1 
2 /*
3  * Mesa 3-D graphics library
4  * Version:  3.5
5  *
6  * Copyright (C) 1999-2001  Brian Paul   All Rights Reserved.
7  *
8  * Permission is hereby granted, free of charge, to any person obtaining a
9  * copy of this software and associated documentation files (the "Software"),
10  * to deal in the Software without restriction, including without limitation
11  * the rights to use, copy, modify, merge, publish, distribute, sublicense,
12  * and/or sell copies of the Software, and to permit persons to whom the
13  * Software is furnished to do so, subject to the following conditions:
14  *
15  * The above copyright notice and this permission notice shall be included
16  * in all copies or substantial portions of the Software.
17  *
18  * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
19  * OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
20  * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL
21  * BRIAN PAUL BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN
22  * AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
23  * CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
24  */
25 
26 
27 /*
28  * eval.c was written by
29  * Bernd Barsuhn (bdbarsuh@cip.informatik.uni-erlangen.de) and
30  * Volker Weiss (vrweiss@cip.informatik.uni-erlangen.de).
31  *
32  * My original implementation of evaluators was simplistic and didn't
33  * compute surface normal vectors properly.  Bernd and Volker applied
34  * used more sophisticated methods to get better results.
35  *
36  * Thanks guys!
37  */
38 
39 
40 #include "main/glheader.h"
41 #include "main/config.h"
42 #include "m_eval.h"
43 
44 static GLfloat inv_tab[MAX_EVAL_ORDER];
45 
46 
47 
48 /*
49  * Horner scheme for Bezier curves
50  *
51  * Bezier curves can be computed via a Horner scheme.
52  * Horner is numerically less stable than the de Casteljau
53  * algorithm, but it is faster. For curves of degree n
54  * the complexity of Horner is O(n) and de Casteljau is O(n^2).
55  * Since stability is not important for displaying curve
56  * points I decided to use the Horner scheme.
57  *
58  * A cubic Bezier curve with control points b0, b1, b2, b3 can be
59  * written as
60  *
61  *        (([3]        [3]     )     [3]       )     [3]
62  * c(t) = (([0]*s*b0 + [1]*t*b1)*s + [2]*t^2*b2)*s + [3]*t^2*b3
63  *
64  *                                           [n]
65  * where s=1-t and the binomial coefficients [i]. These can
66  * be computed iteratively using the identity:
67  *
68  * [n]               [n  ]             [n]
69  * [i] = (n-i+1)/i * [i-1]     and     [0] = 1
70  */
71 
72 
73 void
_math_horner_bezier_curve(const GLfloat * cp,GLfloat * out,GLfloat t,GLuint dim,GLuint order)74 _math_horner_bezier_curve(const GLfloat * cp, GLfloat * out, GLfloat t,
75 			  GLuint dim, GLuint order)
76 {
77    GLfloat s, powert, bincoeff;
78    GLuint i, k;
79 
80    if (order >= 2) {
81       bincoeff = (GLfloat) (order - 1);
82       s = 1.0F - t;
83 
84       for (k = 0; k < dim; k++)
85 	 out[k] = s * cp[k] + bincoeff * t * cp[dim + k];
86 
87       for (i = 2, cp += 2 * dim, powert = t * t; i < order;
88 	   i++, powert *= t, cp += dim) {
89 	 bincoeff *= (GLfloat) (order - i);
90 	 bincoeff *= inv_tab[i];
91 
92 	 for (k = 0; k < dim; k++)
93 	    out[k] = s * out[k] + bincoeff * powert * cp[k];
94       }
95    }
96    else {			/* order=1 -> constant curve */
97 
98       for (k = 0; k < dim; k++)
99 	 out[k] = cp[k];
100    }
101 }
102 
103 /*
104  * Tensor product Bezier surfaces
105  *
106  * Again the Horner scheme is used to compute a point on a
107  * TP Bezier surface. First a control polygon for a curve
108  * on the surface in one parameter direction is computed,
109  * then the point on the curve for the other parameter
110  * direction is evaluated.
111  *
112  * To store the curve control polygon additional storage
113  * for max(uorder,vorder) points is needed in the
114  * control net cn.
115  */
116 
117 void
_math_horner_bezier_surf(GLfloat * cn,GLfloat * out,GLfloat u,GLfloat v,GLuint dim,GLuint uorder,GLuint vorder)118 _math_horner_bezier_surf(GLfloat * cn, GLfloat * out, GLfloat u, GLfloat v,
119 			 GLuint dim, GLuint uorder, GLuint vorder)
120 {
121    GLfloat *cp = cn + uorder * vorder * dim;
122    GLuint i, uinc = vorder * dim;
123 
124    if (vorder > uorder) {
125       if (uorder >= 2) {
126 	 GLfloat s, poweru, bincoeff;
127 	 GLuint j, k;
128 
129 	 /* Compute the control polygon for the surface-curve in u-direction */
130 	 for (j = 0; j < vorder; j++) {
131 	    GLfloat *ucp = &cn[j * dim];
132 
133 	    /* Each control point is the point for parameter u on a */
134 	    /* curve defined by the control polygons in u-direction */
135 	    bincoeff = (GLfloat) (uorder - 1);
136 	    s = 1.0F - u;
137 
138 	    for (k = 0; k < dim; k++)
139 	       cp[j * dim + k] = s * ucp[k] + bincoeff * u * ucp[uinc + k];
140 
141 	    for (i = 2, ucp += 2 * uinc, poweru = u * u; i < uorder;
142 		 i++, poweru *= u, ucp += uinc) {
143 	       bincoeff *= (GLfloat) (uorder - i);
144 	       bincoeff *= inv_tab[i];
145 
146 	       for (k = 0; k < dim; k++)
147 		  cp[j * dim + k] =
148 		     s * cp[j * dim + k] + bincoeff * poweru * ucp[k];
149 	    }
150 	 }
151 
152 	 /* Evaluate curve point in v */
153 	 _math_horner_bezier_curve(cp, out, v, dim, vorder);
154       }
155       else			/* uorder=1 -> cn defines a curve in v */
156 	 _math_horner_bezier_curve(cn, out, v, dim, vorder);
157    }
158    else {			/* vorder <= uorder */
159 
160       if (vorder > 1) {
161 	 GLuint i;
162 
163 	 /* Compute the control polygon for the surface-curve in u-direction */
164 	 for (i = 0; i < uorder; i++, cn += uinc) {
165 	    /* For constant i all cn[i][j] (j=0..vorder) are located */
166 	    /* on consecutive memory locations, so we can use        */
167 	    /* horner_bezier_curve to compute the control points     */
168 
169 	    _math_horner_bezier_curve(cn, &cp[i * dim], v, dim, vorder);
170 	 }
171 
172 	 /* Evaluate curve point in u */
173 	 _math_horner_bezier_curve(cp, out, u, dim, uorder);
174       }
175       else			/* vorder=1 -> cn defines a curve in u */
176 	 _math_horner_bezier_curve(cn, out, u, dim, uorder);
177    }
178 }
179 
180 /*
181  * The direct de Casteljau algorithm is used when a point on the
182  * surface and the tangent directions spanning the tangent plane
183  * should be computed (this is needed to compute normals to the
184  * surface). In this case the de Casteljau algorithm approach is
185  * nicer because a point and the partial derivatives can be computed
186  * at the same time. To get the correct tangent length du and dv
187  * must be multiplied with the (u2-u1)/uorder-1 and (v2-v1)/vorder-1.
188  * Since only the directions are needed, this scaling step is omitted.
189  *
190  * De Casteljau needs additional storage for uorder*vorder
191  * values in the control net cn.
192  */
193 
194 void
_math_de_casteljau_surf(GLfloat * cn,GLfloat * out,GLfloat * du,GLfloat * dv,GLfloat u,GLfloat v,GLuint dim,GLuint uorder,GLuint vorder)195 _math_de_casteljau_surf(GLfloat * cn, GLfloat * out, GLfloat * du,
196 			GLfloat * dv, GLfloat u, GLfloat v, GLuint dim,
197 			GLuint uorder, GLuint vorder)
198 {
199    GLfloat *dcn = cn + uorder * vorder * dim;
200    GLfloat us = 1.0F - u, vs = 1.0F - v;
201    GLuint h, i, j, k;
202    GLuint minorder = uorder < vorder ? uorder : vorder;
203    GLuint uinc = vorder * dim;
204    GLuint dcuinc = vorder;
205 
206    /* Each component is evaluated separately to save buffer space  */
207    /* This does not drasticaly decrease the performance of the     */
208    /* algorithm. If additional storage for (uorder-1)*(vorder-1)   */
209    /* points would be available, the components could be accessed  */
210    /* in the innermost loop which could lead to less cache misses. */
211 
212 #define CN(I,J,K) cn[(I)*uinc+(J)*dim+(K)]
213 #define DCN(I, J) dcn[(I)*dcuinc+(J)]
214    if (minorder < 3) {
215       if (uorder == vorder) {
216 	 for (k = 0; k < dim; k++) {
217 	    /* Derivative direction in u */
218 	    du[k] = vs * (CN(1, 0, k) - CN(0, 0, k)) +
219 	       v * (CN(1, 1, k) - CN(0, 1, k));
220 
221 	    /* Derivative direction in v */
222 	    dv[k] = us * (CN(0, 1, k) - CN(0, 0, k)) +
223 	       u * (CN(1, 1, k) - CN(1, 0, k));
224 
225 	    /* bilinear de Casteljau step */
226 	    out[k] = us * (vs * CN(0, 0, k) + v * CN(0, 1, k)) +
227 	       u * (vs * CN(1, 0, k) + v * CN(1, 1, k));
228 	 }
229       }
230       else if (minorder == uorder) {
231 	 for (k = 0; k < dim; k++) {
232 	    /* bilinear de Casteljau step */
233 	    DCN(1, 0) = CN(1, 0, k) - CN(0, 0, k);
234 	    DCN(0, 0) = us * CN(0, 0, k) + u * CN(1, 0, k);
235 
236 	    for (j = 0; j < vorder - 1; j++) {
237 	       /* for the derivative in u */
238 	       DCN(1, j + 1) = CN(1, j + 1, k) - CN(0, j + 1, k);
239 	       DCN(1, j) = vs * DCN(1, j) + v * DCN(1, j + 1);
240 
241 	       /* for the `point' */
242 	       DCN(0, j + 1) = us * CN(0, j + 1, k) + u * CN(1, j + 1, k);
243 	       DCN(0, j) = vs * DCN(0, j) + v * DCN(0, j + 1);
244 	    }
245 
246 	    /* remaining linear de Casteljau steps until the second last step */
247 	    for (h = minorder; h < vorder - 1; h++)
248 	       for (j = 0; j < vorder - h; j++) {
249 		  /* for the derivative in u */
250 		  DCN(1, j) = vs * DCN(1, j) + v * DCN(1, j + 1);
251 
252 		  /* for the `point' */
253 		  DCN(0, j) = vs * DCN(0, j) + v * DCN(0, j + 1);
254 	       }
255 
256 	    /* derivative direction in v */
257 	    dv[k] = DCN(0, 1) - DCN(0, 0);
258 
259 	    /* derivative direction in u */
260 	    du[k] = vs * DCN(1, 0) + v * DCN(1, 1);
261 
262 	    /* last linear de Casteljau step */
263 	    out[k] = vs * DCN(0, 0) + v * DCN(0, 1);
264 	 }
265       }
266       else {			/* minorder == vorder */
267 
268 	 for (k = 0; k < dim; k++) {
269 	    /* bilinear de Casteljau step */
270 	    DCN(0, 1) = CN(0, 1, k) - CN(0, 0, k);
271 	    DCN(0, 0) = vs * CN(0, 0, k) + v * CN(0, 1, k);
272 	    for (i = 0; i < uorder - 1; i++) {
273 	       /* for the derivative in v */
274 	       DCN(i + 1, 1) = CN(i + 1, 1, k) - CN(i + 1, 0, k);
275 	       DCN(i, 1) = us * DCN(i, 1) + u * DCN(i + 1, 1);
276 
277 	       /* for the `point' */
278 	       DCN(i + 1, 0) = vs * CN(i + 1, 0, k) + v * CN(i + 1, 1, k);
279 	       DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
280 	    }
281 
282 	    /* remaining linear de Casteljau steps until the second last step */
283 	    for (h = minorder; h < uorder - 1; h++)
284 	       for (i = 0; i < uorder - h; i++) {
285 		  /* for the derivative in v */
286 		  DCN(i, 1) = us * DCN(i, 1) + u * DCN(i + 1, 1);
287 
288 		  /* for the `point' */
289 		  DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
290 	       }
291 
292 	    /* derivative direction in u */
293 	    du[k] = DCN(1, 0) - DCN(0, 0);
294 
295 	    /* derivative direction in v */
296 	    dv[k] = us * DCN(0, 1) + u * DCN(1, 1);
297 
298 	    /* last linear de Casteljau step */
299 	    out[k] = us * DCN(0, 0) + u * DCN(1, 0);
300 	 }
301       }
302    }
303    else if (uorder == vorder) {
304       for (k = 0; k < dim; k++) {
305 	 /* first bilinear de Casteljau step */
306 	 for (i = 0; i < uorder - 1; i++) {
307 	    DCN(i, 0) = us * CN(i, 0, k) + u * CN(i + 1, 0, k);
308 	    for (j = 0; j < vorder - 1; j++) {
309 	       DCN(i, j + 1) = us * CN(i, j + 1, k) + u * CN(i + 1, j + 1, k);
310 	       DCN(i, j) = vs * DCN(i, j) + v * DCN(i, j + 1);
311 	    }
312 	 }
313 
314 	 /* remaining bilinear de Casteljau steps until the second last step */
315 	 for (h = 2; h < minorder - 1; h++)
316 	    for (i = 0; i < uorder - h; i++) {
317 	       DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
318 	       for (j = 0; j < vorder - h; j++) {
319 		  DCN(i, j + 1) = us * DCN(i, j + 1) + u * DCN(i + 1, j + 1);
320 		  DCN(i, j) = vs * DCN(i, j) + v * DCN(i, j + 1);
321 	       }
322 	    }
323 
324 	 /* derivative direction in u */
325 	 du[k] = vs * (DCN(1, 0) - DCN(0, 0)) + v * (DCN(1, 1) - DCN(0, 1));
326 
327 	 /* derivative direction in v */
328 	 dv[k] = us * (DCN(0, 1) - DCN(0, 0)) + u * (DCN(1, 1) - DCN(1, 0));
329 
330 	 /* last bilinear de Casteljau step */
331 	 out[k] = us * (vs * DCN(0, 0) + v * DCN(0, 1)) +
332 	    u * (vs * DCN(1, 0) + v * DCN(1, 1));
333       }
334    }
335    else if (minorder == uorder) {
336       for (k = 0; k < dim; k++) {
337 	 /* first bilinear de Casteljau step */
338 	 for (i = 0; i < uorder - 1; i++) {
339 	    DCN(i, 0) = us * CN(i, 0, k) + u * CN(i + 1, 0, k);
340 	    for (j = 0; j < vorder - 1; j++) {
341 	       DCN(i, j + 1) = us * CN(i, j + 1, k) + u * CN(i + 1, j + 1, k);
342 	       DCN(i, j) = vs * DCN(i, j) + v * DCN(i, j + 1);
343 	    }
344 	 }
345 
346 	 /* remaining bilinear de Casteljau steps until the second last step */
347 	 for (h = 2; h < minorder - 1; h++)
348 	    for (i = 0; i < uorder - h; i++) {
349 	       DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
350 	       for (j = 0; j < vorder - h; j++) {
351 		  DCN(i, j + 1) = us * DCN(i, j + 1) + u * DCN(i + 1, j + 1);
352 		  DCN(i, j) = vs * DCN(i, j) + v * DCN(i, j + 1);
353 	       }
354 	    }
355 
356 	 /* last bilinear de Casteljau step */
357 	 DCN(2, 0) = DCN(1, 0) - DCN(0, 0);
358 	 DCN(0, 0) = us * DCN(0, 0) + u * DCN(1, 0);
359 	 for (j = 0; j < vorder - 1; j++) {
360 	    /* for the derivative in u */
361 	    DCN(2, j + 1) = DCN(1, j + 1) - DCN(0, j + 1);
362 	    DCN(2, j) = vs * DCN(2, j) + v * DCN(2, j + 1);
363 
364 	    /* for the `point' */
365 	    DCN(0, j + 1) = us * DCN(0, j + 1) + u * DCN(1, j + 1);
366 	    DCN(0, j) = vs * DCN(0, j) + v * DCN(0, j + 1);
367 	 }
368 
369 	 /* remaining linear de Casteljau steps until the second last step */
370 	 for (h = minorder; h < vorder - 1; h++)
371 	    for (j = 0; j < vorder - h; j++) {
372 	       /* for the derivative in u */
373 	       DCN(2, j) = vs * DCN(2, j) + v * DCN(2, j + 1);
374 
375 	       /* for the `point' */
376 	       DCN(0, j) = vs * DCN(0, j) + v * DCN(0, j + 1);
377 	    }
378 
379 	 /* derivative direction in v */
380 	 dv[k] = DCN(0, 1) - DCN(0, 0);
381 
382 	 /* derivative direction in u */
383 	 du[k] = vs * DCN(2, 0) + v * DCN(2, 1);
384 
385 	 /* last linear de Casteljau step */
386 	 out[k] = vs * DCN(0, 0) + v * DCN(0, 1);
387       }
388    }
389    else {			/* minorder == vorder */
390 
391       for (k = 0; k < dim; k++) {
392 	 /* first bilinear de Casteljau step */
393 	 for (i = 0; i < uorder - 1; i++) {
394 	    DCN(i, 0) = us * CN(i, 0, k) + u * CN(i + 1, 0, k);
395 	    for (j = 0; j < vorder - 1; j++) {
396 	       DCN(i, j + 1) = us * CN(i, j + 1, k) + u * CN(i + 1, j + 1, k);
397 	       DCN(i, j) = vs * DCN(i, j) + v * DCN(i, j + 1);
398 	    }
399 	 }
400 
401 	 /* remaining bilinear de Casteljau steps until the second last step */
402 	 for (h = 2; h < minorder - 1; h++)
403 	    for (i = 0; i < uorder - h; i++) {
404 	       DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
405 	       for (j = 0; j < vorder - h; j++) {
406 		  DCN(i, j + 1) = us * DCN(i, j + 1) + u * DCN(i + 1, j + 1);
407 		  DCN(i, j) = vs * DCN(i, j) + v * DCN(i, j + 1);
408 	       }
409 	    }
410 
411 	 /* last bilinear de Casteljau step */
412 	 DCN(0, 2) = DCN(0, 1) - DCN(0, 0);
413 	 DCN(0, 0) = vs * DCN(0, 0) + v * DCN(0, 1);
414 	 for (i = 0; i < uorder - 1; i++) {
415 	    /* for the derivative in v */
416 	    DCN(i + 1, 2) = DCN(i + 1, 1) - DCN(i + 1, 0);
417 	    DCN(i, 2) = us * DCN(i, 2) + u * DCN(i + 1, 2);
418 
419 	    /* for the `point' */
420 	    DCN(i + 1, 0) = vs * DCN(i + 1, 0) + v * DCN(i + 1, 1);
421 	    DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
422 	 }
423 
424 	 /* remaining linear de Casteljau steps until the second last step */
425 	 for (h = minorder; h < uorder - 1; h++)
426 	    for (i = 0; i < uorder - h; i++) {
427 	       /* for the derivative in v */
428 	       DCN(i, 2) = us * DCN(i, 2) + u * DCN(i + 1, 2);
429 
430 	       /* for the `point' */
431 	       DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
432 	    }
433 
434 	 /* derivative direction in u */
435 	 du[k] = DCN(1, 0) - DCN(0, 0);
436 
437 	 /* derivative direction in v */
438 	 dv[k] = us * DCN(0, 2) + u * DCN(1, 2);
439 
440 	 /* last linear de Casteljau step */
441 	 out[k] = us * DCN(0, 0) + u * DCN(1, 0);
442       }
443    }
444 #undef DCN
445 #undef CN
446 }
447 
448 
449 /*
450  * Do one-time initialization for evaluators.
451  */
452 void
_math_init_eval(void)453 _math_init_eval(void)
454 {
455    GLuint i;
456 
457    /* KW: precompute 1/x for useful x.
458     */
459    for (i = 1; i < MAX_EVAL_ORDER; i++)
460       inv_tab[i] = 1.0F / i;
461 }
462