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/* $Id: m_eval.c,v 1.1 2003-02-28 11:48:05 pj Exp $ */
2
 
3
/*
4
 * Mesa 3-D graphics library
5
 * Version:  3.5
6
 *
7
 * Copyright (C) 1999-2001  Brian Paul   All Rights Reserved.
8
 *
9
 * Permission is hereby granted, free of charge, to any person obtaining a
10
 * copy of this software and associated documentation files (the "Software"),
11
 * to deal in the Software without restriction, including without limitation
12
 * the rights to use, copy, modify, merge, publish, distribute, sublicense,
13
 * and/or sell copies of the Software, and to permit persons to whom the
14
 * Software is furnished to do so, subject to the following conditions:
15
 *
16
 * The above copyright notice and this permission notice shall be included
17
 * in all copies or substantial portions of the Software.
18
 *
19
 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
20
 * OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
21
 * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL
22
 * BRIAN PAUL BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN
23
 * AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
24
 * CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
25
 */
26
 
27
 
28
/*
29
 * eval.c was written by
30
 * Bernd Barsuhn (bdbarsuh@cip.informatik.uni-erlangen.de) and
31
 * Volker Weiss (vrweiss@cip.informatik.uni-erlangen.de).
32
 *
33
 * My original implementation of evaluators was simplistic and didn't
34
 * compute surface normal vectors properly.  Bernd and Volker applied
35
 * used more sophisticated methods to get better results.
36
 *
37
 * Thanks guys!
38
 */
39
 
40
 
41
#include "glheader.h"
42
#include "config.h"
43
#include "m_eval.h"
44
 
45
static GLfloat inv_tab[MAX_EVAL_ORDER];
46
 
47
 
48
 
49
/*
50
 * Horner scheme for Bezier curves
51
 *
52
 * Bezier curves can be computed via a Horner scheme.
53
 * Horner is numerically less stable than the de Casteljau
54
 * algorithm, but it is faster. For curves of degree n
55
 * the complexity of Horner is O(n) and de Casteljau is O(n^2).
56
 * Since stability is not important for displaying curve
57
 * points I decided to use the Horner scheme.
58
 *
59
 * A cubic Bezier curve with control points b0, b1, b2, b3 can be
60
 * written as
61
 *
62
 *        (([3]        [3]     )     [3]       )     [3]
63
 * c(t) = (([0]*s*b0 + [1]*t*b1)*s + [2]*t^2*b2)*s + [3]*t^2*b3
64
 *
65
 *                                           [n]
66
 * where s=1-t and the binomial coefficients [i]. These can
67
 * be computed iteratively using the identity:
68
 *
69
 * [n]               [n  ]             [n]
70
 * [i] = (n-i+1)/i * [i-1]     and     [0] = 1
71
 */
72
 
73
 
74
void
75
_math_horner_bezier_curve(const GLfloat * cp, GLfloat * out, GLfloat t,
76
                          GLuint dim, GLuint order)
77
{
78
   GLfloat s, powert, bincoeff;
79
   GLuint i, k;
80
 
81
   if (order >= 2) {
82
      bincoeff = (GLfloat) (order - 1);
83
      s = 1.0F - t;
84
 
85
      for (k = 0; k < dim; k++)
86
         out[k] = s * cp[k] + bincoeff * t * cp[dim + k];
87
 
88
      for (i = 2, cp += 2 * dim, powert = t * t; i < order;
89
           i++, powert *= t, cp += dim) {
90
         bincoeff *= (GLfloat) (order - i);
91
         bincoeff *= inv_tab[i];
92
 
93
         for (k = 0; k < dim; k++)
94
            out[k] = s * out[k] + bincoeff * powert * cp[k];
95
      }
96
   }
97
   else {                       /* order=1 -> constant curve */
98
 
99
      for (k = 0; k < dim; k++)
100
         out[k] = cp[k];
101
   }
102
}
103
 
104
/*
105
 * Tensor product Bezier surfaces
106
 *
107
 * Again the Horner scheme is used to compute a point on a
108
 * TP Bezier surface. First a control polygon for a curve
109
 * on the surface in one parameter direction is computed,
110
 * then the point on the curve for the other parameter
111
 * direction is evaluated.
112
 *
113
 * To store the curve control polygon additional storage
114
 * for max(uorder,vorder) points is needed in the
115
 * control net cn.
116
 */
117
 
118
void
119
_math_horner_bezier_surf(GLfloat * cn, GLfloat * out, GLfloat u, GLfloat v,
120
                         GLuint dim, GLuint uorder, GLuint vorder)
121
{
122
   GLfloat *cp = cn + uorder * vorder * dim;
123
   GLuint i, uinc = vorder * dim;
124
 
125
   if (vorder > uorder) {
126
      if (uorder >= 2) {
127
         GLfloat s, poweru, bincoeff;
128
         GLuint j, k;
129
 
130
         /* Compute the control polygon for the surface-curve in u-direction */
131
         for (j = 0; j < vorder; j++) {
132
            GLfloat *ucp = &cn[j * dim];
133
 
134
            /* Each control point is the point for parameter u on a */
135
            /* curve defined by the control polygons in u-direction */
136
            bincoeff = (GLfloat) (uorder - 1);
137
            s = 1.0F - u;
138
 
139
            for (k = 0; k < dim; k++)
140
               cp[j * dim + k] = s * ucp[k] + bincoeff * u * ucp[uinc + k];
141
 
142
            for (i = 2, ucp += 2 * uinc, poweru = u * u; i < uorder;
143
                 i++, poweru *= u, ucp += uinc) {
144
               bincoeff *= (GLfloat) (uorder - i);
145
               bincoeff *= inv_tab[i];
146
 
147
               for (k = 0; k < dim; k++)
148
                  cp[j * dim + k] =
149
                     s * cp[j * dim + k] + bincoeff * poweru * ucp[k];
150
            }
151
         }
152
 
153
         /* Evaluate curve point in v */
154
         _math_horner_bezier_curve(cp, out, v, dim, vorder);
155
      }
156
      else                      /* uorder=1 -> cn defines a curve in v */
157
         _math_horner_bezier_curve(cn, out, v, dim, vorder);
158
   }
159
   else {                       /* vorder <= uorder */
160
 
161
      if (vorder > 1) {
162
         GLuint i;
163
 
164
         /* Compute the control polygon for the surface-curve in u-direction */
165
         for (i = 0; i < uorder; i++, cn += uinc) {
166
            /* For constant i all cn[i][j] (j=0..vorder) are located */
167
            /* on consecutive memory locations, so we can use        */
168
            /* horner_bezier_curve to compute the control points     */
169
 
170
            _math_horner_bezier_curve(cn, &cp[i * dim], v, dim, vorder);
171
         }
172
 
173
         /* Evaluate curve point in u */
174
         _math_horner_bezier_curve(cp, out, u, dim, uorder);
175
      }
176
      else                      /* vorder=1 -> cn defines a curve in u */
177
         _math_horner_bezier_curve(cn, out, u, dim, uorder);
178
   }
179
}
180
 
181
/*
182
 * The direct de Casteljau algorithm is used when a point on the
183
 * surface and the tangent directions spanning the tangent plane
184
 * should be computed (this is needed to compute normals to the
185
 * surface). In this case the de Casteljau algorithm approach is
186
 * nicer because a point and the partial derivatives can be computed
187
 * at the same time. To get the correct tangent length du and dv
188
 * must be multiplied with the (u2-u1)/uorder-1 and (v2-v1)/vorder-1.
189
 * Since only the directions are needed, this scaling step is omitted.
190
 *
191
 * De Casteljau needs additional storage for uorder*vorder
192
 * values in the control net cn.
193
 */
194
 
195
void
196
_math_de_casteljau_surf(GLfloat * cn, GLfloat * out, GLfloat * du,
197
                        GLfloat * dv, GLfloat u, GLfloat v, GLuint dim,
198
                        GLuint uorder, GLuint vorder)
199
{
200
   GLfloat *dcn = cn + uorder * vorder * dim;
201
   GLfloat us = 1.0F - u, vs = 1.0F - v;
202
   GLuint h, i, j, k;
203
   GLuint minorder = uorder < vorder ? uorder : vorder;
204
   GLuint uinc = vorder * dim;
205
   GLuint dcuinc = vorder;
206
 
207
   /* Each component is evaluated separately to save buffer space  */
208
   /* This does not drasticaly decrease the performance of the     */
209
   /* algorithm. If additional storage for (uorder-1)*(vorder-1)   */
210
   /* points would be available, the components could be accessed  */
211
   /* in the innermost loop which could lead to less cache misses. */
212
 
213
#define CN(I,J,K) cn[(I)*uinc+(J)*dim+(K)]
214
#define DCN(I, J) dcn[(I)*dcuinc+(J)]
215
   if (minorder < 3) {
216
      if (uorder == vorder) {
217
         for (k = 0; k < dim; k++) {
218
            /* Derivative direction in u */
219
            du[k] = vs * (CN(1, 0, k) - CN(0, 0, k)) +
220
               v * (CN(1, 1, k) - CN(0, 1, k));
221
 
222
            /* Derivative direction in v */
223
            dv[k] = us * (CN(0, 1, k) - CN(0, 0, k)) +
224
               u * (CN(1, 1, k) - CN(1, 0, k));
225
 
226
            /* bilinear de Casteljau step */
227
            out[k] = us * (vs * CN(0, 0, k) + v * CN(0, 1, k)) +
228
               u * (vs * CN(1, 0, k) + v * CN(1, 1, k));
229
         }
230
      }
231
      else if (minorder == uorder) {
232
         for (k = 0; k < dim; k++) {
233
            /* bilinear de Casteljau step */
234
            DCN(1, 0) = CN(1, 0, k) - CN(0, 0, k);
235
            DCN(0, 0) = us * CN(0, 0, k) + u * CN(1, 0, k);
236
 
237
            for (j = 0; j < vorder - 1; j++) {
238
               /* for the derivative in u */
239
               DCN(1, j + 1) = CN(1, j + 1, k) - CN(0, j + 1, k);
240
               DCN(1, j) = vs * DCN(1, j) + v * DCN(1, j + 1);
241
 
242
               /* for the `point' */
243
               DCN(0, j + 1) = us * CN(0, j + 1, k) + u * CN(1, j + 1, k);
244
               DCN(0, j) = vs * DCN(0, j) + v * DCN(0, j + 1);
245
            }
246
 
247
            /* remaining linear de Casteljau steps until the second last step */
248
            for (h = minorder; h < vorder - 1; h++)
249
               for (j = 0; j < vorder - h; j++) {
250
                  /* for the derivative in u */
251
                  DCN(1, j) = vs * DCN(1, j) + v * DCN(1, j + 1);
252
 
253
                  /* for the `point' */
254
                  DCN(0, j) = vs * DCN(0, j) + v * DCN(0, j + 1);
255
               }
256
 
257
            /* derivative direction in v */
258
            dv[k] = DCN(0, 1) - DCN(0, 0);
259
 
260
            /* derivative direction in u */
261
            du[k] = vs * DCN(1, 0) + v * DCN(1, 1);
262
 
263
            /* last linear de Casteljau step */
264
            out[k] = vs * DCN(0, 0) + v * DCN(0, 1);
265
         }
266
      }
267
      else {                    /* minorder == vorder */
268
 
269
         for (k = 0; k < dim; k++) {
270
            /* bilinear de Casteljau step */
271
            DCN(0, 1) = CN(0, 1, k) - CN(0, 0, k);
272
            DCN(0, 0) = vs * CN(0, 0, k) + v * CN(0, 1, k);
273
            for (i = 0; i < uorder - 1; i++) {
274
               /* for the derivative in v */
275
               DCN(i + 1, 1) = CN(i + 1, 1, k) - CN(i + 1, 0, k);
276
               DCN(i, 1) = us * DCN(i, 1) + u * DCN(i + 1, 1);
277
 
278
               /* for the `point' */
279
               DCN(i + 1, 0) = vs * CN(i + 1, 0, k) + v * CN(i + 1, 1, k);
280
               DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
281
            }
282
 
283
            /* remaining linear de Casteljau steps until the second last step */
284
            for (h = minorder; h < uorder - 1; h++)
285
               for (i = 0; i < uorder - h; i++) {
286
                  /* for the derivative in v */
287
                  DCN(i, 1) = us * DCN(i, 1) + u * DCN(i + 1, 1);
288
 
289
                  /* for the `point' */
290
                  DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
291
               }
292
 
293
            /* derivative direction in u */
294
            du[k] = DCN(1, 0) - DCN(0, 0);
295
 
296
            /* derivative direction in v */
297
            dv[k] = us * DCN(0, 1) + u * DCN(1, 1);
298
 
299
            /* last linear de Casteljau step */
300
            out[k] = us * DCN(0, 0) + u * DCN(1, 0);
301
         }
302
      }
303
   }
304
   else if (uorder == vorder) {
305
      for (k = 0; k < dim; k++) {
306
         /* first bilinear de Casteljau step */
307
         for (i = 0; i < uorder - 1; i++) {
308
            DCN(i, 0) = us * CN(i, 0, k) + u * CN(i + 1, 0, k);
309
            for (j = 0; j < vorder - 1; j++) {
310
               DCN(i, j + 1) = us * CN(i, j + 1, k) + u * CN(i + 1, j + 1, k);
311
               DCN(i, j) = vs * DCN(i, j) + v * DCN(i, j + 1);
312
            }
313
         }
314
 
315
         /* remaining bilinear de Casteljau steps until the second last step */
316
         for (h = 2; h < minorder - 1; h++)
317
            for (i = 0; i < uorder - h; i++) {
318
               DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
319
               for (j = 0; j < vorder - h; j++) {
320
                  DCN(i, j + 1) = us * DCN(i, j + 1) + u * DCN(i + 1, j + 1);
321
                  DCN(i, j) = vs * DCN(i, j) + v * DCN(i, j + 1);
322
               }
323
            }
324
 
325
         /* derivative direction in u */
326
         du[k] = vs * (DCN(1, 0) - DCN(0, 0)) + v * (DCN(1, 1) - DCN(0, 1));
327
 
328
         /* derivative direction in v */
329
         dv[k] = us * (DCN(0, 1) - DCN(0, 0)) + u * (DCN(1, 1) - DCN(1, 0));
330
 
331
         /* last bilinear de Casteljau step */
332
         out[k] = us * (vs * DCN(0, 0) + v * DCN(0, 1)) +
333
            u * (vs * DCN(1, 0) + v * DCN(1, 1));
334
      }
335
   }
336
   else if (minorder == uorder) {
337
      for (k = 0; k < dim; k++) {
338
         /* first bilinear de Casteljau step */
339
         for (i = 0; i < uorder - 1; i++) {
340
            DCN(i, 0) = us * CN(i, 0, k) + u * CN(i + 1, 0, k);
341
            for (j = 0; j < vorder - 1; j++) {
342
               DCN(i, j + 1) = us * CN(i, j + 1, k) + u * CN(i + 1, j + 1, k);
343
               DCN(i, j) = vs * DCN(i, j) + v * DCN(i, j + 1);
344
            }
345
         }
346
 
347
         /* remaining bilinear de Casteljau steps until the second last step */
348
         for (h = 2; h < minorder - 1; h++)
349
            for (i = 0; i < uorder - h; i++) {
350
               DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
351
               for (j = 0; j < vorder - h; j++) {
352
                  DCN(i, j + 1) = us * DCN(i, j + 1) + u * DCN(i + 1, j + 1);
353
                  DCN(i, j) = vs * DCN(i, j) + v * DCN(i, j + 1);
354
               }
355
            }
356
 
357
         /* last bilinear de Casteljau step */
358
         DCN(2, 0) = DCN(1, 0) - DCN(0, 0);
359
         DCN(0, 0) = us * DCN(0, 0) + u * DCN(1, 0);
360
         for (j = 0; j < vorder - 1; j++) {
361
            /* for the derivative in u */
362
            DCN(2, j + 1) = DCN(1, j + 1) - DCN(0, j + 1);
363
            DCN(2, j) = vs * DCN(2, j) + v * DCN(2, j + 1);
364
 
365
            /* for the `point' */
366
            DCN(0, j + 1) = us * DCN(0, j + 1) + u * DCN(1, j + 1);
367
            DCN(0, j) = vs * DCN(0, j) + v * DCN(0, j + 1);
368
         }
369
 
370
         /* remaining linear de Casteljau steps until the second last step */
371
         for (h = minorder; h < vorder - 1; h++)
372
            for (j = 0; j < vorder - h; j++) {
373
               /* for the derivative in u */
374
               DCN(2, j) = vs * DCN(2, j) + v * DCN(2, j + 1);
375
 
376
               /* for the `point' */
377
               DCN(0, j) = vs * DCN(0, j) + v * DCN(0, j + 1);
378
            }
379
 
380
         /* derivative direction in v */
381
         dv[k] = DCN(0, 1) - DCN(0, 0);
382
 
383
         /* derivative direction in u */
384
         du[k] = vs * DCN(2, 0) + v * DCN(2, 1);
385
 
386
         /* last linear de Casteljau step */
387
         out[k] = vs * DCN(0, 0) + v * DCN(0, 1);
388
      }
389
   }
390
   else {                       /* minorder == vorder */
391
 
392
      for (k = 0; k < dim; k++) {
393
         /* first bilinear de Casteljau step */
394
         for (i = 0; i < uorder - 1; i++) {
395
            DCN(i, 0) = us * CN(i, 0, k) + u * CN(i + 1, 0, k);
396
            for (j = 0; j < vorder - 1; j++) {
397
               DCN(i, j + 1) = us * CN(i, j + 1, k) + u * CN(i + 1, j + 1, k);
398
               DCN(i, j) = vs * DCN(i, j) + v * DCN(i, j + 1);
399
            }
400
         }
401
 
402
         /* remaining bilinear de Casteljau steps until the second last step */
403
         for (h = 2; h < minorder - 1; h++)
404
            for (i = 0; i < uorder - h; i++) {
405
               DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
406
               for (j = 0; j < vorder - h; j++) {
407
                  DCN(i, j + 1) = us * DCN(i, j + 1) + u * DCN(i + 1, j + 1);
408
                  DCN(i, j) = vs * DCN(i, j) + v * DCN(i, j + 1);
409
               }
410
            }
411
 
412
         /* last bilinear de Casteljau step */
413
         DCN(0, 2) = DCN(0, 1) - DCN(0, 0);
414
         DCN(0, 0) = vs * DCN(0, 0) + v * DCN(0, 1);
415
         for (i = 0; i < uorder - 1; i++) {
416
            /* for the derivative in v */
417
            DCN(i + 1, 2) = DCN(i + 1, 1) - DCN(i + 1, 0);
418
            DCN(i, 2) = us * DCN(i, 2) + u * DCN(i + 1, 2);
419
 
420
            /* for the `point' */
421
            DCN(i + 1, 0) = vs * DCN(i + 1, 0) + v * DCN(i + 1, 1);
422
            DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
423
         }
424
 
425
         /* remaining linear de Casteljau steps until the second last step */
426
         for (h = minorder; h < uorder - 1; h++)
427
            for (i = 0; i < uorder - h; i++) {
428
               /* for the derivative in v */
429
               DCN(i, 2) = us * DCN(i, 2) + u * DCN(i + 1, 2);
430
 
431
               /* for the `point' */
432
               DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
433
            }
434
 
435
         /* derivative direction in u */
436
         du[k] = DCN(1, 0) - DCN(0, 0);
437
 
438
         /* derivative direction in v */
439
         dv[k] = us * DCN(0, 2) + u * DCN(1, 2);
440
 
441
         /* last linear de Casteljau step */
442
         out[k] = us * DCN(0, 0) + u * DCN(1, 0);
443
      }
444
   }
445
#undef DCN
446
#undef CN
447
}
448
 
449
 
450
/*
451
 * Do one-time initialization for evaluators.
452
 */
453
void
454
_math_init_eval(void)
455
{
456
   GLuint i;
457
 
458
   /* KW: precompute 1/x for useful x.
459
    */
460
   for (i = 1; i < MAX_EVAL_ORDER; i++)
461
      inv_tab[i] = 1.0F / i;
462
}