
%    https://github.com/latex3/latex3
%
% for those people who are interested.
%
%<*driver>
\RequirePackage{expl3}
\documentclass[full]{l3doc}
\begin{document}
  \DocInput{\jobname.dtx}
\end{document}
%</driver>
% \fi
%
% \title{^^A
%   The \pkg{l3draw-points} package\\ Calculating points^^A
% }
%
% \author{^^A
%  The \LaTeX3 Project\thanks
%    {^^A
%      E-mail:
%        \href{mailto:latex-team@latex-project.org}
%          {latex-team@latex-project.org}^^A
%    }^^A
% }
%
% \date{Released 2018/03/05}
%
% \maketitle
%
% \begin{implementation}
%
% \section{\pkg{l3draw-points} implementation}
%
%    \begin{macrocode}
%<*initex|package>
%    \end{macrocode}
%
%    \begin{macrocode}
%<@@=draw>
%    \end{macrocode}
%
% This sub-module covers more-or-less the same ideas as
% \texttt{pgfcorepoints.code.tex}, though the approach taken to returning
% values is different: point expressions here are processed by expansion
% and return a co-ordinate pair in the form |{|\meta{x}|}{|\meta{y}|}|.
% Equivalents of following \pkg{pgf} functions are deliberately omitted:
% \begin{itemize}
%   \item \cs{pgfpointorigin}: Can be given explicitly as |{0pt}{0pt}|.
%   \item \cs{pgfextractx}, \cs{pgfextracty}: Available by applying
%     \cs{use_i:nn}/\cs{use_ii:nn} or similar to the \texttt{x}-type
%     expansion of a point expression.
%    \item \cs{pgfgetlastxy}: Unused in the entire \pkg{pgf} core, may be
%      emulated by \texttt{x}-type expansion of a point expression, then using
%      the result.
% \end{itemize}
% In addition, equivalents of the following \emph{may} be added in future but
% are currently absent:
% \begin{itemize}
%   \item \cs{pgfpointcylindrical}, \cs{pgfpointspherical}: The usefulness
%     of these commands is not currently clear.
%   \item \cs{pgfpointborderrectangle}, \cs{pgfpointborderellipse}: To be
%     revisited once the semantics and use cases are clear.
%   \item \cs{pgfqpoint}, \cs{pgfqpointscale}, \cs{pgfqpointpolar},
%     \cs{pgfqpointxy}, \cs{pgfqpointxyz}: The expandable approach taken in
%     the code here, along with the absolute requirement for \eTeX{}, means
%     it is likely many use cases for these commands may be covered in other
%     ways. This may be revisited as higher-level structures are constructed.
% \end{itemize}
%
% \subsection{Support functions}
%
% \begin{macro}[EXP]{\@@_point_process:nn}
% \begin{macro}[EXP]{\@@_point_process_auxi:nn, \@@_point_process_auxi:fn}
% \begin{macro}[EXP]{\@@_point_process_auxii:nw}
% \begin{macro}[EXP]{\@@_point_process:nnn}
% \begin{macro}[EXP]{\@@_point_process_auxiii:nnn, \@@_point_process_auxiii:ffn}
% \begin{macro}[EXP]{\@@_point_process_auxiv:nw}
%   Execute whatever code is passed to extract the $x$ and $y$ co-ordinates.
%   The first argument here should itself absorb two arguments. There is
%   also a version to deal with two co-ordinates: common enough to justify a
%   separate function.
%    \begin{macrocode}
\cs_new:Npn \@@_point_process:nn #1#2
  {
    \@@_point_process_auxi:fn
      { \@@_point_to_dim:n {#2} }
      {#1}
  }
\cs_new:Npn \@@_point_process_auxi:nn #1#2
  { \@@_point_process_auxii:nw {#2} #1 \q_stop }
\cs_generate_variant:Nn \@@_point_process_auxi:nn { f }
\cs_new:Npn \@@_point_process_auxii:nw #1 #2 , #3 \q_stop
  { #1 {#2} {#3} }
\cs_new:Npn \@@_point_process:nnn #1#2#3
  {
    \@@_point_process_auxiii:ffn
      { \@@_point_to_dim:n {#2} }
      { \@@_point_to_dim:n {#3} }
      {#1}
  }
\cs_new:Npn \@@_point_process_auxiii:nnn #1#2#3
  { \@@_point_process_auxiv:nw {#3} #1 \q_mark #2 \q_stop }
\cs_generate_variant:Nn \@@_point_process_auxiii:nnn { ff }
\cs_new:Npn \@@_point_process_auxiv:nw #1 #2 , #3 \q_mark #4 , #5 \q_stop
  { #1 {#2} {#3} {#4} {#5} }
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_point_to_dim:n}
% \begin{macro}[EXP]{\@@_point_to_dim_aux:n, \@@_point_to_dim_aux:f}
% \begin{macro}[EXP]{\@@_point_to_dim_aux:w} 
%   Co-ordinates are always returned as two dimensions. 
%    \begin{macrocode} 
\cs_new:Npn \@@_point_to_dim:n #1 
  { \@@_point_to_dim_aux:f { \fp_eval:n {#1} } }
\cs_new:Npn \@@_point_to_dim_aux:n #1 
  { \@@_point_to_dim_aux:w #1 }
\cs_generate_variant:Nn \@@_point_to_dim_aux:n { f }
\cs_new:Npn \@@_point_to_dim_aux:w ( #1 , ~ #2 ) { #1pt , #2pt }
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \subsection{Co-ordinates}	
%
% The most basic way of giving points is as simple $(x,y)$ co-ordinates.
%
% \begin{macro}[EXP]{\draw_point:nn}
%   Simply turn the given values into dimensions.
%    \begin{macrocode}
\cs_new:Npn \draw_point:nn #1#2
  { \@@_point_to_dim:n { #1 , #2 } }
%    \end{macrocode}
% \end{macro}
%
% \subsection{Polar co-ordinates}
%
% \begin{macro}[EXP]{\draw_point_polar:nn}
% \begin{macro}[EXP]{\draw_point_polar:nnn}
% \begin{macro}[EXP]{\@@_draw_polar:nnn, \@@_draw_polar:fnn}
%   Polar co-ordinates may have either one or two lengths, so there is a need
%   to do a simple split before the calculation. As the angle gets used twice,
%   save on any expression evaluation there and force expansion.
%    \begin{macrocode}
\cs_new:Npn \draw_point_polar:nn #1#2
  { \draw_point_polar:nnn {#1} {#2} {#2} }
\cs_new:Npn \draw_point_polar:nnn #1#2#3
  { \@@_draw_polar:fnn { \fp_eval:n {#1} } {#2} {#3} }
\cs_new:Npn \@@_draw_polar:nnn #1#2#3
  { \@@_point_to_dim:n { cosd(#1) * (#2) , sind(#1) * (#3) } }
\cs_generate_variant:Nn \@@_draw_polar:nnn { f }
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
%
% \subsection{Point expression arithmetic}
%
% These functions all take point expressions as arguments.
%
% \begin{macro}[EXP]
%   {\draw_point_add:nn, \draw_point_diff:nn, \draw_point_scale:nn}
%   Simple mathematics.
%    \begin{macrocode}
\cs_new:Npn \draw_point_add:nn #1#2
  { \@@_point_to_dim:n { (#1) + (#2) } }
\cs_new:Npn \draw_point_diff:nn #1#2
  { \@@_point_to_dim:n { (#2) - (#1) } }
\cs_new:Npn \draw_point_scale:nn #1#2
  { \@@_point_to_dim:n { #1 * (#2) } }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\draw_point_unit_vector:n}
% \begin{macro}[EXP]{\@@_point_unit_vector:nn}
%   Only a single point expression so the expansion is done here. The
%   outcome is the normalised vector from $(0,0)$ in the direction of
%   the point, \emph{i.e.}
%   \[
%     P_{x} = \frac{x}{\sqrt{x^{2} + y^{2}}} \quad
%     P_{y} = \frac{y}{\sqrt{x^{2} + y^{2}}}
%   \]
%    \begin{macrocode}
\cs_new:Npn \draw_point_unit_vector:n #1
  { \@@_point_process:nn { \@@_point_unit_vector:nn } {#1} }
\cs_new:Npn \@@_point_unit_vector:nn #1#2
  {
    \@@_point_to_dim:n
      { ( #1 , #2 ) / (sqrt(#1 * #1 + #2 * #2)) }
  }
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \subsection{Intersection calculations}
%
% \begin{macro}[EXP]{\draw_point_intersect_lines:nnnn}
% \begin{macro}[EXP]{\@@_point_intersect_lines:nnnnnn}
% \begin{macro}[EXP]{\@@_point_intersect_lines:nnnnnnnn}
% \begin{macro}[EXP]
%   {\@@_point_intersect_lines_aux:nnnnnn, \@@_point_intersect_lines_aux:ffffff}
%    The intersection point~$P$ between a line joining points $(x_{1}, y_{1})$
%    and $(x_{2}, y_{2})$ with a second line joining points $(x_{3}, y_{3})$
%    and $(x_{4}, y_{4})$ can be calculated using the formulae
%    \[
%      P_{x} =
%          \frac{(x_{1}y_{2} - y_{1}x_{2})(x_{3} - x_{4})
%            - (x_{3}y_{4} - y_{3}x_{4})(x_{1} - x_{2})}
%          {(x_{1} - x_{2})(y_{3} - y_{4}) - (y_{1} - y_{2})(x_{3} - x_{4})}
%    \]
%    and
%    \[
%      P_{y} =
%          \frac{(x_{1}y_{2} - y_{1}x_{2})(y_{3} - y_{5})
%          - (x_{3}y_{4} - y_{3}x_{4})(y_{1} - y_{2})}
%          {(x_{1} - x_{2})(y_{3} - y_{4}) - (y_{1} - y_{2})(x_{3} - x_{4})} 
%    \]
%    The work therefore comes down to expanding the incoming data, then
%    pre-calculating as many parts as possible before the final work to find
%    the intersection. (Expansion and argument re-ordering is much less work
%    than additional floating point calculations.)
%    \begin{macrocode}
\cs_new:Npn \draw_point_intersect_lines:nnnn #1#2#3#4
  {
    \@@_point_process:nnn
      {
        \@@_point_process:nnn
          { \@@_point_intersect_lines:nnnnnnnn } {#3} {#4}
      }
      {#1} {#2}
  }
%    \end{macrocode}
%   At this stage we have all of the information we need, fully expanded:
%   \begin{enumerate}[label = \#\arabic*, font = \ttfamily]
%     \item $x_{3}$
%     \item $y_{3}$
%     \item $x_{4}$
%     \item $y_{4}$
%     \item $x_{1}$
%     \item $y_{1}$
%     \item $x_{2}$
%     \item $y_{2}$
%   \end{enumerate}
%   so now just have to do all of the calculation.
%    \begin{macrocode}
\cs_new:Npn \@@_point_intersect_lines:nnnnnnnn #1#2#3#4#5#6#7#8
  {
    \@@_point_intersect_lines_aux:ffffff
      { \fp_eval:n { #1 * #4 - #2 * #3 } }
      { \fp_eval:n { #5 * #8 - #6 * #7 } }
      { \fp_eval:n { #1 - #3 } }
      { \fp_eval:n { #5 - #7 } }
      { \fp_eval:n { #2 - #4 } }
      { \fp_eval:n { #6 - #8 } }
  }
\cs_new:Npn \@@_point_intersect_lines_aux:nnnnnn #1#2#3#4#5#6
  {
    \@@_point_to_dim:n
      {
        ( #2 * #3 - #1 * #4 , #2 * #5 - #1 * #6 )
          / ( #4 * #5 - #6 * #3 )
      }
  }
\cs_generate_variant:Nn \@@_point_intersect_lines_aux:nnnnnn { ffffff }
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
%
% \begin{macro}[EXP]{\draw_point_intersect_circles:nnnnn}
% \begin{macro}[EXP]{\@@_point_intersect_circles_auxi:nnnnnnn}
% \begin{macro}[EXP]
%   {
%     \@@_point_intersect_circles_auxii:nnnnnnn,
%     \@@_point_intersect_circles_auxii:ffnnnnn,
%     \@@_point_intersect_circles_auxiii:nnnnnnn,
%     \@@_point_intersect_circles_auxiii:ffnnnnn
%   }
%  \begin{macro}[EXP]
%    {
%      \@@_point_intersect_circles_auxiv:nnnnnnnn,
%      \@@_point_intersect_circles_auxiv:fnnnnnnn
%    }
%  \begin{macro}[EXP]
%    {
%      \@@_point_intersect_circles_auxv:nnnnnnnnn,
%      \@@_point_intersect_circles_auxv:ffnnnnnnn
%    }
%  \begin{macro}[EXP]
%    {
%      \@@_point_intersect_circles_auxvi:nnnnnnnn,
%      \@@_point_intersect_circles_auxvi:fnnnnnnn
%    }
% \begin{macro}[EXP]
%   {
%     \@@_point_intersect_circles_auxvii:nnnnnnn,
%     \@@_point_intersect_circles_auxvii:fffnnnn
%   }
%   Another long expansion chain to get the values in the right places.
%   We have two circles, the first with center $(a, b)$ and radius~$r$,
%   the second with center $(c, d)$ and radius~$s$. We use the intermediate
%   values
%   \begin{align*}
%     e &= c - a \\
%     f &= d - b \\
%     p &= \sqrt{e^{2} + f^{2}} \\
%     k &= \frac{p^{2} + r^{2} - s^{2}}{2p}
%   \end{align*}
%   in either
%   \begin{align*}
%     P_{x} &= a + \frac{ek}{p} + \frac{f}{p}\sqrt{r^{2} - k^{2}} \\
%     P_{y} &= b + \frac{fk}{p} - \frac{e}{p}\sqrt{r^{2} - k^{2}}
%   \end{align*}
%   or
%   \begin{align*}
%     P_{x} &= a + \frac{ek}{p} - \frac{f}{p}\sqrt{r^{2} - k^{2}} \\
%     P_{y} &= b + \frac{fk}{p} + \frac{e}{p}\sqrt{r^{2} - k^{2}}
%   \end{align*}
%   depending on which solution is required. The rest of the work is simply
%   forcing the appropriate expansion and shuffling arguments.
%    \begin{macrocode}
\cs_new:Npn \draw_point_intersect_circles:nnnnn #1#2#3#4#5
  {
    \@@_point_process:nnn
      { \@@_point_intersect_circles_auxi:nnnnnnn {#2} {#4} {#5} }
      {#1} {#3}
  }
\cs_new:Npn \@@_point_intersect_circles_auxi:nnnnnnn #1#2#3#4#5#6#7
  {
    \@@_point_intersect_circles_auxii:ffnnnnn
      { \fp_eval:n {#1} } { \fp_eval:n {#2} } {#4} {#5} {#6} {#7} {#3}
  }
%    \end{macrocode}
%   At this stage we have all of the information we need, fully expanded:
%   \begin{enumerate}[label = \#\arabic*, font = \ttfamily]
%     \item $r$
%     \item $s$
%     \item $a$
%     \item $b$
%     \item $c$
%     \item $d$
%     \item $n$
%   \end{enumerate}
%   Once we evaluate $e$ and $f$, the co-ordinate $(c,d)$ is no longer
%   required: handy as we will need various intermediate values in the
%   following.
%    \begin{macrocode}
\cs_new:Npn \@@_point_intersect_circles_auxii:nnnnnnn #1#2#3#4#5#6#7
  {
    \@@_point_intersect_circles_auxiii:ffnnnnn
      { \fp_eval:n { #5 - #3 } }
      { \fp_eval:n { #6 - #4 } }
      {#1} {#2} {#3} {#4} {#7}
  }
\cs_generate_variant:Nn \@@_point_intersect_circles_auxii:nnnnnnn { ff }
\cs_new:Npn \@@_point_intersect_circles_auxiii:nnnnnnn #1#2#3#4#5#6#7
  {
    \@@_point_intersect_circles_auxiv:fnnnnnnn
      { \fp_eval:n { sqrt( #1 * #1 + #2 * #2 ) } }
      {#1} {#2} {#3} {#4} {#5} {#6} {#7}
  }
\cs_generate_variant:Nn \@@_point_intersect_circles_auxiii:nnnnnnn { ff }
%    \end{macrocode}
%   We now have $p$: we pre-calculate $1/p$ as it is needed a few times and
%   is relatively expensive. We also need $r^{2}$ twice so deal with that
%   here too.
%    \begin{macrocode}
\cs_new:Npn \@@_point_intersect_circles_auxiv:nnnnnnnn #1#2#3#4#5#6#7#8
  {
    \@@_point_intersect_circles_auxv:ffnnnnnnn
      { \fp_eval:n { 1 / #1 } }
      { \fp_eval:n { #4 * #4 } }
      {#1} {#2} {#3} {#5} {#6} {#7} {#8}
  }
\cs_generate_variant:Nn \@@_point_intersect_circles_auxiv:nnnnnnnn { f }
\cs_new:Npn \@@_point_intersect_circles_auxv:nnnnnnnnn #1#2#3#4#5#6#7#8#9
  {
    \@@_point_intersect_circles_auxvi:fnnnnnnn
      { \fp_eval:n { 0.5 * #1 * ( #2 + #3 * #3 - #6 * #6 ) } }
      {#1} {#2} {#4} {#5} {#7} {#8} {#9}
  }
\cs_generate_variant:Nn \@@_point_intersect_circles_auxv:nnnnnnnnn { ff }
%    \end{macrocode}
%   We now have all of the intermediate values we require, with one division
%   carried out up-front to avoid doing this expensive step twice:
%   \begin{enumerate}[label = \#\arabic*, font = \ttfamily]
%     \item $k$
%     \item $1/p$
%     \item $r^{2}$
%     \item $e$
%     \item $f$
%     \item $a$
%     \item $b$
%     \item $n$
%   \end{enumerate}
%   There are some final pre-calculations, $k/p$,
%   $\frac{\sqrt{r^{2} - k^{2}}}{p}$ and the usage of $n$, then we
%   can yield a result.
%    \begin{macrocode}
\cs_new:Npn \@@_point_intersect_circles_auxvi:nnnnnnnn #1#2#3#4#5#6#7#8
  {
    \@@_point_intersect_circles_auxvii:fffnnnn
      { \fp_eval:n { #1 * #2 } }
      { \int_if_odd:nTF {#8} { 1 } { -1 } }
      { \fp_eval:n { sqrt ( #3 - #1 * #1 ) * #2 } }
      {#4} {#5} {#6} {#7}
  }
\cs_generate_variant:Nn \@@_point_intersect_circles_auxvi:nnnnnnnn { f }
\cs_new:Npn \@@_point_intersect_circles_auxvii:nnnnnnn #1#2#3#4#5#6#7
  {
    \@@_point_to_dim:n
      { #6 + #4 * #1 + #2 * #3 * #5 , #7 + #5 * #1 + -1 * #2 * #3 * #4 }
  }
\cs_generate_variant:Nn \@@_point_intersect_circles_auxvii:nnnnnnn { fff }
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
%
% \subsection{Interpolation on a line (vector) or arc}
%
% \begin{macro}[EXP]{\draw_point_interpolate_line:nnn}
% \begin{macro}[EXP]
%   {
%     \@@_point_interpolate_line_aux:nnnnn,
%     \@@_point_interpolate_line_aux:fnnnn,
%   }
% \begin{macro}[EXP]
%   {
%     \@@_point_interpolate_line_aux:nnnnnn,
%     \@@_point_interpolate_line_aux:fnnnnn,
%   }
%   Simple maths after expansion.
%    \begin{macrocode}
\cs_new:Npn \draw_point_interpolate_line:nnn #1#2#3
  {
    \@@_point_process:nnn
      { \@@_point_interpolate_line_aux:fnnnn { \fp_eval:n {#1} } }
      {#2} {#3}
  }
\cs_new:Npn \@@_point_interpolate_line_aux:nnnnn #1#2#3#4#5
  {
    \@@_point_interpolate_line_aux:fnnnnn { \fp_eval:n { 1 - #1 } }
      {#1} {#2} {#3} {#4} {#5}
  }
\cs_generate_variant:Nn \@@_point_interpolate_line_aux:nnnnn { f }
\cs_new:Npn \@@_point_interpolate_line_aux:nnnnnn #1#2#3#4#5#6
  { \@@_point_to_dim:n { #2 * #3 + #1 * #5 , #2 * #4 + #1 * #6 } }
\cs_generate_variant:Nn \@@_point_interpolate_line_aux:nnnnnn { f }
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
%
% \begin{macro}[EXP]{\draw_point_interpolate_distance:nnn}
% \begin{macro}[EXP]{\@@_point_interpolate_distance:nnnnn}
% \begin{macro}[EXP]
%   {
%     \@@_point_interpolate_distance:nnnnnn,
%     \@@_point_interpolate_distance:fnnnnn,
%   }
%   Same idea but using the normalised length to obtain the scale factor.
%   The start point is needed twice, so we force evaluation, but the end
%   point is needed only the once.
%    \begin{macrocode}
\cs_new:Npn \draw_point_interpolate_distance:nnn #1#2#3
  {
    \@@_point_process:nn
      { \@@_point_interpolate_distance:nnnn {#1} {#3} }
      {#2}
  }
\cs_new:Npn \@@_point_interpolate_distance:nnnn #1#2#3#4
  {
    \@@_point_process:nn
      {
        \@@_point_interpolate_distance:fnnnn
          { \fp_eval:n {#1} } {#3} {#4}
      }
      { \draw_point_unit_vector:n { ( #2 ) - ( #3 , #4 ) } }
  }
\cs_new:Npn \@@_point_interpolate_distance:nnnnn #1#2#3#4#5
  { \@@_point_to_dim:n { #2 + #1 * #4 , #3 + #1 * #5 } }
\cs_generate_variant:Nn \@@_point_interpolate_distance:nnnnn { f }
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
%
% \begin{macro}[EXP]{\draw_point_interpolate_arcaxes:nnnnnn}
% \begin{macro}[EXP]{\@@_point_interpolate_arcaxes_auxi:nnnnnnnnn}
% \begin{macro}[EXP]
%   {
%     \@@_point_interpolate_arcaxes_auxii:nnnnnnnnn,
%     \@@_point_interpolate_arcaxes_auxii:fnnnnnnnn
%   }
% \begin{macro}[EXP]
%   {
%     \@@_point_interpolate_arcaxes_auxiii:nnnnnnn,
%     \@@_point_interpolate_arcaxes_auxiii:fnnnnnn
%   }
% \begin{macro}[EXP]
%   {
%     \@@_point_interpolate_arcaxes_auxiv:nnnnnnnn,
%     \@@_point_interpolate_arcaxes_auxiv:ffnnnnnn
%   }
%   Finding a point on an ellipse arc is relatively easy: find the correct
%   angle between the two given, use the sine and cosine of that angle,
%   apply to the axes. We just have to work a bit with the co-ordinate
%   expansion.
%    \begin{macrocode}
\cs_new:Npn \draw_point_interpolate_arcaxes:nnnnnn #1#2#3#4#5#6
  {
    \@@_point_process:nnn
      {
        \@@_point_process:nn
          { \@@_point_interpolate_arcaxes_auxi:nnnnnnnnn {#1} {#5} {#6} }
          {#4}
      }
      {#2} {#3}
  }
\cs_new:Npn \@@_point_interpolate_arcaxes_auxi:nnnnnnnnn #1#2#3#4#5#6#7#8#9
  {
    \@@_point_interpolate_arcaxes_auxii:fnnnnnnnn
      { \fp_eval:n {#1} } {#2} {#3} {#6} {#7} {#8} {#9} {#4} {#5}
  }
%    \end{macrocode}
%   At this stage, the three co-ordinate pairs are fully expanded but somewhat
%   re-ordered:
%   \begin{enumerate}[label = \#\arabic*, font = \ttfamily]
%     \item $p$
%     \item $\theta_{1}$
%     \item $\theta_{2}$
%     \item $x_{c}$
%     \item $y_{c}$
%     \item $x_{a1}$
%     \item $y_{a1}$
%     \item $x_{a2}$
%     \item $y_{a2}$
%   \end{enumerate}
%   We are now in a position to find the target angle, and from that
%   the sine and cosine required.
%    \begin{macrocode}
\cs_new:Npn \@@_point_interpolate_arcaxes_auxii:nnnnnnnnn #1#2#3#4#5#6#7#8#9
  {
    \@@_point_interpolate_arcaxes_auxiii:fnnnnnn
      { \fp_eval:n { #1 * (#3) + ( 1 - #1 ) * (#2) } }
      {#4} {#5} {#6} {#7} {#8} {#9}
  }
\cs_generate_variant:Nn \@@_point_interpolate_arcaxes_auxii:nnnnnnnnn { f }
\cs_new:Npn \@@_point_interpolate_arcaxes_auxiii:nnnnnnn #1#2#3#4#5#6#7
  {
    \@@_point_interpolate_arcaxes_auxiv:ffnnnnnn
      { \fp_eval:n { cosd (#1) } }
      { \fp_eval:n { sind (#1) } }
      {#2} {#3} {#4} {#5} {#6} {#7}
  }
\cs_generate_variant:Nn \@@_point_interpolate_arcaxes_auxiii:nnnnnnn { f }
\cs_new:Npn \@@_point_interpolate_arcaxes_auxiv:nnnnnnnn #1#2#3#4#5#6#7#8
  {
    \@@_point_to_dim:n
      { #3 + #1 * #5 + #2 * #7 , #4 + #1 * #6 + #2 * #8 }
  }
\cs_generate_variant:Nn \@@_point_interpolate_arcaxes_auxiv:nnnnnnnn { ff }
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
%
% \begin{macro}[EXP]{\draw_point_interpolate_curve:nnnnn}
% \begin{macro}[EXP]{\draw_point_interpolate_curve_auxi:nnnnnnnnn}
% \begin{macro}[EXP]
%   {
%     \draw_point_interpolate_curve_auxii:nnnnnnnnn,
%     \draw_point_interpolate_curve_auxii:fnnnnnnnn,
%   }
% \begin{macro}[EXP]
%   {
%     \draw_point_interpolate_curve_auxiii:nnnnnn,
%     \draw_point_interpolate_curve_auxiii:fnnnnn,
%   }
% \begin{macro}[EXP]{\draw_point_interpolate_curve_auxiv:nnnnnn}
% \begin{macro}[EXP]
%   {
%     \draw_point_interpolate_curve_auxv:nnw,
%     \draw_point_interpolate_curve_auxv:ffw,
%   }
% \begin{macro}[EXP]{\draw_point_interpolate_curve_auxvi:n}
% \begin{macro}[EXP]{\draw_point_interpolate_curve_auxvii:nnnnnnnn}
% \begin{macro}[EXP]
%   {
%     \draw_point_interpolate_curve_auxviii:nnnnnn,
%     \draw_point_interpolate_curve_auxviii:ffnnnn,
%   }
%   Here we start with a proportion of the curve ($p$) and four points
%   \begin{enumerate}
%     \item The initial point $(x_{1},y_{1})$
%     \item The first control point $(x_{2},y_{2})$
%     \item The second control point $(x_{3},y_{3})$
%     \item The final point $(x_{4},y_{4})$
%   \end{enumerate}
%   The first phase is to expand out all of these values.
%    \begin{macrocode}
\cs_new:Npn \draw_point_interpolate_curve:nnnnnn #1#2#3#4#5
  {
    \@@_point_process:nnn
      {
        \@@_point_process:nnn
          { \@@_point_interpolate_curve_auxi:nnnnnnnnn {#1} }
          {#4} {#5}
      }
      {#2} {#3}
  }
\cs_new:Npn \@@_point_interpolate_curve_auxi:nnnnnnnnn #1#2#3#4#5#6#7#8#9
  {
    \@@_point_interpolate_curve_auxii:fnnnnnnnn
      { \fp_eval:n {#1} }
      {#6} {#7} {#8} {#9} {#2} {#3} {#4} {#5}
  }
%    \end{macrocode}
%   At this stage, everything is fully expanded and back in the input order.
%   The approach to finding the required point is iterative. We carry out
%   three phases. In phase one, we need all of the input co-ordinates
%   \begin{align*}
%     x_{1}' &= (1 - p)x_{1} + px_{2} \\
%     y_{1}' &= (1 - p)y_{1} + py_{2} \\
%     x_{2}' &= (1 - p)x_{2} + px_{3} \\
%     y_{2}' &= (1 - p)y_{2} + py_{3} \\
%     x_{3}' &= (1 - p)x_{3} + px_{4} \\
%     y_{3}' &= (1 - p)y_{3} + py_{4}
%   \end{align*}
%   In the second stage, we can drop the final point
%   \begin{align*}
%     x_{1}'' &= (1 - p)x_{1}' + px_{2}' \\
%     y_{1}'' &= (1 - p)y_{1}' + py_{2}' \\
%     x_{2}'' &= (1 - p)x_{2}' + px_{3}' \\
%     y_{2}'' &= (1 - p)y_{2}' + py_{3}'
%   \end{align*}
%   and for the final stage only need one set of calculations
%   \begin{align*}
%     P_{x} &= (1 - p)x_{1}'' + px_{2}'' \\
%     P_{y} &= (1 - p)y_{1}'' + py_{2}''
%   \end{align*}
%   Of course, this does mean a lot of calculations and expansion!
%    \begin{macrocode}
\cs_new:Npn \@@_point_interpolate_curve_auxii:nnnnnnnnn
  #1#2#3#4#5#6#7#8#9
  {
    \@@_point_interpolate_curve_auxiii:fnnnnn
      { \fp_eval:n { 1 - #1 } }
      {#1}
      { {#2} {#3} } { {#4} {#5} } { {#6} {#7} } { {#8} {#9} }
  }
\cs_generate_variant:Nn \@@_point_interpolate_curve_auxii:nnnnnnnnn { f }
%    \begin{macrocode}
%   We need to do the first cycle, but haven't got enough arguments to keep
%   everything in play at once. So her ewe use a but of argument re-ordering
%   and a single auxiliary to get the job done. 
%    \begin{macrocode}
\cs_new:Npn \@@_point_interpolate_curve_auxiii:nnnnnn #1#2#3#4#5#6
  {
    \@@_point_interpolate_curve_auxiv:nnnnnn {#1} {#2} #3 #4
    \@@_point_interpolate_curve_auxiv:nnnnnn {#1} {#2} #4 #5
    \@@_point_interpolate_curve_auxiv:nnnnnn {#1} {#2} #5 #6
    \prg_do_nothing:
    \@@_point_interpolate_curve_auxvi:n { {#1} {#2} }
  }
\cs_generate_variant:Nn \@@_point_interpolate_curve_auxiii:nnnnnn { f }
\cs_new:Npn \@@_point_interpolate_curve_auxiv:nnnnnn #1#2#3#4#5#6
  {
    \@@_point_interpolate_curve_auxv:ffw
      { \fp_eval:n { #1 * #3 + #2 * #5 } }
      { \fp_eval:n { #1 * #4 + #2 * #6 } }
  }
\cs_new:Npn \@@_point_interpolate_curve_auxv:nnw
  #1#2#3 \prg_do_nothing: #4#5
  {
    #3
    \prg_do_nothing:
    #4 { #5 {#1} {#2} }
  }
\cs_generate_variant:Nn \@@_point_interpolate_curve_auxv:nnw { ff }
%    \begin{macrocode}
%   Get the arguments back into the right places and to the second and
%   third cycles directly.
%    \begin{macrocode}
\cs_new:Npn \@@_point_interpolate_curve_auxvi:n #1
  { \@@_point_interpolate_curve_auxvii:nnnnnnnn #1 }
\cs_new:Npn \@@_point_interpolate_curve_auxvii:nnnnnnnn #1#2#3#4#5#6#7#8
  {
    \@@_point_interpolate_curve_auxviii:ffffnn
      { \fp_eval:n { #1 * #5 + #2 * #3 } }
      { \fp_eval:n { #1 * #6 + #2 * #4 } }
      { \fp_eval:n { #1 * #7 + #2 * #5 } }
      { \fp_eval:n { #1 * #8 + #2 * #6 } }
      {#1} {#2}
  }
\cs_new:Npn \@@_point_interpolate_curve_auxviii:nnnnnn #1#2#3#4#5#6
  {
    \@@_point_to_dim:n
      { #5 * #3 + #6 * #1 , #5 * #4 + #6 * #2 }
  }
\cs_generate_variant:Nn \@@_point_interpolate_curve_auxviii:nnnnnn { ffff }
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
%
% \subsection{Vector support}
%
% As well as co-ordinates relative to the drawing 
%
% \begin{variable}
%   {
%     \l_@@_xvec_x_dim,
%     \l_@@_xvec_y_dim,
%     \l_@@_yvec_x_dim,
%     \l_@@_yvec_y_dim,
%     \l_@@_zvec_x_dim,
%     \l_@@_zvec_y_dim
%   }
%   Base vectors to map to the underlying two-dimensional drawing space.
%    \begin{macrocode}
\dim_new:N \l_@@_xvec_x_dim
\dim_new:N \l_@@_xvec_y_dim
\dim_new:N \l_@@_yvec_x_dim
\dim_new:N \l_@@_yvec_y_dim
\dim_new:N \l_@@_zvec_x_dim
\dim_new:N \l_@@_zvec_y_dim
%    \end{macrocode}
% \end{variable}
%
% \begin{macro}{\draw_xvec:n, \draw_yvec:n, \draw_zvec:n}
% \begin{macro}{\@@_vec:nn}
% \begin{macro}{\@@_vec:nnn}
%   Calculate the underlying position and store it.
%    \begin{macrocode}
\cs_new_protected:Npn \draw_xvec:n #1
  { \@@_vec:nn { x } {#1} }
\cs_new_protected:Npn \draw_yvec:n #1
  { \@@_vec:nn { y } {#1} }
\cs_new_protected:Npn \draw_zvec:n #1
  { \@@_vec:nn { z } {#1} }
\cs_new_protected:Npn \@@_vec:nn #1#2
  {
    \@@_point_process:nn { \@@_vec:nnn {#1} } {#2}
  }
\cs_new_protected:Npn \@@_vec:nnn #1#2#3
  {
    \dim_set:cn { l_@@_ #1 vec_x_dim } {#2}
    \dim_set:cn { l_@@_ #1 vec_y_dim } {#3}
  }
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
%
% Initialise the vectors.
%    \begin{macrocode}
\draw_xvec:n { 1cm , 0cm }
\draw_yvec:n { 0cm , 1cm }
\draw_zvec:n { -0.385cm , -0.385cm }
%    \end{macrocode}
%
% \begin{macro}[EXP]{\draw_point_vec:nn}
% \begin{macro}[EXP]{\@@_point_vec:nn, \@@_point_vec:ff}
% \begin{macro}[EXP]{\draw_point_vec:nnn}
% \begin{macro}[EXP]{\@@_point_vec:nnn, \@@_point_vec:fff}
%   Force a single evaluation of each factor, then use these to work out the
%   underlying point.
%    \begin{macrocode}
\cs_new:Npn \draw_point_vec:nn #1#2
  { \@@_point_vec:ff { \fp_eval:n {#1} } { \fp_eval:n {#2} } }
\cs_new:Npn \@@_point_vec:nn #1#2
  {
    \@@_point_to_dim:n
      {
        #1 * \l_@@_xvec_x_dim + #2 * \l_@@_yvec_x_dim ,
        #1 * \l_@@_xvec_y_dim + #2 * \l_@@_yvec_y_dim
      }
  }
\cs_generate_variant:Nn \@@_point_vec:nn { ff }
\cs_new:Npn \draw_point_vec:nnn #1#2#3
  {
    \@@_point_vec:fff
      { \fp_eval:n {#1} } { \fp_eval:n {#2} } { \fp_eval:n {#3} }
  }
\cs_new:Npn \@@_point_vec:nnn #1#2#3
  {
    \@@_point_to_dim:n
      {
             #1 * \l_@@_xvec_x_dim
           + #2 * \l_@@_yvec_x_dim
           + #3 * \l_@@_zvec_x_dim
        ,
            #1 * \l_@@_xvec_y_dim
          + #2 * \l_@@_yvec_y_dim
          + #3 * \l_@@_zvec_y_dim
    }
  }
\cs_generate_variant:Nn \@@_point_vec:nnn { fff }
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
%
% \begin{macro}[EXP]{\draw_point_vec_polar:nn}
% \begin{macro}[EXP]{\draw_point_vec_polar:nnn}
% \begin{macro}[EXP]{\@@_point_vec_polar:nnn, \@@_point_vec_polar:fnn}
%   Much the same as the core polar approach.
%    \begin{macrocode}
\cs_new:Npn \draw_point_vec_polar:nn #1#2
  { \draw_point_vec_polar:nnn {#1} {#2} {#2} }
\cs_new:Npn \draw_point_vec_polar:nnn #1#2#3
  { \@@_draw_vec_polar:fnn { \fp_eval:n {#1} } {#2} {#3} }
\cs_new:Npn \@@_draw_vec_polar:nnn #1#2#3
  {
    \@@_point_to_dim:n
      {
        cosd(#1) * (#2) * \l_@@_xvec_x_dim ,
        sind(#1) * (#3) * \l_@@_yvec_y_dim
     }
  }
\cs_generate_variant:Nn \@@_draw_vec_polar:nnn { f }
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
%
% \subsection{Transformations}
%
% \begin{macro}[EXP]{\draw_point_transform:n}
% \begin{macro}[EXP]{\@@_point_transform:nn}
%   Applies a transformation matrix to a point: see \texttt{l3draw-transforms}
%   for the business end. Where possible, we avoid the relatively expensive
%   multiplication step.
%    \begin{macrocode}
\cs_new:Npn \draw_point_transform:n #1
  {
    \@@_point_process:nn
      { \@@_point_transform:nn } {#1}
  }
\cs_new:Npn \@@_point_transform:nn #1#2
  {
    \bool_if:NTF \l_@@_matrix_active_bool
      {
        \@@_point_to_dim:n
          {
            (
                \l_@@_matrix_a_fp * #1
              + \l_@@_matrix_c_fp * #2
              + \l_@@_xshift_dim
            )
            ,
            (
                \l_@@_matrix_b_fp * #1
              + \l_@@_matrix_d_fp * #2
              + \l_@@_yshift_dim
            )
        }
      }
      {
        \@@_point_to_dim:n
          {
              (#1, #2)
            + ( \l_@@_xshift_dim , \l_@@_yshift_dim )
          }
      }
  }
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_point_transform_noshift:n}
% \begin{macro}[EXP]{\@@_point_transform_noshift:nn}
%   A version with no shift: used for internal purposes.
%    \begin{macrocode}
\cs_new:Npn \@@_point_transform_noshift:n #1
  {
    \@@_point_process:nn
      { \@@_point_transform_noshift:nn } {#1}
  }
\cs_new:Npn \@@_point_transform_noshift:nn #1#2
  {
    \bool_if:NTF \l_@@_matrix_active_bool
      {
        \@@_point_to_dim:n
          {
            (
                \l_@@_matrix_a_fp * #1
              + \l_@@_matrix_c_fp * #2
            )
            ,
            (
                \l_@@_matrix_b_fp * #1
              + \l_@@_matrix_d_fp * #2
            )
        }
      }
      { \@@_point_to_dim:n { (#1, #2) } }
  }
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
%    \begin{macrocode}
%</initex|package>
%    \end{macrocode}
%
% \end{implementation}
%
% \PrintIndex