aboutsummaryrefslogtreecommitdiff
path: root/src/algebra/draw.spad.pamphlet
blob: b5e7bae896f21f9e1ef33a9b44642f19adbc2372 (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
\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra draw.spad}
\author{Clifton J. Williamson, Scott Morrison, Jon Steinbach, Mike Dewar}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package DRAWCFUN TopLevelDrawFunctionsForCompiledFunctions}
<<package DRAWCFUN TopLevelDrawFunctionsForCompiledFunctions>>=
)abbrev package DRAWCFUN TopLevelDrawFunctionsForCompiledFunctions
++ Author: Clifton J. Williamson
++ Date Created: 22 June 1990
++ Date Last Updated: January 1992 by Scott Morrison
++ Basic Operations: draw, recolor
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description: TopLevelDrawFunctionsForCompiledFunctions provides top level 
++ functions for drawing graphics of expressions.
TopLevelDrawFunctionsForCompiledFunctions():
 Exports == Implementation where
  ANY1 ==> AnyFunctions1
  B    ==> Boolean
  F    ==> Float
  L    ==> List
  SEG  ==> Segment Float
  SF   ==> DoubleFloat
  DROP ==> DrawOption
  PLOT ==> Plot
  PPC  ==> ParametricPlaneCurve(SF -> SF)
  PSC  ==> ParametricSpaceCurve(SF -> SF)
  PSF  ==> ParametricSurface((SF,SF) -> SF)
  Pt   ==> Point SF
  PSFUN ==> (SF, SF) -> Pt
  PCFUN ==> SF -> Pt
  SPACE3 ==> ThreeSpace(SF)
  VIEW2 ==> TwoDimensionalViewport
  VIEW3 ==> ThreeDimensionalViewport

  Exports ==> with

--% Two Dimensional Function Plots

    draw: (SF -> SF,SEG,L DROP) -> VIEW2
      ++ draw(f,a..b,l) draws the graph of \spad{y = f(x)} as x
      ++ ranges from \spad{min(a,b)} to \spad{max(a,b)}.
      ++ The options contained in the list l of
      ++ the domain \spad{DrawOption} are applied.
    draw: (SF -> SF,SEG) -> VIEW2
      ++ draw(f,a..b) draws the graph of \spad{y = f(x)} as x
      ++ ranges from \spad{min(a,b)} to \spad{max(a,b)}.

--% Parametric Plane Curves

    draw: (PPC,SEG,L DROP) -> VIEW2
      ++ draw(curve(f,g),a..b,l) draws the graph of the parametric
      ++ curve \spad{x = f(t), y = g(t)} as t ranges from \spad{min(a,b)} to 
      ++ \spad{max(a,b)}.
      ++ The options contained in the list l of the domain \spad{DrawOption}
      ++ are applied.
    draw: (PPC,SEG) -> VIEW2
      ++ draw(curve(f,g),a..b) draws the graph of the parametric
      ++ curve \spad{x = f(t), y = g(t)} as t ranges from \spad{min(a,b)} to 
      ++ \spad{max(a,b)}.

--% Parametric Space Curves

    draw: (PSC,SEG,L DROP) -> VIEW3
      ++ draw(curve(f,g,h),a..b,l) draws the graph of the parametric
      ++ curve \spad{x = f(t), y = g(t), z = h(t)} as t ranges from 
      ++ \spad{min(a,b)} to \spad{max(a,b)}.
      ++ The options contained in the list l of the domain
      ++ \spad{DrawOption} are applied.
    draw: (PSC,SEG) -> VIEW3
      ++ draw(curve(f,g,h),a..b,l) draws the graph of the parametric
      ++ curve \spad{x = f(t), y = g(t), z = h(t)} as t ranges from 
      ++ \spad{min(a,b)} to \spad{max(a,b)}.
    draw: (PCFUN,SEG,L DROP) -> VIEW3
      ++ draw(f,a..b,l) draws the graph of the parametric
      ++ curve \spad{f} as t ranges from 
      ++ \spad{min(a,b)} to \spad{max(a,b)}.
      ++ The options contained in the list l of the domain
      ++ \spad{DrawOption} are applied.
    draw: (PCFUN,SEG) -> VIEW3
      ++ draw(f,a..b,l) draws the graph of the parametric
      ++ curve \spad{f} as t ranges from 
      ++ \spad{min(a,b)} to \spad{max(a,b)}.

    makeObject: (PSC,SEG,L DROP) -> SPACE3
      ++ makeObject(curve(f,g,h),a..b,l) returns a space of the
      ++ domain \spadtype{ThreeSpace} which contains the graph of the
      ++ parametric curve \spad{x = f(t), y = g(t), z = h(t)} as t ranges from 
      ++ \spad{min(a,b)} to \spad{max(a,b)};
      ++ The options contained in the list l of the domain
      ++ \spad{DrawOption} are applied.
    makeObject: (PSC,SEG) -> SPACE3
      ++ makeObject(sp,curve(f,g,h),a..b) returns the space \spad{sp}
      ++ of the domain \spadtype{ThreeSpace} with the addition of the graph
      ++ of the parametric curve \spad{x = f(t), y = g(t), z = h(t)} as t
      ++ ranges from \spad{min(a,b)} to \spad{max(a,b)}.
    makeObject: (PCFUN,SEG,L DROP) -> SPACE3
      ++ makeObject(curve(f,g,h),a..b,l) returns a space of the
      ++ domain \spadtype{ThreeSpace} which contains the graph of the
      ++ parametric curve \spad{x = f(t), y = g(t), z = h(t)} as t ranges from 
      ++ \spad{min(a,b)} to \spad{max(a,b)}.
      ++ The options contained in the list l of the domain
      ++ \spad{DrawOption} are applied.
    makeObject: (PCFUN,SEG) -> SPACE3
      ++ makeObject(sp,curve(f,g,h),a..b) returns the space \spad{sp}
      ++ of the domain \spadtype{ThreeSpace} with the addition of the graph
      ++ of the parametric curve \spad{x = f(t), y = g(t), z = h(t)} as t
      ++ ranges from \spad{min(a,b)} to \spad{max(a,b)}.

--% Three Dimensional Function Plots

    draw: ((SF,SF) -> SF,SEG,SEG,L DROP) -> VIEW3
      ++ draw(f,a..b,c..d,l) draws the graph of \spad{z = f(x,y)}
      ++ as x ranges from \spad{min(a,b)} to \spad{max(a,b)} and y ranges from
      ++ \spad{min(c,d)} to \spad{max(c,d)}.
      ++ and the options contained in the list l of the domain
      ++ \spad{DrawOption} are applied.
    draw: ((SF,SF) -> SF,SEG,SEG) -> VIEW3
      ++ draw(f,a..b,c..d) draws the graph of \spad{z = f(x,y)}
      ++ as x ranges from \spad{min(a,b)} to \spad{max(a,b)} and y ranges from
      ++ \spad{min(c,d)} to \spad{max(c,d)}.
    makeObject: ((SF,SF) -> SF,SEG,SEG,L DROP) -> SPACE3
      ++ makeObject(f,a..b,c..d,l) returns a space of the domain
      ++ \spadtype{ThreeSpace} which contains the graph of \spad{z = f(x,y)}
      ++ as x ranges from \spad{min(a,b)} to \spad{max(a,b)} and y ranges from
      ++ \spad{min(c,d)} to \spad{max(c,d)}, and the options contained in the
      ++ list l of the domain \spad{DrawOption} are applied.
    makeObject: ((SF,SF) -> SF,SEG,SEG) -> SPACE3
      ++ makeObject(f,a..b,c..d) returns a space of the domain
      ++ \spadtype{ThreeSpace} which contains the graph of \spad{z = f(x,y)}
      ++ as x ranges from \spad{min(a,b)} to \spad{max(a,b)} and y ranges from
      ++ \spad{min(c,d)} to \spad{max(c,d)}.

--% Parametric Surfaces

    draw: (PSFUN, SEG, SEG, L DROP) -> VIEW3
      ++ draw(f,a..b,c..d) draws the
      ++ graph of the parametric surface \spad{f(u,v)}
      ++ as u ranges from \spad{min(a,b)} to \spad{max(a,b)}
      ++ and v ranges from \spad{min(c,d)} to \spad{max(c,d)}.
      ++ The options contained in the
      ++ list l of the domain \spad{DrawOption} are applied.
    draw: (PSFUN, SEG, SEG) -> VIEW3
      ++ draw(f,a..b,c..d) draws the
      ++ graph of the parametric surface \spad{f(u,v)}
      ++ as u ranges from \spad{min(a,b)} to \spad{max(a,b)}
      ++ and v ranges from \spad{min(c,d)} to \spad{max(c,d)}
      ++ The options contained in the list
      ++ l of the domain \spad{DrawOption} are applied.
    makeObject: (PSFUN, SEG, SEG, L DROP) -> SPACE3
      ++ makeObject(f,a..b,c..d,l) returns a
      ++ space of the domain \spadtype{ThreeSpace} which contains the
      ++ graph of the parametric surface \spad{f(u,v)}
      ++ as u ranges from \spad{min(a,b)} to
      ++ \spad{max(a,b)} and v ranges from \spad{min(c,d)} to \spad{max(c,d)};
      ++ The options contained in the
      ++ list l of the domain \spad{DrawOption} are applied.
    makeObject: (PSFUN, SEG, SEG) -> SPACE3
      ++ makeObject(f,a..b,c..d,l) returns a
      ++ space of the domain \spadtype{ThreeSpace} which contains the
      ++ graph of the parametric surface \spad{f(u,v)}
      ++ as u ranges from \spad{min(a,b)} to
      ++ \spad{max(a,b)} and v ranges from \spad{min(c,d)} to \spad{max(c,d)}.
    draw: (PSF,SEG,SEG,L DROP) -> VIEW3
      ++ draw(surface(f,g,h),a..b,c..d) draws the
      ++ graph of the parametric surface \spad{x = f(u,v)}, \spad{y = g(u,v)},
      ++ \spad{z = h(u,v)} as u ranges from \spad{min(a,b)} to \spad{max(a,b)}
      ++ and v ranges from \spad{min(c,d)} to \spad{max(c,d)};
      ++ The options contained in the
      ++ list l of the domain \spad{DrawOption} are applied.
    draw: (PSF,SEG,SEG) -> VIEW3
      ++ draw(surface(f,g,h),a..b,c..d) draws the
      ++ graph of the parametric surface \spad{x = f(u,v)}, \spad{y = g(u,v)},
      ++ \spad{z = h(u,v)} as u ranges from \spad{min(a,b)} to \spad{max(a,b)}
      ++ and v ranges from \spad{min(c,d)} to \spad{max(c,d)};
    makeObject: (PSF,SEG,SEG,L DROP) -> SPACE3
      ++ makeObject(surface(f,g,h),a..b,c..d,l) returns a
      ++ space of the domain \spadtype{ThreeSpace} which contains the
      ++ graph of the parametric surface \spad{x = f(u,v)}, \spad{y = g(u,v)},
      ++ \spad{z = h(u,v)} as u ranges from \spad{min(a,b)} to
      ++ \spad{max(a,b)} and v ranges from \spad{min(c,d)} to \spad{max(c,d)}.
      ++ The options contained in the
      ++ list l of the domain \spad{DrawOption} are applied.
    makeObject: (PSF,SEG,SEG) -> SPACE3
      ++ makeObject(surface(f,g,h),a..b,c..d,l) returns a
      ++ space of the domain \spadtype{ThreeSpace} which contains the
      ++ graph of the parametric surface \spad{x = f(u,v)}, \spad{y = g(u,v)},
      ++ \spad{z = h(u,v)} as u ranges from \spad{min(a,b)} to
      ++ \spad{max(a,b)} and v ranges from \spad{min(c,d)} to \spad{max(c,d)}.
    recolor: ((SF,SF) -> Pt,(SF,SF,SF) -> SF) -> ((SF,SF) -> Pt)
      ++ recolor(), uninteresting to top level user; exported in order to 
      ++ compile package.

  Implementation ==> add
    --!!  I have had to work my way around the following bug in the compiler:
    --!!  When a local variable is given a mapping as a value, e.g.
    --!!  foo : SF -> SF := makeFloatFunction(f,t),
    --!!  the compiler cannot distinguish that local variable from a local
    --!!  function defined elsewhere in the package.  Thus, when 'foo' is
    --!!  passed to a function, e.g.
    --!!  bird := fcn(foo),
    --!!  foo will often be compiled as |DRAW;foo| rather than |foo|. This,
    --!!  of course, causes a run-time error.
    --!!  To avoid this problem, local variables are not given mappings as
    --!!  values, but rather (singleton) lists of mappings.  The first element
    --!!  of the list can always be extracted and everything goes through
    --!!  as before.  There is no major loss in efficiency, as the computation
    --!!  of points will always dominate the computation time.
    --!!                                     - cjw,  22 June MCMXC

    import PLOT
    import TwoDimensionalPlotClipping
    import GraphicsDefaults
    import ViewportPackage
    import ThreeDimensionalViewport
    import DrawOptionFunctions0
    import MakeFloatCompiledFunction(Ex)
    import MeshCreationRoutinesForThreeDimensions
    import SegmentFunctions2(SF,Float)
    import ViewDefaultsPackage
    import AnyFunctions1(Pt -> Pt)
    import AnyFunctions1((SF,SF,SF) -> SF)
    import DrawOptionFunctions0
    import SPACE3
    import DROP

    EXTOVARERROR : String := _
      "draw: when specifying function, left hand side must be a variable"
    SMALLRANGEERROR : String := _
      "draw: range is in interval with only one point"
    DEPVARERROR : String := _
      "draw: independent variable appears on lhs of function definition"

------------------------------------------------------------------------
--                     2D - draw's  
------------------------------------------------------------------------

    drawToScaleRanges: (Segment SF,Segment SF) -> L SEG
    drawToScaleRanges(xVals,yVals) ==
      -- warning: assumes window is square
      xHi := convert(hi xVals)@Float; xLo := convert(lo xVals)@Float
      yHi := convert(hi yVals)@Float; yLo := convert(lo yVals)@Float
      xDiff := xHi - xLo; yDiff := yHi - yLo
      pad := abs(yDiff - xDiff)/2
      yDiff > xDiff =>
        [segment(xLo - pad,xHi + pad),map(convert(#1)@Float,yVals)]
      [map(convert(#1)@Float,xVals),segment(yLo - pad,yHi + pad)]

    drawPlot: (PLOT,L DROP) -> VIEW2
    drawPlot(plot,l) ==
      branches := listBranches plot
      xRange := xRange plot; yRange := yRange plot
      -- process clipping information
      if (cl := option(l,"clipSegment" :: Symbol)) case "failed" then
        if clipBoolean(l,clipPointsDefault()) then
          clipInfo :=
            parametric? plot => clipParametric plot
            clip plot
          branches := clipInfo.brans
          xRange := clipInfo.xValues; yRange := clipInfo.yValues
        else
          "No explicit user-specified clipping"
      else
        segList := retract(cl :: Any)$ANY1(L SEG)
        empty? segList =>
          error "draw: you may specify at least 1 segment for 2D clipping"
        #segList > 2 =>
          error "draw: you may specify at most 2 segments for 2D clipping"
        xLo : SF := 0; xHi : SF := 0; yLo : SF := 0; yHi : SF := 0
        if empty? rest segList then
          xLo := lo xRange; xHi := hi xRange
          yRangeF := first segList
          yLo := convert(lo yRangeF)@SF; yHi := convert(hi yRangeF)@SF
        else
          xRangeF := first segList
          xLo := convert(lo xRangeF)@SF; xHi := convert(hi xRangeF)@SF
          yRangeF := second segList
          yLo := convert(lo yRangeF)@SF; yHi := convert(hi yRangeF)@SF
        clipInfo := clipWithRanges(branches,xLo,xHi,yLo,yHi)
        branches := clipInfo.brans
        xRange := clipInfo.xValues; yRange := clipInfo.yValues
      -- process scaling information
      if toScale(l,drawToScale()) then
        scaledRanges := drawToScaleRanges(xRange,yRange)
        -- add scaled ranges to list of options
        l := concat(ranges scaledRanges,l)
      else
        xRangeFloat : SEG := map(convert(#1)@Float,xRange)
        yRangeFloat : SEG := map(convert(#1)@Float,yRange)
        -- add ranges to list of options
        l := concat(ranges(ll : L SEG := [xRangeFloat,yRangeFloat]),l)
      -- process color information
      ptCol := pointColorPalette(l,pointColorDefault())
      crCol := curveColorPalette(l,lineColorDefault())
      -- draw
      drawCurves(branches,ptCol,crCol,pointSizeDefault(),l)

    normalize: SEG -> Segment SF
    normalize seg ==
      -- normalize [a,b]:
      -- error if a = b, returns [a,b] if a < b, returns [b,a] if b > a
      a := convert(lo seg)@SF; b := convert(hi seg)@SF
      a = b => error SMALLRANGEERROR
      a < b => segment(a,b)
      segment(b,a)

--% functions for creation of maps SF -> Point SF (two dimensional)

    myTrap1: (SF-> SF, SF) -> SF
    myTrap1(ff:SF-> SF, f:SF):SF ==
      s := trapNumericErrors(ff(f))$Lisp :: Union(SF, "failed")
      s case "failed" => quietDoubleNaN()$Foreign(Builtin)
      r:=s::SF
      r >max()$SF or r < min()$SF => quietDoubleNaN()$Foreign(Builtin)
      r

    makePt2: (SF,SF) -> Point SF
    makePt2(x,y) == point(l : List SF := [x,y])

--% Two Dimensional Function Plots
 
    draw(f:SF -> SF,seg:SEG,l:L DROP) ==
      -- set adaptive plotting off or on
      oldAdaptive := adaptive?()$PLOT
      setAdaptive(adaptive(l,oldAdaptive))$PLOT
      -- create function SF -> Point SF
      ff : L(SF -> Point SF) := [makePt2(myTrap1(f,#1),#1)]
      -- process change of coordinates
      if (c := option(l,"coordinates" :: Symbol)) case "failed" then
        -- default coordinate transformation
        ff := [makePt2(#1,myTrap1(f,#1))]
      else
        cc : L(Pt -> Pt) := [retract(c :: Any)$ANY1(Pt -> Pt)]
        ff := [(first cc)((first ff)(#1))]
      -- create PLOT
      pl := pointPlot(first ff,normalize seg)
      -- reset adaptive plotting
      setAdaptive(oldAdaptive)$PLOT
      -- draw
      drawPlot(pl,l)
 
    draw(f:SF -> SF,seg:SEG) == draw(f,seg,nil())
 
--% Parametric Plane Curves

    draw(ppc:PPC,seg:SEG,l:L DROP) ==
      -- set adaptive plotting off or on
      oldAdaptive := adaptive?()$PLOT
      setAdaptive(adaptive(l,oldAdaptive))$PLOT
      -- create function SF -> Point SF
      f := coordinate(ppc,1); g := coordinate(ppc,2)
      fcn : L(SF -> Pt) := [makePt2(myTrap1(f,#1),myTrap1(g,#1))]
      -- process change of coordinates
      if not (c := option(l,"coordinates" :: Symbol)) case "failed" then
        cc : L(Pt -> Pt) := [retract(c :: Any)$ANY1(Pt -> Pt)]
        fcn := [(first cc)((first fcn)(#1))]
      -- create PLOT
      pl := pointPlot(first fcn,normalize seg)
      -- reset adaptive plotting
      setAdaptive(oldAdaptive)$PLOT
      -- draw
      drawPlot(pl,l)
 
    draw(ppc:PPC,seg:SEG) == draw(ppc,seg,nil())

------------------------------------------------------------------------
--                     3D - Curves  
------------------------------------------------------------------------

--% functions for creation of maps SF -> Point SF (three dimensional)

    makePt4: (SF,SF,SF,SF) -> Point SF
    makePt4(x,y,z,c) == point(l : List SF := [x,y,z,c])

--% Parametric Space Curves

    id: SF -> SF
    id x == x

    zCoord: (SF,SF,SF) -> SF
    zCoord(x,y,z) == z

    colorPoints: (List List Pt,(SF,SF,SF) -> SF) -> List List Pt
    colorPoints(llp,func) ==
      for lp in llp repeat for p in lp repeat
        p.4 := func(p.1,p.2,p.3)
      llp

    makeObject(psc:PSC,seg:SEG,l:L DROP) ==
      sp := space l
      -- obtain dependent variable and coordinate functions
      f := coordinate(psc,1); g := coordinate(psc,2); h := coordinate(psc,3)
      -- create function SF -> Point SF with default or user-specified
      -- color function
      fcn : L(SF -> Pt) := [makePt4(myTrap1(f,#1),myTrap1(g,#1),myTrap1(h,#1),_
                            myTrap1(id,#1))]
      pointsColored? : Boolean := false
      if not (c1 := option(l,"colorFunction1" :: Symbol)) case "failed" then
        pointsColored? := true
        fcn := [makePt4(myTrap1(f,#1),myTrap1(g,#1),myTrap1(h,#1),_
                retract(c1 :: Any)$ANY1(SF -> SF)(#1))]
      -- process change of coordinates
      if not (c := option(l,"coordinates" :: Symbol)) case "failed" then
        cc : L(Pt -> Pt) := [retract(c :: Any)$ANY1(Pt -> Pt)]
        fcn := [(first cc)((first fcn)(#1))]
      -- create PLOT
      pl := pointPlot(first fcn,normalize seg)$Plot3D
      -- create ThreeSpace
      s := sp
      -- draw Tube
--      print(pl::OutputForm)
      option?(l,"tubeRadius" :: Symbol) =>
        pts := tubePoints(l,8)
        rad := convert(tubeRadius(l,0.25))@DoubleFloat
        tub := tube(pl,rad,pts)$NumericTubePlot(Plot3D)
        loops := listLoops tub
        -- color points if this has not been done already
        if not pointsColored? then
          if (c3 := option(l,"colorFunction3" :: Symbol)) case "failed"
            then colorPoints(loops,zCoord)  -- default color function
            else colorPoints(loops,retract(c3 :: Any)$ANY1((SF,SF,SF) -> SF))
        mesh(s,loops,false,false)
        s
      -- draw curve
      br := listBranches pl
      for b in br repeat curve(s,b)
      s

    makeObject(psc:PCFUN,seg:SEG,l:L DROP) ==
      sp := space l
      -- create function SF -> Point SF with default or user-specified
      -- color function
      fcn : L(SF -> Pt) := [psc]
      pointsColored? : Boolean := false
      if not (c1 := option(l,"colorFunction1" :: Symbol)) case "failed" then
        pointsColored? := true
        fcn := [concat(psc(#1), retract(c1 :: Any)$ANY1(SF -> SF)(#1))]
      -- process change of coordinates
      if not (c := option(l,"coordinates" :: Symbol)) case "failed" then
        cc : L(Pt -> Pt) := [retract(c :: Any)$ANY1(Pt -> Pt)]
        fcn := [(first cc)((first fcn)(#1))]
      -- create PLOT
      pl := pointPlot(first fcn,normalize seg)$Plot3D
      -- create ThreeSpace
      s := sp
      -- draw Tube
      option?(l,"tubeRadius" :: Symbol) =>
        pts := tubePoints(l,8)
        rad := convert(tubeRadius(l,0.25))@DoubleFloat
        tub := tube(pl,rad,pts)$NumericTubePlot(Plot3D)
        loops := listLoops tub
        -- color points if this has not been done already
        mesh(s,loops,false,false)
        s
      -- draw curve
      br := listBranches pl
      for b in br repeat curve(s,b)
      s

    makeObject(psc:PSC,seg:SEG) ==
      makeObject(psc,seg,nil())

    makeObject(psc:PCFUN,seg:SEG) ==
      makeObject(psc,seg,nil())

    draw(psc:PSC,seg:SEG,l:L DROP) ==
      sp := makeObject(psc,seg,l)
      makeViewport3D(sp, l)

    draw(psc:PSC,seg:SEG) ==
      draw(psc,seg,nil())

    draw(psc:PCFUN,seg:SEG,l:L DROP) ==
      sp := makeObject(psc,seg,l)
      makeViewport3D(sp, l)

    draw(psc:PCFUN,seg:SEG) ==
      draw(psc,seg,nil())

------------------------------------------------------------------------
--                     3D - Surfaces  
------------------------------------------------------------------------

    myTrap2: ((SF, SF) -> SF, SF, SF) -> SF
    myTrap2(ff:(SF, SF) -> SF, u:SF, v:SF):SF ==
      s := trapNumericErrors(ff(u, v))$Lisp :: Union(SF, "failed")
      s case "failed" =>  quietDoubleNaN()$Foreign(Builtin)
      r:SF := s::SF
      r >max()$SF or r < min()$SF => quietDoubleNaN()$Foreign(Builtin)
      r

    recolor(ptFunc,colFunc) ==
      pt := ptFunc(#1,#2)
      pt.4 := colFunc(pt.1,pt.2,pt.3)
      pt

    xCoord: (SF,SF) -> SF
    xCoord(x,y) == x

--% Three Dimensional Function Plots

    makeObject(f:(SF,SF) -> SF,xSeg:SEG,ySeg:SEG,l:L DROP) ==
      sp := space l
      -- process color function of two variables
      col2 : L((SF,SF) -> SF) := [xCoord]     -- dummy color function
      pointsColored? : Boolean := false
      if not (c2 := option(l,"colorFunction2" :: Symbol)) case "failed" then
        pointsColored? := true
        col2 := [retract(c2 :: Any)$ANY1((SF,SF) -> SF)]
      fcn : L((SF,SF) -> Pt) :=
        [makePt4(myTrap2(f,#1,#2),#1,#2,(first col2)(#1,#2))]
      -- process change of coordinates
      if (c := option(l,"coordinates" :: Symbol)) case "failed" then
        -- default coordinate transformation
        fcn := [makePt4(#1,#2,myTrap2(f,#1,#2),(first col2)(#1,#2))]
      else
        cc : L(Pt -> Pt) := [retract(c :: Any)$ANY1(Pt -> Pt)]
        fcn := [(first cc)((first fcn)(#1,#2))]
      -- process color function of three variables, if there was no
      -- color function of two variables
      if not pointsColored? then
        c := option(l,"colorFunction3" :: Symbol)
        fcn := 
          c case "failed" => [recolor((first fcn),zCoord)]
          [recolor((first fcn),retract(c :: Any)$ANY1((SF,SF,SF) -> SF))]
      -- create mesh
      mesh := meshPar2Var(sp,first fcn,normalize xSeg,normalize ySeg,l)
      mesh

    makeObject(f:(SF,SF) -> SF,xSeg:SEG,ySeg:SEG) ==
      makeObject(f,xSeg,ySeg,nil())

    draw(f:(SF,SF) -> SF,xSeg:SEG,ySeg:SEG,l:L DROP) ==
      sp := makeObject(f, xSeg, ySeg, l)
      makeViewport3D(sp, l)

    draw(f:(SF,SF) -> SF,xSeg:SEG,ySeg:SEG) ==
      draw(f,xSeg,ySeg,nil())

--% parametric surface

    makeObject(s:PSF,uSeg:SEG,vSeg:SEG,l:L DROP) ==
      sp := space l
      -- create functions from expressions
      f : L((SF,SF) -> SF) := [coordinate(s,1)]
      g : L((SF,SF) -> SF) := [coordinate(s,2)]
      h : L((SF,SF) -> SF) := [coordinate(s,3)]
      -- process color function of two variables
      col2 : L((SF,SF) -> SF) := [xCoord]     -- dummy color function
      pointsColored? : Boolean := false
      if not (c2 := option(l,"colorFunction2" :: Symbol)) case "failed" then
        pointsColored? := true
        col2 := [retract(c2 :: Any)$ANY1((SF,SF) -> SF)]
      fcn : L((SF,SF) -> Pt) := 
        [makePt4(myTrap2((first f),#1,#2),myTrap2((first g),#1,#2),myTrap2((first h),#1,#2),_
                 myTrap2((first col2),#1,#2))]
      -- process change of coordinates
      if not (c := option(l,"coordinates" :: Symbol)) case "failed" then
        cc : L(Pt -> Pt) := [retract(c :: Any)$ANY1(Pt -> Pt)]
        fcn := [(first cc)((first fcn)(#1,#2))]
      -- process color function of three variables, if there was no
      -- color function of two variables
      if not pointsColored? then
        col3 : L((SF,SF,SF) -> SF) := [zCoord]  -- default color function
        if not (c := option(l,"colorFunction3" :: Symbol)) case "failed" then 
          col3 := [retract(c :: Any)$ANY1((SF,SF,SF) -> SF)]
        fcn := [recolor((first fcn),(first col3))]
      -- create mesh
      mesh := meshPar2Var(sp,first fcn,normalize uSeg,normalize vSeg,l)
      mesh

    makeObject(s:PSFUN,uSeg:SEG,vSeg:SEG,l:L DROP) ==
      sp := space l
      -- process color function of two variables
      col2 : L((SF,SF) -> SF) := [xCoord]     -- dummy color function
      pointsColored? : Boolean := false
      if not (c2 := option(l,"colorFunction2" :: Symbol)) case "failed" then
        pointsColored? := true
        col2 := [retract(c2 :: Any)$ANY1((SF,SF) -> SF)]
      fcn : L((SF,SF) -> Pt) := 
        pointsColored? => [concat(s(#1, #2), (first col2)(#1, #2))]
        [s]
      -- process change of coordinates
      if not (c := option(l,"coordinates" :: Symbol)) case "failed" then
        cc : L(Pt -> Pt) := [retract(c :: Any)$ANY1(Pt -> Pt)]
        fcn := [(first cc)((first fcn)(#1,#2))]
      -- create mesh
      mesh := meshPar2Var(sp,first fcn,normalize uSeg,normalize vSeg,l)
      mesh

    makeObject(s:PSF,uSeg:SEG,vSeg:SEG) ==
      makeObject(s,uSeg,vSeg,nil())

    draw(s:PSF,uSeg:SEG,vSeg:SEG,l:L DROP) ==
      -- draw
      mesh := makeObject(s,uSeg,vSeg,l)
      makeViewport3D(mesh,l)

    draw(s:PSF,uSeg:SEG,vSeg:SEG) ==
      draw(s,uSeg,vSeg,nil())
 
    makeObject(s:PSFUN,uSeg:SEG,vSeg:SEG) ==
      makeObject(s,uSeg,vSeg,nil())

    draw(s:PSFUN,uSeg:SEG,vSeg:SEG,l:L DROP) ==
      -- draw
      mesh := makeObject(s,uSeg,vSeg,l)
      makeViewport3D(mesh,l)

    draw(s:PSFUN,uSeg:SEG,vSeg:SEG) ==
      draw(s,uSeg,vSeg,nil())
 
@
\section{package DRAW TopLevelDrawFunctions}
<<package DRAW TopLevelDrawFunctions>>=
)abbrev package DRAW TopLevelDrawFunctions
++ Author: Clifton J. Williamson
++ Date Created: 23 January 1990
++ Date Last Updated: October 1991 by Jon Steinbach
++ Basic Operations: draw
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description: TopLevelDrawFunctions provides top level functions for 
++ drawing graphics of expressions.
TopLevelDrawFunctions(Ex:Join(ConvertibleTo InputForm,SetCategory)):
 Exports == Implementation where
  B    ==> Boolean
  BIND ==> SegmentBinding Float
  L    ==> List
  SF   ==> DoubleFloat
  DROP ==> DrawOption

  PPC  ==> ParametricPlaneCurve Ex
  PPCF ==> ParametricPlaneCurve(SF -> SF)
  PSC  ==> ParametricSpaceCurve Ex
  PSCF ==> ParametricSpaceCurve(SF -> SF)
  PSF  ==> ParametricSurface Ex
  PSFF ==> ParametricSurface((SF,SF) -> SF)
  SPACE3 ==> ThreeSpace(SF)
  VIEW2 ==> TwoDimensionalViewport
  VIEW3 ==> ThreeDimensionalViewport

  Exports ==> with

--% Two Dimensional Function Plots

    draw: (Ex,BIND,L DROP) -> VIEW2
      ++ draw(f(x),x = a..b,l) draws the graph of \spad{y = f(x)} as x
      ++ ranges from \spad{min(a,b)} to \spad{max(a,b)}; \spad{f(x)} is the 
      ++ default title, and the options contained in the list l of
      ++ the domain \spad{DrawOption} are applied.
    draw: (Ex,BIND) -> VIEW2
      ++ draw(f(x),x = a..b) draws the graph of \spad{y = f(x)} as x
      ++ ranges from \spad{min(a,b)} to \spad{max(a,b)}; \spad{f(x)} appears 
      ++ in the title bar.

--% Parametric Plane Curves

    draw: (PPC,BIND,L DROP) -> VIEW2
      ++ draw(curve(f(t),g(t)),t = a..b,l) draws the graph of the parametric
      ++ curve \spad{x = f(t), y = g(t)} as t ranges from \spad{min(a,b)} to 
      ++ \spad{max(a,b)}; \spad{(f(t),g(t))} is the default title, and the
      ++ options contained in the list l of the domain \spad{DrawOption}
      ++ are applied.
    draw: (PPC,BIND) -> VIEW2
      ++ draw(curve(f(t),g(t)),t = a..b) draws the graph of the parametric
      ++ curve \spad{x = f(t), y = g(t)} as t ranges from \spad{min(a,b)} to 
      ++ \spad{max(a,b)}; \spad{(f(t),g(t))} appears in the title bar.

--% Parametric Space Curves

    draw: (PSC,BIND,L DROP) -> VIEW3
      ++ draw(curve(f(t),g(t),h(t)),t = a..b,l) draws the graph of the
      ++ parametric curve \spad{x = f(t)}, \spad{y = g(t)}, \spad{z = h(t)}
      ++ as t ranges from \spad{min(a,b)} to \spad{max(a,b)}; \spad{h(t)}
      ++ is the default title, and the options contained in the list l of
      ++ the domain \spad{DrawOption} are applied.
    draw: (PSC,BIND) -> VIEW3
      ++ draw(curve(f(t),g(t),h(t)),t = a..b) draws the graph of the parametric
      ++ curve \spad{x = f(t)}, \spad{y = g(t)}, \spad{z = h(t)} as t ranges
      ++ from \spad{min(a,b)} to \spad{max(a,b)}; \spad{h(t)} is the default
      ++ title.
    makeObject: (PSC,BIND,L DROP) -> SPACE3
      ++ makeObject(curve(f(t),g(t),h(t)),t = a..b,l) returns a space of
      ++ the domain \spadtype{ThreeSpace} which contains the graph of the
      ++ parametric curve \spad{x = f(t)}, \spad{y = g(t)}, \spad{z = h(t)}
      ++ as t ranges from \spad{min(a,b)} to \spad{max(a,b)}; \spad{h(t)}
      ++ is the default title, and the options contained in the list l of
      ++ the domain \spad{DrawOption} are applied.
    makeObject: (PSC,BIND) -> SPACE3
      ++ makeObject(curve(f(t),g(t),h(t)),t = a..b) returns a space of the
      ++ domain \spadtype{ThreeSpace} which contains the graph of the
      ++ parametric curve \spad{x = f(t)}, \spad{y = g(t)}, \spad{z = h(t)}
      ++ as t ranges from \spad{min(a,b)} to \spad{max(a,b)}; \spad{h(t)} is
      ++ the default title.

--% Three Dimensional Function Plots

    draw: (Ex,BIND,BIND,L DROP) -> VIEW3
      ++ draw(f(x,y),x = a..b,y = c..d,l) draws the graph of \spad{z = f(x,y)}
      ++ as x ranges from \spad{min(a,b)} to \spad{max(a,b)} and y ranges from
      ++ \spad{min(c,d)} to \spad{max(c,d)}; \spad{f(x,y)} is the default
      ++ title, and the options contained in the list l of the domain
      ++ \spad{DrawOption} are applied.
    draw: (Ex,BIND,BIND) -> VIEW3
      ++ draw(f(x,y),x = a..b,y = c..d) draws the graph of \spad{z = f(x,y)}
      ++ as x ranges from \spad{min(a,b)} to \spad{max(a,b)} and y ranges from
      ++ \spad{min(c,d)} to \spad{max(c,d)}; \spad{f(x,y)} appears in the title bar.
    makeObject: (Ex,BIND,BIND,L DROP) -> SPACE3
      ++ makeObject(f(x,y),x = a..b,y = c..d,l) returns a space of the
      ++ domain \spadtype{ThreeSpace} which contains the graph of
      ++ \spad{z = f(x,y)} as x ranges from \spad{min(a,b)} to \spad{max(a,b)}
      ++ and y ranges from \spad{min(c,d)} to \spad{max(c,d)}; \spad{f(x,y)}
      ++ is the default title, and the options contained in the list l of the
      ++ domain \spad{DrawOption} are applied.
    makeObject: (Ex,BIND,BIND) -> SPACE3
      ++ makeObject(f(x,y),x = a..b,y = c..d) returns a space of the domain
      ++ \spadtype{ThreeSpace} which contains the graph of \spad{z = f(x,y)}
      ++ as x ranges from \spad{min(a,b)} to \spad{max(a,b)} and y ranges from
      ++ \spad{min(c,d)} to \spad{max(c,d)}; \spad{f(x,y)} appears as the
      ++ default title.

--% Parametric Surfaces

    draw: (PSF,BIND,BIND,L DROP) -> VIEW3
      ++ draw(surface(f(u,v),g(u,v),h(u,v)),u = a..b,v = c..d,l) draws the
      ++ graph of the parametric surface \spad{x = f(u,v)}, \spad{y = g(u,v)},
      ++ \spad{z = h(u,v)} as u ranges from \spad{min(a,b)} to \spad{max(a,b)}
      ++ and v ranges from \spad{min(c,d)} to \spad{max(c,d)}; \spad{h(t)}
      ++ is the default title, and the options contained in the list l of
      ++ the domain \spad{DrawOption} are applied.
    draw: (PSF,BIND,BIND) -> VIEW3
      ++ draw(surface(f(u,v),g(u,v),h(u,v)),u = a..b,v = c..d) draws the
      ++ graph of the parametric surface \spad{x = f(u,v)}, \spad{y = g(u,v)},
      ++ \spad{z = h(u,v)} as u ranges from \spad{min(a,b)} to \spad{max(a,b)}
      ++ and v ranges from \spad{min(c,d)} to \spad{max(c,d)}; \spad{h(t)} is
      ++ the default title.
    makeObject: (PSF,BIND,BIND,L DROP) -> SPACE3
      ++ makeObject(surface(f(u,v),g(u,v),h(u,v)),u = a..b,v = c..d,l) returns
      ++ a space of the domain \spadtype{ThreeSpace} which contains the graph
      ++ of the parametric surface \spad{x = f(u,v)}, \spad{y = g(u,v)},
      ++ \spad{z = h(u,v)} as u ranges from \spad{min(a,b)} to \spad{max(a,b)}
      ++ and v ranges from \spad{min(c,d)} to \spad{max(c,d)}; \spad{h(t)} is
      ++ the default title, and the options contained in the list l of
      ++ the domain \spad{DrawOption} are applied.
    makeObject: (PSF,BIND,BIND) -> SPACE3
      ++ makeObject(surface(f(u,v),g(u,v),h(u,v)),u = a..b,v = c..d) returns
      ++ a space of the domain \spadtype{ThreeSpace} which contains the
      ++ graph of the parametric surface \spad{x = f(u,v)}, \spad{y = g(u,v)},
      ++ \spad{z = h(u,v)} as u ranges from \spad{min(a,b)} to \spad{max(a,b)}
      ++ and v ranges from \spad{min(c,d)} to \spad{max(c,d)}; \spad{h(t)} is
      ++ the default title.

  Implementation ==> add
    import TopLevelDrawFunctionsForCompiledFunctions
    import MakeFloatCompiledFunction(Ex)
    import ParametricPlaneCurve(SF -> SF)
    import ParametricSpaceCurve(SF -> SF)
    import ParametricSurface((SF,SF) -> SF)
    import ThreeSpace(SF)

------------------------------------------------------------------------
--                     2D - draw's  (given by formulae)
------------------------------------------------------------------------

--% Two Dimensional Function Plots
 
    draw(f:Ex,bind:BIND,l:L DROP) ==
      import DROP
      -- create title if necessary
      if not option?(l,"title" :: Symbol) then
        s:String := unparse(convert(f)@InputForm)
        if sayLength(s)$DisplayPackage > 50 then
          l := concat(title "AXIOM2D",l)
        else l := concat(title s,l)
      -- call 'draw'
      draw(makeFloatFunction(f,variable bind),segment bind,l)
 
    draw(f:Ex,bind:BIND) == draw(f,bind,nil())
 
--% Parametric Plane Curves

    draw(ppc:PPC,bind:BIND,l:L DROP) ==
      import DROP
      f := coordinate(ppc,1); g := coordinate(ppc,2)
      -- create title if necessary
      if not option?(l,"title" :: Symbol) then
        s:String := unparse(convert(f)@InputForm)
        if sayLength(s)$DisplayPackage > 50 then
          l := concat(title "AXIOM2D",l)
        else l := concat(title s,l)
      -- create curve with functions as coordinates
      curve : PPCF := curve(makeFloatFunction(f,variable bind),_
                            makeFloatFunction(g,variable bind))$PPCF
      -- call 'draw'
      draw(curve,segment bind,l)
 
    draw(ppc:PPC,bind:BIND) == draw(ppc,bind,nil())

------------------------------------------------------------------------
--                     3D - Curves  (given by formulas)
------------------------------------------------------------------------

    makeObject(psc:PSC,tBind:BIND,l:L DROP) ==
      import DROP
      -- obtain dependent variable and coordinate functions
      t := variable tBind; tSeg := segment tBind
      f := coordinate(psc,1); g := coordinate(psc,2); h := coordinate(psc,3)
      -- create title if necessary
      if not option?(l,"title" :: Symbol) then
        s:String := unparse(convert(f)@InputForm)
        if sayLength(s)$DisplayPackage > 50 then
          l := concat(title "AXIOM3D",l)
        else l := concat(title s,l)
      -- indicate draw style if necessary
      if not option?(l,"style" :: Symbol) then
        l := concat(style unparse(convert(f)@InputForm),l)
      -- create curve with functions as coordinates
      curve : PSCF := curve(makeFloatFunction(f,t),_
                            makeFloatFunction(g,t),_
                            makeFloatFunction(h,t))
      -- call 'draw'
      makeObject(curve,tSeg,l)

    makeObject(psc:PSC,tBind:BIND) ==
      makeObject(psc,tBind,nil())

    draw(psc:PSC,tBind:BIND,l:L DROP) ==
      import DROP
      -- obtain dependent variable and coordinate functions
      t := variable tBind; tSeg := segment tBind
      f := coordinate(psc,1); g := coordinate(psc,2); h := coordinate(psc,3)
      -- create title if necessary
      if not option?(l,"title" :: Symbol) then
        s:String := unparse(convert(f)@InputForm)
        if sayLength(s)$DisplayPackage > 50 then
          l := concat(title "AXIOM3D",l)
        else l := concat(title s,l)
      -- indicate draw style if necessary
      if not option?(l,"style" :: Symbol) then
        l := concat(style unparse(convert(f)@InputForm),l)
      -- create curve with functions as coordinates
      curve : PSCF := curve(makeFloatFunction(f,t),_
                            makeFloatFunction(g,t),_
                            makeFloatFunction(h,t))
      -- call 'draw'
      draw(curve,tSeg,l)

    draw(psc:PSC,tBind:BIND) ==
      draw(psc,tBind,nil())

------------------------------------------------------------------------
--                     3D - Surfaces  (given by formulas)
------------------------------------------------------------------------

--% Three Dimensional Function Plots

    makeObject(f:Ex,xBind:BIND,yBind:BIND,l:L DROP) ==
      import DROP
      -- create title if necessary
      if not option?(l,"title" :: Symbol) then
        s:String := unparse(convert(f)@InputForm)
        if sayLength(s)$DisplayPackage > 50 then
          l := concat(title "AXIOM3D",l)
        else l := concat(title s,l)
      -- indicate draw style if necessary
      if not option?(l,"style" :: Symbol) then
        l := concat(style unparse(convert(f)@InputForm),l)
      -- obtain dependent variables and their ranges
      x := variable xBind; xSeg := segment xBind
      y := variable yBind; ySeg := segment yBind
      -- call 'draw'
      makeObject(makeFloatFunction(f,x,y),xSeg,ySeg,l)

    makeObject(f:Ex,xBind:BIND,yBind:BIND) ==
      makeObject(f,xBind,yBind,nil())

    draw(f:Ex,xBind:BIND,yBind:BIND,l:L DROP) ==
      import DROP
      -- create title if necessary
      if not option?(l,"title" :: Symbol) then
        s:String := unparse(convert(f)@InputForm)
        if sayLength(s)$DisplayPackage > 50 then
          l := concat(title "AXIOM3D",l)
        else l := concat(title s,l)
      -- indicate draw style if necessary
      if not option?(l,"style" :: Symbol) then
        l := concat(style unparse(convert(f)@InputForm),l)
      -- obtain dependent variables and their ranges
      x := variable xBind; xSeg := segment xBind
      y := variable yBind; ySeg := segment yBind
      -- call 'draw'
      draw(makeFloatFunction(f,x,y),xSeg,ySeg,l)

    draw(f:Ex,xBind:BIND,yBind:BIND) ==
      draw(f,xBind,yBind,nil())

--% parametric surface

    makeObject(s:PSF,uBind:BIND,vBind:BIND,l:L DROP) ==
      import DROP
      f := coordinate(s,1); g := coordinate(s,2); h := coordinate(s,3)
      if not option?(l,"title" :: Symbol) then
        s:String := unparse(convert(f)@InputForm)
        if sayLength(s)$DisplayPackage > 50 then
          l := concat(title "AXIOM3D",l)
        else l := concat(title s,l)
      if not option?(l,"style" :: Symbol) then
        l := concat(style unparse(convert(f)@InputForm),l)
      u := variable uBind; uSeg := segment uBind
      v := variable vBind; vSeg := segment vBind
      surf : PSFF := surface(makeFloatFunction(f,u,v),_
                             makeFloatFunction(g,u,v),_
                             makeFloatFunction(h,u,v))
      makeObject(surf,uSeg,vSeg,l)

    makeObject(s:PSF,uBind:BIND,vBind:BIND) ==
      makeObject(s,uBind,vBind,nil())

    draw(s:PSF,uBind:BIND,vBind:BIND,l:L DROP) ==
      import DROP
      f := coordinate(s,1); g := coordinate(s,2); h := coordinate(s,3)
      -- create title if necessary
      if not option?(l,"title" :: Symbol) then
        s:String := unparse(convert(f)@InputForm)
        if sayLength(s)$DisplayPackage > 50 then
          l := concat(title "AXIOM3D",l)
        else l := concat(title s,l)
      -- indicate draw style if necessary
      if not option?(l,"style" :: Symbol) then
        l := concat(style unparse(convert(f)@InputForm),l)
      -- obtain dependent variables and their ranges
      u := variable uBind; uSeg := segment uBind
      v := variable vBind; vSeg := segment vBind
      -- create surface with functions as coordinates
      surf : PSFF := surface(makeFloatFunction(f,u,v),_
                             makeFloatFunction(g,u,v),_
                             makeFloatFunction(h,u,v))
      -- call 'draw'
      draw(surf,uSeg,vSeg,l)

    draw(s:PSF,uBind:BIND,vBind:BIND) ==
      draw(s,uBind,vBind,nil())

@
\section{package DRAWCURV TopLevelDrawFunctionsForAlgebraicCurves}
<<package DRAWCURV TopLevelDrawFunctionsForAlgebraicCurves>>=
)abbrev package DRAWCURV TopLevelDrawFunctionsForAlgebraicCurves
++ Author: Clifton J. Williamson
++ Date Created: 26 June 1990
++ Date Last Updated:  October 1991 by Jon Steinbach
++ Basic Operations: draw
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description: TopLevelDrawFunctionsForAlgebraicCurves provides top level 
++ functions for drawing non-singular algebraic curves.

TopLevelDrawFunctionsForAlgebraicCurves(R,Ex): Exports == Implementation where
  R  : Join(IntegralDomain, OrderedSet, RetractableTo Integer)
  Ex : FunctionSpace(R)

  ANY1  ==> AnyFunctions1
  DROP  ==> DrawOption
  EQ    ==> Equation
  F     ==> Float
  FRAC  ==> Fraction
  I     ==> Integer
  L     ==> List
  P     ==> Polynomial
  RN    ==> Fraction Integer
  SEG   ==> Segment
  SY    ==> Symbol
  VIEW2 ==> TwoDimensionalViewport

  Exports ==> with

    draw: (EQ Ex,SY,SY,L DROP) -> VIEW2
      ++ draw(f(x,y) = g(x,y),x,y,l) draws the graph of a polynomial
      ++ equation.  The list l of draw options must specify a region
      ++ in the plane in which the curve is to sketched.

  Implementation ==> add
    import ViewportPackage
    import PlaneAlgebraicCurvePlot
    import ViewDefaultsPackage
    import GraphicsDefaults
    import DrawOptionFunctions0
    import SegmentFunctions2(RN,F)
    import SegmentFunctions2(F,RN)
    import AnyFunctions1(L SEG RN)
    import DROP

    drawToScaleRanges: (SEG F,SEG F) -> L SEG F
    drawToScaleRanges(xVals,yVals) ==
      -- warning: assumes window is square
      xHi := hi xVals; xLo := lo xVals
      yHi := hi yVals; yLo := lo yVals
      xDiff := xHi - xLo; yDiff := yHi - yLo
      pad := abs(yDiff - xDiff)/2
      yDiff > xDiff =>
        [segment(xLo - pad,xHi + pad),yVals]
      [xVals,segment(yLo - pad,yHi + pad)]

    intConvert: R -> I
    intConvert r ==
      (nn := retractIfCan(r)@Union(I,"failed")) case "failed" =>
        error "draw: polynomial must have rational coefficients"
      nn :: I

    polyEquation: EQ Ex -> P I
    polyEquation eq ==
      ff := lhs(eq) - rhs(eq)
      (r := retractIfCan(ff)@Union(FRAC P R,"failed")) case "failed" =>
        error "draw: not a polynomial equation"
      rat := r :: FRAC P R
      retractIfCan(denom rat)@Union(R,"failed") case "failed" =>
        error "draw: non-constant denominator"
      map(intConvert,numer rat)$PolynomialFunctions2(R,I)

    draw(eq,x,y,l) ==
      -- obtain polynomial equation
      p := polyEquation eq
      -- extract ranges from option list
      floatRange := option(l,"rangeFloat" :: Symbol)
      ratRange := option(l,"rangeRat" :: Symbol)
      (floatRange case "failed") and (ratRange case "failed") =>
        error "draw: you must specify ranges for an implicit plot"
      ranges : L SEG RN := nil()             -- dummy value
      floatRanges : L SEG F := nil()         -- dummy value
      xRange : SEG RN := segment(0,0)        -- dummy value
      yRange : SEG RN := segment(0,0)        -- dummy value
      xRangeFloat : SEG F := segment(0,0)    -- dummy value
      yRangeFloat : SEG F := segment(0,0)    -- dummy value
      if not ratRange case "failed" then
        ranges := retract(ratRange :: Any)$ANY1(L SEG RN)
        #ranges ~= 2 => error "draw: you must specify two ranges"
        xRange := first ranges; yRange := second ranges
        xRangeFloat := map(convert(#1)@Float,xRange)@(SEG F)
        yRangeFloat := map(convert(#1)@Float,yRange)@(SEG F)
        floatRanges := [xRangeFloat,yRangeFloat]
      else
        floatRanges := retract(floatRange :: Any)$ANY1(L SEG F)
        #floatRanges ~= 2 =>
          error "draw: you must specify two ranges"
        xRangeFloat := first floatRanges
        yRangeFloat := second floatRanges
        xRange := map(retract(#1)@RN,xRangeFloat)@(SEG RN)
        yRange := map(retract(#1)@RN,yRangeFloat)@(SEG RN)
        ranges := [xRange,yRange]
      -- create curve plot
      acplot := makeSketch(p,x,y,xRange,yRange)
      -- process scaling information
      if toScale(l,drawToScale()) then
        scaledRanges := drawToScaleRanges(xRangeFloat,yRangeFloat)
        -- add scaled ranges to list of options
        l := concat(ranges scaledRanges,l)
      else
        -- add ranges to list of options
        l := concat(ranges floatRanges,l)
      -- process color information
      ptCol := pointColorPalette(l,pointColorDefault())
      crCol := curveColorPalette(l,lineColorDefault())
      -- draw
      drawCurves(listBranches acplot,ptCol,crCol,pointSizeDefault(),l)

@
\section{package DRAWPT TopLevelDrawFunctionsForPoints}
<<package DRAWPT TopLevelDrawFunctionsForPoints>>=
)abbrev package DRAWPT TopLevelDrawFunctionsForPoints
++ Author: Mike Dewar
++ Date Created: 24 May 1995
++ Date Last Updated: 25 November 1996
++ Basic Operations: draw
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description: TopLevelDrawFunctionsForPoints provides top level functions for 
++ drawing curves and surfaces described by sets of points.
 
TopLevelDrawFunctionsForPoints(): Exports == Implementation where

  DROP  ==> DrawOption
  L     ==> List
  SF    ==> DoubleFloat
  Pt    ==> Point SF
  VIEW2 ==> TwoDimensionalViewport
  VIEW3 ==> ThreeDimensionalViewport

  Exports ==> with
    draw: (L SF,L SF) -> VIEW2
      ++ draw(lx,ly) plots the curve constructed of points (x,y) for x
      ++ in \spad{lx} for y in \spad{ly}.
    draw: (L SF,L SF,L DROP) -> VIEW2
      ++ draw(lx,ly,l) plots the curve constructed of points (x,y) for x
      ++ in \spad{lx} for y in \spad{ly}.
      ++ The options contained in the list l of
      ++ the domain \spad{DrawOption} are applied.
    draw: (L Pt) -> VIEW2
      ++ draw(lp) plots the curve constructed from the list of points lp.
    draw: (L Pt,L DROP) -> VIEW2
      ++ draw(lp,l) plots the curve constructed from the list of points lp.
      ++ The options contained in the list l of the domain \spad{DrawOption}
      ++ are applied.
    draw: (L SF, L SF, L SF) -> VIEW3
      ++ draw(lx,ly,lz) draws the surface constructed by projecting the values
      ++ in the \axiom{lz} list onto the rectangular grid formed by the 
      ++ \axiom{lx X ly}.
    draw: (L SF, L SF, L SF, L DROP) -> VIEW3
      ++ draw(lx,ly,lz,l) draws the surface constructed by projecting the values
      ++ in the \axiom{lz} list onto the rectangular grid formed by the 
      ++ The options contained in the list l of the domain \spad{DrawOption}
      ++ are applied.

  Implementation ==> add

    draw(lp:L Pt,l:L DROP):VIEW2 ==
      makeViewport2D(makeGraphImage([lp])$GraphImage,l)$VIEW2

    draw(lp:L Pt):VIEW2 == draw(lp,[])

    draw(lx: L SF, ly: L SF, l:L DROP):VIEW2 ==
      draw([point([x,y])$Pt for x in lx for y in ly],l)

    draw(lx: L SF, ly: L SF):VIEW2 == draw(lx,ly,[])

    draw(x:L SF,y:L SF,z:L SF):VIEW3 == draw(x,y,z,[])

    draw(x:L SF,y:L SF,z:L SF,l:L DROP):VIEW3 ==
      m  : Integer := #x
      zero? m => error "No X values"
      n  : Integer := #y
      zero? n => error "No Y values"
      zLen : Integer := #z
      zLen ~= (m*n) => 
        zLen > (m*n) => error "Too many Z-values to fit grid"
        error "Not enough Z-values to fit grid"
      points : L L Pt := []
      for j in n..1 by -1 repeat
        row : L Pt := []
        for i in m..1 by -1 repeat
          zval := (j-1)*m+i
          row := cons(point([x.i,y.j,z.zval,z.zval]),row)
        points := cons(row,points)
      makeViewport3D(mesh points,l)

@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
--    - Redistributions of source code must retain the above copyright
--      notice, this list of conditions and the following disclaimer.
--
--    - Redistributions in binary form must reproduce the above copyright
--      notice, this list of conditions and the following disclaimer in
--      the documentation and/or other materials provided with the
--      distribution.
--
--    - Neither the name of The Numerical ALgorithms Group Ltd. nor the
--      names of its contributors may be used to endorse or promote products
--      derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>

<<package DRAWCFUN TopLevelDrawFunctionsForCompiledFunctions>>
<<package DRAW TopLevelDrawFunctions>>
<<package DRAWCURV TopLevelDrawFunctionsForAlgebraicCurves>>
<<package DRAWPT TopLevelDrawFunctionsForPoints>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}