summaryrefslogtreecommitdiff
path: root/Master/texmf-dist/tex/lualatex/bezierplot/bezierplot.lua
blob: da702f987fcfc23ca259564fc11fbc3e10cd0777 (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
#!/usr/bin/env lua
-- Linus Romer, published 2018 under LPPL Version 1.3c
-- version 1.0 2018-04-12
abs = math.abs
acos = math.acos
asin = math.asin
atan = math.atan
cos = math.cos
exp = math.exp
e = math.exp(1)
log = math.log
pi = math.pi
sin = math.sin
sqrt = math.sqrt
tan = math.tan

-- cube root defined for all real numbers x
function cbrt(x)
	if x < 0 then
		return -(-x)^(1/3)
	else
		return x^(1/3)
	end
end

function sgn(x)
	if x<0 then
		return -1
	elseif x>0 then
		return 1
	else
		return 0
	end
end

function round(num, decimals)
	local result = tonumber(string.format("%." .. (decimals or 0) .. "f", num))
	if abs(result) == 0 then
		return 0
	else
		return result
	end
end

-- 5-stencil method
-- return from a graph from f in the form {{x,y},...}
-- the derivatives in form {{x,y,dy/dx,ddy/ddx},...}
function diffgraph(func,graph,h)
	local dgraph = {}	
	local yh = func(graph[1][1]-h)
	local yhh = func(graph[1][1]-2*h)
	if yhh > -math.huge and yhh < math.huge  -- if defined at all
	and yh > -math.huge and yh < math.huge then
		dgraph[1] = {graph[1][1],graph[1][2],
		(yhh-8*yh+8*graph[2][2]-graph[3][2])/(12*h),
		(-yhh+16*yh-30*graph[1][2]+16*graph[2][2]-graph[3][2])
		/(12*h^2)}
		dgraph[2] = {graph[2][1],graph[2][2],
		(yh-8*graph[1][2]+8*graph[3][2]-graph[4][2])/(12*h),
		(-yh+16*graph[1][2]-30*graph[2][2]+16*graph[3][2]-graph[4][2])
		/(12*h^2)}
	else -- take neighbour values
		dgraph[1] = {graph[1][1],graph[1][2],
		(graph[1][2]-8*graph[2][2]+8*graph[4][2]-graph[5][2])/(12*h),
		(-graph[1][2]+16*graph[2][2]-30*graph[3][2]
		+16*graph[4][2]-graph[5][2])/(12*h^2)}
		dgraph[2] = {graph[2][1],graph[2][2],
		(graph[1][2]-8*graph[2][2]+8*graph[4][2]-graph[5][2])/(12*h),
		(-graph[1][2]+16*graph[2][2]-30*graph[3][2]
		+16*graph[4][2]-graph[5][2])/(12*h^2)}
	end
	local l = #graph
	for i = 3, l-2 do 
		table.insert(dgraph,{graph[i][1],graph[i][2],
		(graph[i-2][2]-8*graph[i-1][2]+8*graph[i+1][2]-graph[i+2][2])
		/(12*h),
		(-graph[i-2][2]+16*graph[i-1][2]-30*graph[i][2]
		+16*graph[i+1][2]-graph[i+2][2])
		/(12*h^2)})
	end
	yh = func(graph[l][1]+h)
	yhh = func(graph[l][1]+2*h)
	if yhh > -math.huge and yhh < math.huge  -- if defined at all
	and yh > -math.huge and yh < math.huge then
		dgraph[l-1] = {graph[l-1][1],graph[l-1][2],
		(graph[l-3][2]-8*graph[l-2][2]+8*graph[l][2]-yh)/(12*h),
		(-graph[l-3][2]+16*graph[l-2][2]-30*graph[l-1][2]
		+16*graph[l][2]-yh)/(12*h^2)}
		dgraph[l] = {graph[l][1],graph[l][2],
		(graph[l-2][2]-8*graph[l-1][2]+8*yh-yhh)/(12*h),
		(-graph[l-2][2]+16*graph[l-1][2]-30*graph[l][2]
		+16*yh-yhh)/(12*h^2)}
	else	
		-- take neighbour values
		dgraph[l] = {graph[l][1],graph[l][2],
		(graph[l-4][2]-8*graph[l-3][2]+8*graph[l-1][2]-graph[l][2])
		/(12*h),
		(-graph[l-4][2]+16*graph[l-3][2]-30*graph[l-2][2]
		+16*graph[l-1][2]-graph[l][2])/(12*h^2)}
		dgraph[l-1] = {graph[l-1][1],graph[l-2][2],
		(graph[l-4][2]-8*graph[l-3][2]+8*graph[l-1][2]-graph[l][2])
		/(12*h),
		(-graph[l-4][2]+16*graph[l-3][2]-30*graph[l-2][2]
		+16*graph[l-1][2]-graph[l][2])/(12*h^2)}
	end
	-- add information about being extremum / inflection point (true/false)
	for i = 1, l do 
		dgraph[i][5] = false -- dy/dx == 0 ? default, may change later
		dgraph[i][6] = false -- ddy/ddx == 0 ? default, may change later
	end
	for i = 1, l-1 do 
		-- if no gap is inbetween
		if not (dgraph[i+1][1] - dgraph[i][1] > 1.5*h) then
			-- check for dy/dx == 0 
			-- if not already determined as near dy/dx=0
			if not dgraph[i][5] then 
				if dgraph[i][3] == 0 then
					dgraph[i][5] = true
				elseif abs(dgraph[i][3]*dgraph[i+1][3]) 
				~= dgraph[i][3]*dgraph[i+1][3] then -- this must be near
					if abs(dgraph[i][4]) <= abs(dgraph[i+1][4]) then
						dgraph[i][5] = true
					else
						dgraph[i+1][5] = true
					end
				end
			end
			-- check for ddy/ddx == 0 
			-- if not already determined as near ddy/ddx=0
			if not dgraph[i][6] then 
				if abs(dgraph[i][4]*dgraph[i+1][4]) 
				~= dgraph[i][4]*dgraph[i+1][4] then -- this must be near
					if abs(dgraph[i][4]) <= abs(dgraph[i+1][4]) then
						dgraph[i][6] = true
					else
						dgraph[i+1][6] = true
					end
				end
			end
		end
	end
	return dgraph
end

-- checks for 100 x, if the function given by funcstring
-- fits the graph g (up to maxerror) after filling in
-- the parameters a, b, c, d
-- if the graph is inverted, then isinverse has to be set true
function do_parameters_fit(a,b,c,d,funcstring,funcgraph,maxerror,isinverse)
	local funcx = string.gsub(funcstring, "a", a)
	local funcx = string.gsub(funcx, "b", b)
	local funcx = string.gsub(funcx, "c", c)
	local funcx = string.gsub(funcx, "d", d)
	local func = assert(load("local x = ...; return "..funcx))
	for i = 1, #funcgraph, math.max(1,math.floor(0.01*#funcgraph)) do 
		if isinverse then
			if abs(func(funcgraph[i][2])-funcgraph[i][1]) 
			> maxerror then
				return false
			end
		else
			if abs(func(funcgraph[i][1])-funcgraph[i][2]) 
			> maxerror then
				return false
			end
		end
	end
	return true
end

-- f(x)=a*x^3+b*x+c
function parameters_cubic(xp,yp,xq,yq,xr,yr,xs,ys)
	local a = (((xp^2 * xq) * yr) - ((xp^2 * xq) * ys) 
	- ((xp^2 * xr) * yq) + ((xp^2 * xr) * ys) + ((xp^2 * xs) * yq)
	- ((xp^2 * xs) * yr) - ((xp * xq^2) * yr) + ((xp * xq^2) * ys) 
	+ ((xp * xr^2) * yq) - ((xp * xr^2) * ys) - ((xp * xs^2) * yq) 
	+ ((xp * xs^2) * yr) + ((xq^2 * xr) * yp) - ((xq^2 * xr) * ys) 
	- ((xq^2 * xs) * yp) + ((xq^2 * xs) * yr) - ((xq * xr^2) * yp) 
	+ ((xq * xr^2) * ys) + ((xq * xs^2) * yp) - ((xq * xs^2) * yr) 
	+ ((xr^2 * xs) * yp) - ((xr^2 * xs) * yq) - ((xr * xs^2) * yp) 
	+ ((xr * xs^2) * yq)) / 
	(((xp^3 * xq^2) * xr) - ((xp^3 * xq^2) * xs) 
	- ((xp^3 * xq) * xr^2) + ((xp^3 * xq) * xs^2) 
	+ ((xp^3 * xr^2) * xs) - ((xp^3 * xr) * xs^2) 
	- ((xp^2 * xq^3) * xr) + ((xp^2 * xq^3) * xs) 
	+ ((xp^2 * xq) * xr^3) - ((xp^2 * xq) * xs^3) 
	- ((xp^2 * xr^3) * xs) + ((xp^2 * xr) * xs^3) 
	+ ((xp * xq^3) * xr^2) - ((xp * xq^3) * xs^2) 
	- ((xp * xq^2) * xr^3) + ((xp * xq^2) * xs^3) 
	+ ((xp * xr^3) * xs^2) - ((xp * xr^2) * xs^3) 
	- ((xq^3 * xr^2) * xs) + ((xq^3 * xr) * xs^2) 
	+ ((xq^2 * xr^3) * xs) - ((xq^2 * xr) * xs^3) 
	- ((xq * xr^3) * xs^2) + ((xq * xr^2) * xs^3))
	local b = ((((-xp^3) * xq) * yr) + ((xp^3 * xq) * ys) 
	+ ((xp^3 * xr) * yq) - ((xp^3 * xr) * ys) - ((xp^3 * xs) * yq) 
	+ ((xp^3 * xs) * yr) + ((xp * xq^3) * yr) - ((xp * xq^3) * ys) 
	- ((xp * xr^3) * yq) + ((xp * xr^3) * ys) + ((xp * xs^3) * yq) 
	- ((xp * xs^3) * yr) - ((xq^3 * xr) * yp) + ((xq^3 * xr) * ys) 
	+ ((xq^3 * xs) * yp) - ((xq^3 * xs) * yr) + ((xq * xr^3) * yp) 
	- ((xq * xr^3) * ys) - ((xq * xs^3) * yp) + ((xq * xs^3) * yr) 
	- ((xr^3 * xs) * yp) + ((xr^3 * xs) * yq) + ((xr * xs^3) * yp) 
	- ((xr * xs^3) * yq)) / 
	(((xp^3 * xq^2) * xr) - ((xp^3 * xq^2) * xs) 
	- ((xp^3 * xq) * xr^2) + ((xp^3 * xq) * xs^2) 
	+ ((xp^3 * xr^2) * xs) - ((xp^3 * xr) * xs^2) 
	- ((xp^2 * xq^3) * xr) + ((xp^2 * xq^3) * xs) 
	+ ((xp^2 * xq) * xr^3) - ((xp^2 * xq) * xs^3) 
	- ((xp^2 * xr^3) * xs) + ((xp^2 * xr) * xs^3) 
	+ ((xp * xq^3) * xr^2) - ((xp * xq^3) * xs^2) 
	- ((xp * xq^2) * xr^3) + ((xp * xq^2) * xs^3) 
	+ ((xp * xr^3) * xs^2) - ((xp * xr^2) * xs^3) 
	- ((xq^3 * xr^2) * xs) + ((xq^3 * xr) * xs^2) 
	+ ((xq^2 * xr^3) * xs) - ((xq^2 * xr) * xs^3) 
	- ((xq * xr^3) * xs^2) + ((xq * xr^2) * xs^3))
	local c = (((xp^3 * xq^2) * yr) - ((xp^3 * xq^2) * ys) 
	- ((xp^3 * xr^2) * yq) + ((xp^3 * xr^2) * ys) 
	+ ((xp^3 * xs^2) * yq) - ((xp^3 * xs^2) * yr) 
	- ((xp^2 * xq^3) * yr) + ((xp^2 * xq^3) * ys) 
	+ ((xp^2 * xr^3) * yq) - ((xp^2 * xr^3) * ys) 
	- ((xp^2 * xs^3) * yq) + ((xp^2 * xs^3) * yr) 
	+ ((xq^3 * xr^2) * yp) - ((xq^3 * xr^2) * ys) 
	- ((xq^3 * xs^2) * yp) + ((xq^3 * xs^2) * yr) 
	- ((xq^2 * xr^3) * yp) + ((xq^2 * xr^3) * ys) 
	+ ((xq^2 * xs^3) * yp) - ((xq^2 * xs^3) * yr) 
	+ ((xr^3 * xs^2) * yp) - ((xr^3 * xs^2) * yq) 
	- ((xr^2 * xs^3) * yp) + ((xr^2 * xs^3) * yq)) / 
	(((xp^3 * xq^2) * xr) - ((xp^3 * xq^2) * xs) 
	- ((xp^3 * xq) * xr^2) + ((xp^3 * xq) * xs^2) 
	+ ((xp^3 * xr^2) * xs) - ((xp^3 * xr) * xs^2) 
	- ((xp^2 * xq^3) * xr) + ((xp^2 * xq^3) * xs) 
	+ ((xp^2 * xq) * xr^3) - ((xp^2 * xq) * xs^3) 
	- ((xp^2 * xr^3) * xs) + ((xp^2 * xr) * xs^3) 
	+ ((xp * xq^3) * xr^2) - ((xp * xq^3) * xs^2) 
	- ((xp * xq^2) * xr^3) + ((xp * xq^2) * xs^3) 
	+ ((xp * xr^3) * xs^2) - ((xp * xr^2) * xs^3) 
	- ((xq^3 * xr^2) * xs) + ((xq^3 * xr) * xs^2) 
	+ ((xq^2 * xr^3) * xs) - ((xq^2 * xr) * xs^3) 
	- ((xq * xr^3) * xs^2) + ((xq * xr^2) * xs^3))
	local d = ((((xp^(3) * xq^(2)) * xr) * ys) 
	- (((xp^(3) * xq^(2)) * xs) * yr) - (((xp^(3) * xq) * xr^(2)) * ys) 
	+ (((xp^(3) * xq) * xs^(2)) * yr) + (((xp^(3) * xr^(2)) * xs) * yq) 
	- (((xp^(3) * xr) * xs^(2)) * yq) - (((xp^(2) * xq^(3)) * xr) * ys) 
	+ (((xp^(2) * xq^(3)) * xs) * yr) + (((xp^(2) * xq) * xr^(3)) * ys) 
	- (((xp^(2) * xq) * xs^(3)) * yr) - (((xp^(2) * xr^(3)) * xs) * yq) 
	+ (((xp^(2) * xr) * xs^(3)) * yq) + (((xp * xq^(3)) * xr^(2)) * ys) 
	- (((xp * xq^(3)) * xs^(2)) * yr) - (((xp * xq^(2)) * xr^(3)) * ys) 
	+ (((xp * xq^(2)) * xs^(3)) * yr) + (((xp * xr^(3)) * xs^(2)) * yq) 
	- (((xp * xr^(2)) * xs^(3)) * yq) - (((xq^(3) * xr^(2)) * xs) * yp) 
	+ (((xq^(3) * xr) * xs^(2)) * yp) + (((xq^(2) * xr^(3)) * xs) * yp) 
	- (((xq^(2) * xr) * xs^(3)) * yp) - (((xq * xr^(3)) * xs^(2)) * yp) 
	+ (((xq * xr^(2)) * xs^(3)) * yp)) / 
	(((xp^(3) * xq^(2)) * xr) - 
	((xp^(3) * xq^(2)) * xs) - ((xp^(3) * xq) * xr^(2)) 
	+ ((xp^(3) * xq) * xs^(2)) + ((xp^(3) * xr^(2)) * xs) 
	- ((xp^(3) * xr) * xs^(2)) - ((xp^(2) * xq^(3)) * xr) 
	+ ((xp^(2) * xq^(3)) * xs) + ((xp^(2) * xq) * xr^(3)) 
	- ((xp^(2) * xq) * xs^(3)) - ((xp^(2) * xr^(3)) * xs) 
	+ ((xp^(2) * xr) * xs^(3)) + ((xp * xq^(3)) * xr^(2)) 
	- ((xp * xq^(3)) * xs^(2)) - ((xp * xq^(2)) * xr^(3)) 
	+ ((xp * xq^(2)) * xs^(3)) + ((xp * xr^(3)) * xs^(2)) 
	- ((xp * xr^(2)) * xs^(3)) - ((xq^(3) * xr^(2)) * xs) 
	+ ((xq^(3) * xr) * xs^(2)) + ((xq^(2) * xr^(3)) * xs) 
	- ((xq^(2) * xr) * xs^(3)) - ((xq * xr^(3)) * xs^(2)) 
	+ ((xq * xr^(2)) * xs^(3)))
	return a, b, c, d
end

-- f(x)=a*x+b
function parameters_affine(xp,yp,xq,yq)
	local a = (yp - yq) / (xp - xq)
	local b = ((xp * yq) - (xq * yp)) / (xp - xq)
	return a, b
end

-- returns true iff the function is of type f(x)=a*x^3+b*x^2+c*x+d
-- a, b, c, d being real numbers
function is_cubic(graph,maxerror)
	local l = #graph
	local a, b, c, d = parameters_cubic(graph[1][1],graph[1][2],
	graph[math.floor(l/3)][1],graph[math.floor(l/3)][2],
	graph[math.floor(2*l/3)][1],graph[math.floor(2*l/3)][2],
	graph[l][1],graph[l][2])
	return do_parameters_fit(a,b,c,d,"a*x^3+b*x^2+c*x+d",graph,
	maxerror,false)
end

-- returns true iff the function is of type f(x)=a*x^3+b*x^2+c*x+d
-- a, b, c, d being real numbers
-- this takes several graph parts
-- the idea is to have a possibility to avoid tan(x)
function are_cubic(graphs,maxerror)
	if is_cubic(graphs[1],maxerror) then
		if #graphs < 2 then
			return true
		else -- check for the next part
			local a, b, c, d = parameters_cubic(graphs[1][1][1],
			graphs[1][1][2],graphs[1][math.floor(l/3)][1],
			graphs[1][math.floor(l/3)][2],
			graphs[1][math.floor(2*l/3)][1],
			graphs[1][math.floor(2*l/3)][2],
			graphs[1][l][1],graphs[1][l][2])
			return do_parameters_fit(a,b,c,d,"a*x^3+b*x^2+c*x+d",
			graphs[2],maxerror,false)
		end
	else
		return false
	end
end

-- returns true iff the inverse function is of type 
-- f(x)=a*x^3+b*x^2+c*x+d
-- a, b, c, d being real numbers
function is_cuberoot(graph,maxerror)
	local l = #graph
	local a, b, c, d = parameters_cubic(graph[1][2],graph[1][1],
	graph[math.floor(l/3)][2],graph[math.floor(l/3)][1],
	graph[math.floor(2*l/3)][2],graph[math.floor(2*l/3)][1],
	graph[l][2],graph[l][1])
	return do_parameters_fit(a,b,c,d,"a*x^3+b*x^2+c*x+d",graph,
	maxerror,true)
end

-- returns true iff the function is of type f(x)=a*x^3+b*x^2+c*x+d
-- a, b, c, d being real numbers
-- this takes several graph parts
-- the idea is to have a possibility to avoid tan(x)
function are_cuberoot(graphs,maxerror)
	if is_cuberoot(graphs[1],maxerror) then
		if #graphs < 2 then
			return true
		else -- check for the next part
			local a, b, c, d = parameters_cubic(graphs[1][1][2],
			graphs[1][1][1],graphs[1][math.floor(l/3)][2],
			graphs[1][math.floor(l/3)][1],
			graphs[1][math.floor(2*l/3)][2],
			graphs[1][math.floor(2*l/3)][1],
			graphs[1][l][2],graphs[1][l][1])
			return do_parameters_fit(a,b,c,d,"a*x^3+b*x^2+c*x+d",
			graphs[2],maxerror,true)
		end
	else
		return false
	end
end

-- returns true iff function is of type f(x)=a*x+b
-- a, b being real numbers
function is_affine(graph,maxerror)
	l = #graph
	local a, b = parameters_affine(graph[1][1],graph[1][2],
	graph[l][1],graph[l][2])
	return do_parameters_fit(a,b,0,0,"a*x+b",graph,maxerror,false)
end

-- what is the sum of the squared error
-- when comparing the bezier path
-- p.. control q and r .. s
-- with the graph g from index starti to endi
-- (looking at the points at roughly t=.33 and t=.67)
function squareerror(f,g,starti,endi,qx,qy,rx,ry)
	local result = 0
	for t = .33, .7, .34 do
		x = (1-t)^3*g[starti][1]+3*t*(1-t)^2*qx+3*t^2*(1-t)*rx+t^3*g[endi][1]
		y = (1-t)^3*g[starti][2]+3*t*(1-t)^2*qy+3*t^2*(1-t)*ry+t^3*g[endi][2]
		result = result + (y-f(x))^2
	end
	return result
end

function pointstobezier(qx,qy,rx,ry,sx,sy,rndx,rndy)
	return " .. controls (" .. round(qx,rndx) .. "," 
	.. round(qy,rndy) ..") and ("
	.. round(rx,rndx) .. "," 
	.. round(ry,rndy) .. ") .. (" 
	.. round(sx,rndx) .. "," 
	.. round(sy,rndy)..")"
end

-- take end points of a graph g of the function f
-- (from indices starti to endi)
-- without extrema or inflection points inbetween 
-- and try to approximate it with a cubic bezier curve
-- (round to rndx and rndy when printing)
function graphtobezierapprox(f,g,starti,endi,rndx,rndy,maxerror)
	local px = g[starti][1]
	local py = g[starti][2]
	local dp = g[starti][3]
	local sx = g[endi][1]
	local sy = g[endi][2]
	local ds = g[endi][3]
	-- we compute the corner point c, where the controls would meet
	local cx = ((dp * px) - (ds * sx) - py + sy) / (dp - ds)
	local cy = (dp * ((ds * px) - (ds * sx) - py + sy) / (dp - ds)) + py
	-- now we slide q between p and c & r between s and c
	-- and search for the best qx and best rx
	local qx = px+.05*(cx-px)
	local qy = py+.05*(cy-py)
	local rx = sx+.05*(cx-sx)
	local ry = sy+.05*(cy-sy)
	local err = squareerror(f,g,starti,endi,qx,qy,rx,ry)
	for i = 2, 19 do
		for j = 2, 19 do
			xa = px+i*.05*(cx-px)
			ya = py+i*.05*(cy-py)
			xb = sx+j*.05*(cx-sx)
			yb = sy+j*.05*(cy-sy)
			-- now check, if xa and xb fit better
			-- at roughly t=0.33 and t=0.66 for f(x)
			-- than the last qx and rx did
			-- (sum of squares must be smaller)
			if squareerror(f,g,starti,endi,xa,ya,xb,yb) < err then
				qx = xa
				qy = ya
				rx = xb
				ry = yb
				err = squareerror(f,g,starti,endi,qx,qy,rx,ry)
			end
		end
	end
	-- check if it is close enough: (recycling err, xa, ya)
	err = 0
	for t = .1, .9, .1 do
		xa = (1-t)^3*g[starti][1]+3*t*(1-t)^2*qx+3*t^2*(1-t)*rx+t^3*g[endi][1]
		ya = (1-t)^3*g[starti][2]+3*t*(1-t)^2*qy+3*t^2*(1-t)*ry+t^3*g[endi][2]
		if abs(ya-f(xa)) > err then
			err = abs(ya-f(xa))
		end
	end
	if err <= maxerror then
		return pointstobezier(qx,qy,rx,ry,sx,sy,rndx,rndy)
	else
		-- search for an intermediate point where the graph has the same
		-- slope as the line from the start point to the end point:
		local interindex = math.floor(.5*starti+.5*endi) -- will change
		for i = starti + 1, endi - 1 do
			if abs(g[i][3]-(g[endi][2]-g[starti][2])
			/(g[endi][1]-g[starti][1])) 
			< abs(g[interindex][3]-(g[endi][2]-g[starti][2])
			/(g[endi][1]-g[starti][1])) then
				interindex = i
			end
		end
		return graphtobezierapprox(f,g,starti,interindex,rndx,rndy,maxerror)
		.. graphtobezierapprox(f,g,interindex,endi,rndx,rndy,maxerror)
	end
end

-- like above but exact for quadratic and cubic (if not inverse)
-- resp. exact for squareroot and cuberoot (if inverse)
function graphtobezier(g,starti,endi,rndx,rndy,isinverse)
	local px = g[starti][1]
	local py = g[starti][2]
	local dp = g[starti][3]
	local sx = g[endi][1]
	local sy = g[endi][2]
	local ds = g[endi][3]
	local qx = px+(sx-px)/3
	local rx = px+2*(sx-px)/3
	local qy = py+(qx-px)*dp
	local ry = sy+(rx-sx)*ds
	if isinverse then
		return pointstobezier(qy,qx,ry,rx,sy,sx,rndy,rndx)
	else
		return pointstobezier(qx,qy,rx,ry,sx,sy,rndx,rndy)
	end
end

-- reverses a path p e.g. when p = "(0,1) -- (2,3)"
-- it returns "(2,3) -- (0,1)"
-- or when p = "(0,1) .. controls (2,3) and (4,5) .. (6,7)"
-- it returns "(6,7) .. controls (4,5) and (2,3) .. (0,1)"
function reversepath(p)
	local r = "" -- will become the reverse path
	local temppoint  ="" -- will store temporal points like "(0,1)"
	local tempbetween = "" -- will store things like " .. controls "
	for i = 1, #p do
		local c = string.sub(p,i,i)
		if c == "(" then
			if tempbetween == " .. " then
				r = " .. controls " .. r
			elseif tempbetween == " .. controls " then
				r = " .. " .. r
			else
				r = tempbetween .. r
			end
			tempbetween = ""
			temppoint = "("
		elseif c == ")" then
			r = temppoint .. ")" .. r
			temppoint = ""
		else
			if temppoint == "" then -- not reading a point
				tempbetween = tempbetween .. c
			else
				temppoint = temppoint .. c
			end
		end
	end
	return r
end

-- main function
function bezierplot(functionstring,xmin,xmax,ymin,ymax)
	local fstringreplaced = string.gsub(functionstring, "%*%*", "^")
	local f = assert(load("local x = ...; return " .. fstringreplaced)) 
	local isreverse = false
	if xmin > xmax then
		isreverse = true
	end
	xmin, xmax = math.min(xmin,xmax), math.max(xmin,xmax)
	local xstep = (xmax-xmin)/20000
	-- the output of the x coordinates will be rounded to rndx digits
	local rndx = math.max(0,math.floor(4.5-log(xmax-xmin)/log(10)))
	local xerror = abs(xmax-xmin)/(100*10^rndx)
	ymin, ymax = math.min(ymin,ymax), math.max(ymin,ymax)
	-- the output of the x coordinates will be rounded to rndy digits
	local rndy = math.max(0,math.floor(4.5-log(ymax-ymin)/log(10)))
	local yerror = (ymax-ymin)/(100*10^rndy)
	-- determine parts of the graph that are inside window
	local graphs = {}
	local outside = true -- value is outside window
	local i = 0
	local j = 0
	for n = 0, 20000 do
		local x = xmin + n/20000*(xmax-xmin)
		local y = f(x)
		if y >= ymin-yerror and y <= ymax+yerror then -- inside
			if outside then -- if it was outside before
				outside = false
				j = 0
				i = i + 1
				graphs[i] = {}
			end
			j = j + 1
			graphs[i][j] = {x,y}
		else
			outside = true
		end
	end

	local functiontype = "unknown"
	local bezierstring = ""

	-- go through the connected parts
	for part = 1, #graphs do 
		local d = diffgraph(f,graphs[part],xstep)
		-- check for type of function (only for the first part)
		if part == 1 then
			if is_affine(d,yerror) then
				functiontype = "affine"
			elseif are_cubic(graphs,yerror) then
				functiontype = "cubic"
			elseif are_cuberoot(graphs,xerror) then
				functiontype = "cuberoot"
			end
		end
		if functiontype ~= "cuberoot" then -- start with initial point
			bezierstring = bezierstring .. "(" .. round(d[1][1],rndx) 
			.. "," 	.. round(d[1][2],rndy) .. ")"
		end
		if functiontype == "affine" then 
			bezierstring = bezierstring .. " -- (" .. round(d[#d][1],
			rndx) .. "," .. round(d[#d][2],rndy) ..")"
		elseif functiontype == "cubic" then 
			local startindex = 1
			local extremainbetween = false
			for k = 2, #d do
				if d[k][5] then -- extrema 
					extremainbetween = true
					bezierstring = bezierstring 
					.. graphtobezier(d,startindex,k,rndx,rndy,false)
					startindex = k
				end
			end
			if not extremainbetween then
				for k = 2, #d do
					if d[k][6] then -- inflection point
						-- check, if the controlpoints are outside
						-- of the bounding box defined by the vertices
						-- (d[1][1],d[1][2]) and (d[#d][1],d[#d][2])
						local qx = d[1][1]+(d[#d][1]-d[1][1])/3
						local rx = d[1][1]+2*(d[#d][1]-d[1][1])/3
						local qy = d[1][2]+(qx-d[1][1])*d[1][3]
						local ry = d[#d][2]+(rx-d[#d][1])*d[#d][3]
						if math.max(qy,ry) > ymax 
						or math.min(qy,ry) < ymin then
							bezierstring = bezierstring ..graphtobezier(
							d,startindex,k,rndx,rndy,false)
							startindex = k
						end
					end
				end
			end
			if startindex ~= #d then -- if no special points inbetween
				bezierstring = bezierstring 
				.. graphtobezier(d,startindex,#d,rndx,rndy,false)
			end
		elseif functiontype == "cuberoot" then 
			-- we determine a, b, c, d and then
			-- get x' = 3ay^2+2by+c
			local a, b, c, dd = parameters_cubic(
			d[math.floor(.2*l)][2], d[math.floor(.2*l)][1],
			d[math.floor(.4*l)][2], d[math.floor(.4*l)][1],
			d[math.floor(.6*l)][2], d[math.floor(.6*l)][1],
			d[math.floor(.8*l)][2], d[math.floor(.8*l)][1])
			-- now recalculate the graph with the inverse function:
			-- we can increase the accuracy
			xstep = (ymax-ymin)/100000 -- inverse redefinition
			local finverse = assert(load("local x = ...; return "
			..a.."*x^3+"..b.."*x^2+"..c.."*x+"..dd))
			local graphinverse = {}
			local i = 1
			for y = ymin, ymax, xstep do
				local x = finverse(y)
				if x > xmin and x < xmax -- inside
				and abs(y-f(x)) < (ymax-ymin)/(100*10^rndy) then 
					graphinverse[i] = {y,x}
					i = i + 1
				end
			end
			d = diffgraph(finverse,graphinverse,xstep)
			bezierstring = bezierstring .. "(" .. round(d[1][2],rndy) 
			.. "," .. round(d[1][1],rndx) .. ")" -- initial point
			local startindex = 1
			for k = 2, #d do
				if d[k][6] then -- inflection point
					-- check, if the controlpoints are outside
					-- of the bounding box defined by the vertices
					-- (d[1][1],d[1][2]) and (d[#d][1],d[#d][2])
					local qx = d[1][1]+(d[#d][1]-d[1][1])/3
					local rx = d[1][1]+2*(d[#d][1]-d[1][1])/3
					local qy = d[1][2]+(qx-d[1][1])*d[1][3]
					local ry = d[#d][2]+(rx-d[#d][1])*d[#d][3]
					if math.max(qy,ry) > xmax 
					or math.min(qy,ry) < xmin then
						bezierstring = bezierstring..graphtobezier(
						d,startindex,k,rndx,rndy,true)
						startindex = k
					end
				end
			end
			if startindex ~= #d then -- if no special points inbetween
				bezierstring = bezierstring 
				.. graphtobezier(d,startindex,#d,rndx,rndy,true)
			end
		else	
			-- standard case (nothing special)			
			local startindex = 1
			for k = 2, #d do
				if d[k][5] or d[k][6] then -- extrema and inflection points
					bezierstring = bezierstring .. graphtobezierapprox(
					f,d,startindex,k,rndx,rndy,(ymax-ymin)/(0.5*10^rndy))
					startindex = k
				end
			end
			if startindex ~= #d then -- if no special points inbetween
				bezierstring = bezierstring .. graphtobezierapprox(f,d,
				startindex,#d,rndx,rndy,(ymax-ymin)/(0.5*10^rndy))
			end
		end
	end
	if isreverse then
		return reversepath(bezierstring)
	else
		return bezierstring
	end
end

-- main program --

if not pcall(debug.getlocal, 4, 1) then
	if #arg >= 1 then
		local xmin = -5
		local xmax = 5
		if #arg >= 2 then xmin = arg[2] end
		if #arg >= 3 then
			if arg[3] == arg[2] then
				xmax = xmin + 10
			else
				xmax = arg[3]
			end
		end
		local ymin = -5
		local ymax = 5
		if #arg >= 4 then ymin = arg[4] end
		if #arg >= 5 then 
			if arg[5] == arg[4] then
				ymax = ymin + 10
			else
				ymax = arg[5]
			end
		end
		print(bezierplot(arg[1],xmin,xmax,ymin,ymax))
	end
end