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root/radiance/ray/src/rt/normal.c
Revision: 2.58
Committed: Tue Oct 12 19:01:14 2010 UTC (13 years, 6 months ago) by greg
Content type: text/plain
Branch: MAIN
Changes since 2.57: +48 -36 lines
Log Message:
Fixed W-G-M-D weighting based on number of trials

File Contents

# User Rev Content
1 greg 1.1 #ifndef lint
2 greg 2.58 static const char RCSid[] = "$Id: normal.c,v 2.57 2010/10/10 22:31:46 greg Exp $";
3 greg 1.1 #endif
4     /*
5     * normal.c - shading function for normal materials.
6     *
7     * 8/19/85
8     * 12/19/85 - added stuff for metals.
9     * 6/26/87 - improved specular model.
10     * 9/28/87 - added model for translucent materials.
11 greg 2.2 * Later changes described in delta comments.
12 greg 1.1 */
13    
14 greg 2.39 #include "copyright.h"
15 greg 2.38
16 greg 1.1 #include "ray.h"
17 greg 2.46 #include "ambient.h"
18 schorsch 2.47 #include "source.h"
19 greg 1.1 #include "otypes.h"
20 schorsch 2.47 #include "rtotypes.h"
21 greg 2.2 #include "random.h"
22    
23 greg 2.34 #ifndef MAXITER
24     #define MAXITER 10 /* maximum # specular ray attempts */
25     #endif
26 greg 2.38 /* estimate of Fresnel function */
27 greg 2.44 #define FRESNE(ci) (exp(-5.85*(ci)) - 0.00287989916)
28 greg 2.51 #define FRESTHRESH 0.017999 /* minimum specularity for approx. */
29 greg 2.34
30 greg 2.24
31 greg 1.1 /*
32 greg 2.22 * This routine implements the isotropic Gaussian
33     * model described by Ward in Siggraph `92 article.
34 greg 1.1 * We orient the surface towards the incoming ray, so a single
35     * surface can be used to represent an infinitely thin object.
36     *
37     * Arguments for MAT_PLASTIC and MAT_METAL are:
38     * red grn blu specular-frac. facet-slope
39     *
40     * Arguments for MAT_TRANS are:
41     * red grn blu rspec rough trans tspec
42     */
43    
44 greg 2.2 /* specularity flags */
45     #define SP_REFL 01 /* has reflected specular component */
46     #define SP_TRAN 02 /* has transmitted specular */
47 greg 2.11 #define SP_PURE 04 /* purely specular (zero roughness) */
48     #define SP_FLAT 010 /* flat reflecting surface */
49     #define SP_RBLT 020 /* reflection below sample threshold */
50     #define SP_TBLT 040 /* transmission below threshold */
51 greg 1.1
52 greg 1.3 typedef struct {
53     OBJREC *mp; /* material pointer */
54 greg 2.16 RAY *rp; /* ray pointer */
55 greg 2.2 short specfl; /* specularity flags, defined above */
56 greg 1.1 COLOR mcolor; /* color of this material */
57     COLOR scolor; /* color of specular component */
58     FVECT vrefl; /* vector in direction of reflected ray */
59 greg 1.14 FVECT prdir; /* vector in transmitted direction */
60 greg 2.2 double alpha2; /* roughness squared */
61 greg 1.1 double rdiff, rspec; /* reflected specular, diffuse */
62     double trans; /* transmissivity */
63     double tdiff, tspec; /* transmitted specular, diffuse */
64     FVECT pnorm; /* perturbed surface normal */
65     double pdot; /* perturbed dot product */
66 greg 1.3 } NORMDAT; /* normal material data */
67    
68 schorsch 2.47 static srcdirf_t dirnorm;
69     static void gaussamp(RAY *r, NORMDAT *np);
70    
71 greg 1.3
72 greg 2.38 static void
73 schorsch 2.47 dirnorm( /* compute source contribution */
74     COLOR cval, /* returned coefficient */
75     void *nnp, /* material data */
76     FVECT ldir, /* light source direction */
77     double omega /* light source size */
78     )
79 greg 1.3 {
80 schorsch 2.47 register NORMDAT *np = nnp;
81 greg 1.1 double ldot;
82 greg 2.49 double lrdiff, ltdiff;
83 greg 2.54 double dtmp, d2, d3, d4;
84 greg 2.16 FVECT vtmp;
85 greg 1.3 COLOR ctmp;
86    
87     setcolor(cval, 0.0, 0.0, 0.0);
88    
89     ldot = DOT(np->pnorm, ldir);
90    
91     if (ldot < 0.0 ? np->trans <= FTINY : np->trans >= 1.0-FTINY)
92     return; /* wrong side */
93    
94 greg 2.38 /* Fresnel estimate */
95 greg 2.49 lrdiff = np->rdiff;
96     ltdiff = np->tdiff;
97 greg 2.51 if (np->specfl & SP_PURE && np->rspec >= FRESTHRESH &&
98 greg 2.49 (lrdiff > FTINY) | (ltdiff > FTINY)) {
99     dtmp = 1. - FRESNE(fabs(ldot));
100     lrdiff *= dtmp;
101     ltdiff *= dtmp;
102     }
103 greg 2.38
104 greg 2.49 if (ldot > FTINY && lrdiff > FTINY) {
105 greg 1.3 /*
106 greg 1.4 * Compute and add diffuse reflected component to returned
107     * color. The diffuse reflected component will always be
108     * modified by the color of the material.
109 greg 1.3 */
110     copycolor(ctmp, np->mcolor);
111 greg 2.49 dtmp = ldot * omega * lrdiff * (1.0/PI);
112 greg 1.3 scalecolor(ctmp, dtmp);
113     addcolor(cval, ctmp);
114     }
115 greg 2.2 if (ldot > FTINY && (np->specfl&(SP_REFL|SP_PURE)) == SP_REFL) {
116 greg 1.3 /*
117     * Compute specular reflection coefficient using
118 greg 2.54 * Gaussian distribution model.
119 greg 1.3 */
120 greg 2.3 /* roughness */
121 greg 2.16 dtmp = np->alpha2;
122 greg 2.3 /* + source if flat */
123     if (np->specfl & SP_FLAT)
124 greg 2.48 dtmp += omega * (0.25/PI);
125 greg 2.23 /* half vector */
126 greg 2.18 vtmp[0] = ldir[0] - np->rp->rdir[0];
127     vtmp[1] = ldir[1] - np->rp->rdir[1];
128     vtmp[2] = ldir[2] - np->rp->rdir[2];
129 greg 2.16 d2 = DOT(vtmp, np->pnorm);
130 greg 2.23 d2 *= d2;
131 greg 2.54 d3 = DOT(vtmp,vtmp);
132     d4 = (d3 - d2) / d2;
133     /* new W-G-M-D model */
134     dtmp = exp(-d4/dtmp) * d3 / (PI * d2*d2 * dtmp);
135 greg 1.3 /* worth using? */
136     if (dtmp > FTINY) {
137     copycolor(ctmp, np->scolor);
138 greg 2.54 dtmp *= ldot * omega;
139 greg 1.3 scalecolor(ctmp, dtmp);
140     addcolor(cval, ctmp);
141     }
142     }
143 greg 2.49 if (ldot < -FTINY && ltdiff > FTINY) {
144 greg 1.3 /*
145     * Compute diffuse transmission.
146     */
147     copycolor(ctmp, np->mcolor);
148 greg 2.49 dtmp = -ldot * omega * ltdiff * (1.0/PI);
149 greg 1.3 scalecolor(ctmp, dtmp);
150     addcolor(cval, ctmp);
151     }
152 greg 2.2 if (ldot < -FTINY && (np->specfl&(SP_TRAN|SP_PURE)) == SP_TRAN) {
153 greg 1.3 /*
154 greg 1.4 * Compute specular transmission. Specular transmission
155 greg 1.13 * is always modified by material color.
156 greg 1.3 */
157     /* roughness + source */
158 greg 2.48 dtmp = np->alpha2 + omega*(1.0/PI);
159 greg 2.54 /* Gaussian */
160 greg 2.53 dtmp = exp((2.*DOT(np->prdir,ldir)-2.)/dtmp)/(PI*dtmp);
161 greg 1.3 /* worth using? */
162     if (dtmp > FTINY) {
163 greg 1.13 copycolor(ctmp, np->mcolor);
164 greg 2.52 dtmp *= np->tspec * omega * sqrt(-ldot/np->pdot);
165 greg 1.13 scalecolor(ctmp, dtmp);
166 greg 1.3 addcolor(cval, ctmp);
167     }
168     }
169     }
170    
171    
172 schorsch 2.47 extern int
173     m_normal( /* color a ray that hit something normal */
174     register OBJREC *m,
175     register RAY *r
176     )
177 greg 1.3 {
178     NORMDAT nd;
179 greg 2.38 double fest;
180 greg 1.9 double transtest, transdist;
181 greg 2.29 double mirtest, mirdist;
182     int hastexture;
183     double d;
184 greg 1.1 COLOR ctmp;
185     register int i;
186     /* easy shadow test */
187     if (r->crtype & SHADOW && m->otype != MAT_TRANS)
188 greg 2.27 return(1);
189 greg 2.2
190     if (m->oargs.nfargs != (m->otype == MAT_TRANS ? 7 : 5))
191     objerror(m, USER, "bad number of arguments");
192 greg 2.29 /* check for back side */
193     if (r->rod < 0.0) {
194     if (!backvis && m->otype != MAT_TRANS) {
195     raytrans(r);
196     return(1);
197     }
198 greg 2.40 raytexture(r, m->omod);
199 greg 2.29 flipsurface(r); /* reorient if backvis */
200 greg 2.40 } else
201     raytexture(r, m->omod);
202 greg 1.3 nd.mp = m;
203 greg 2.16 nd.rp = r;
204 greg 1.1 /* get material color */
205 greg 1.3 setcolor(nd.mcolor, m->oargs.farg[0],
206 greg 1.1 m->oargs.farg[1],
207     m->oargs.farg[2]);
208     /* get roughness */
209 greg 2.2 nd.specfl = 0;
210 greg 1.3 nd.alpha2 = m->oargs.farg[4];
211 greg 2.2 if ((nd.alpha2 *= nd.alpha2) <= FTINY)
212     nd.specfl |= SP_PURE;
213 greg 2.40
214 schorsch 2.45 if ( (hastexture = (DOT(r->pert,r->pert) > FTINY*FTINY)) ) {
215 greg 2.29 nd.pdot = raynormal(nd.pnorm, r); /* perturb normal */
216 greg 2.41 } else {
217 greg 2.29 VCOPY(nd.pnorm, r->ron);
218     nd.pdot = r->rod;
219     }
220 greg 2.42 if (r->ro != NULL && isflat(r->ro->otype))
221     nd.specfl |= SP_FLAT;
222 greg 1.13 if (nd.pdot < .001)
223     nd.pdot = .001; /* non-zero for dirnorm() */
224 greg 1.3 multcolor(nd.mcolor, r->pcol); /* modify material color */
225 greg 2.29 mirtest = transtest = 0;
226     mirdist = transdist = r->rot;
227 greg 2.30 nd.rspec = m->oargs.farg[3];
228 greg 2.38 /* compute Fresnel approx. */
229 greg 2.51 if (nd.specfl & SP_PURE && nd.rspec >= FRESTHRESH) {
230 greg 2.38 fest = FRESNE(r->rod);
231     nd.rspec += fest*(1. - nd.rspec);
232     } else
233     fest = 0.;
234 greg 1.3 /* compute transmission */
235 greg 1.1 if (m->otype == MAT_TRANS) {
236 greg 1.3 nd.trans = m->oargs.farg[5]*(1.0 - nd.rspec);
237     nd.tspec = nd.trans * m->oargs.farg[6];
238     nd.tdiff = nd.trans - nd.tspec;
239 greg 2.2 if (nd.tspec > FTINY) {
240     nd.specfl |= SP_TRAN;
241 greg 2.5 /* check threshold */
242 greg 2.25 if (!(nd.specfl & SP_PURE) &&
243     specthresh >= nd.tspec-FTINY)
244 greg 2.5 nd.specfl |= SP_TBLT;
245 greg 2.29 if (!hastexture || r->crtype & SHADOW) {
246 greg 2.2 VCOPY(nd.prdir, r->rdir);
247     transtest = 2;
248     } else {
249     for (i = 0; i < 3; i++) /* perturb */
250 greg 2.19 nd.prdir[i] = r->rdir[i] - r->pert[i];
251 greg 2.7 if (DOT(nd.prdir, r->ron) < -FTINY)
252     normalize(nd.prdir); /* OK */
253     else
254     VCOPY(nd.prdir, r->rdir);
255 greg 2.2 }
256 greg 1.14 }
257 greg 1.1 } else
258 greg 1.3 nd.tdiff = nd.tspec = nd.trans = 0.0;
259 greg 1.1 /* transmitted ray */
260 gregl 2.36 if ((nd.specfl&(SP_TRAN|SP_PURE|SP_TBLT)) == (SP_TRAN|SP_PURE)) {
261 greg 1.3 RAY lr;
262 greg 2.50 copycolor(lr.rcoef, nd.mcolor); /* modified by color */
263     scalecolor(lr.rcoef, nd.tspec);
264     if (rayorigin(&lr, TRANS, r, lr.rcoef) == 0) {
265 greg 1.14 VCOPY(lr.rdir, nd.prdir);
266 greg 1.1 rayvalue(&lr);
267 greg 2.50 multcolor(lr.rcol, lr.rcoef);
268 greg 1.1 addcolor(r->rcol, lr.rcol);
269 greg 1.9 transtest *= bright(lr.rcol);
270     transdist = r->rot + lr.rt;
271 greg 1.1 }
272 greg 2.11 } else
273     transtest = 0;
274 greg 2.2
275 greg 2.29 if (r->crtype & SHADOW) { /* the rest is shadow */
276     r->rt = transdist;
277 greg 2.27 return(1);
278 greg 2.30 }
279     /* get specular reflection */
280     if (nd.rspec > FTINY) {
281     nd.specfl |= SP_REFL;
282     /* compute specular color */
283 greg 2.38 if (m->otype != MAT_METAL) {
284     setcolor(nd.scolor, nd.rspec, nd.rspec, nd.rspec);
285     } else if (fest > FTINY) {
286     d = nd.rspec*(1. - fest);
287     for (i = 0; i < 3; i++)
288     nd.scolor[i] = fest + nd.mcolor[i]*d;
289     } else {
290 greg 2.30 copycolor(nd.scolor, nd.mcolor);
291 greg 2.38 scalecolor(nd.scolor, nd.rspec);
292     }
293 greg 2.30 /* check threshold */
294     if (!(nd.specfl & SP_PURE) && specthresh >= nd.rspec-FTINY)
295     nd.specfl |= SP_RBLT;
296     /* compute reflected ray */
297 greg 2.55 VSUM(nd.vrefl, r->rdir, nd.pnorm, 2.*nd.pdot);
298 greg 2.30 /* penetration? */
299     if (hastexture && DOT(nd.vrefl, r->ron) <= FTINY)
300 greg 2.55 VSUM(nd.vrefl, r->rdir, r->ron, 2.*r->rod);
301 greg 2.53 checknorm(nd.vrefl);
302 gregl 2.36 }
303     /* reflected ray */
304     if ((nd.specfl&(SP_REFL|SP_PURE|SP_RBLT)) == (SP_REFL|SP_PURE)) {
305     RAY lr;
306 greg 2.50 if (rayorigin(&lr, REFLECTED, r, nd.scolor) == 0) {
307 gregl 2.36 VCOPY(lr.rdir, nd.vrefl);
308     rayvalue(&lr);
309 greg 2.50 multcolor(lr.rcol, lr.rcoef);
310 gregl 2.36 addcolor(r->rcol, lr.rcol);
311     if (!hastexture && nd.specfl & SP_FLAT) {
312     mirtest = 2.*bright(lr.rcol);
313     mirdist = r->rot + lr.rt;
314 greg 2.30 }
315     }
316 greg 2.29 }
317 greg 1.1 /* diffuse reflection */
318 greg 1.3 nd.rdiff = 1.0 - nd.trans - nd.rspec;
319 greg 1.1
320 greg 2.2 if (nd.specfl & SP_PURE && nd.rdiff <= FTINY && nd.tdiff <= FTINY)
321 greg 2.27 return(1); /* 100% pure specular */
322 greg 2.3
323 gregl 2.36 if (!(nd.specfl & SP_PURE))
324     gaussamp(r, &nd); /* checks *BLT flags */
325 greg 2.2
326 greg 1.3 if (nd.rdiff > FTINY) { /* ambient from this side */
327 greg 2.50 copycolor(ctmp, nd.mcolor); /* modified by material color */
328 greg 2.5 if (nd.specfl & SP_RBLT)
329     scalecolor(ctmp, 1.0-nd.trans);
330     else
331     scalecolor(ctmp, nd.rdiff);
332 greg 2.50 multambient(ctmp, r, hastexture ? nd.pnorm : r->ron);
333 greg 1.2 addcolor(r->rcol, ctmp); /* add to returned color */
334     }
335 greg 1.3 if (nd.tdiff > FTINY) { /* ambient from other side */
336 greg 2.50 copycolor(ctmp, nd.mcolor); /* modified by color */
337     if (nd.specfl & SP_TBLT)
338     scalecolor(ctmp, nd.trans);
339     else
340     scalecolor(ctmp, nd.tdiff);
341 greg 1.1 flipsurface(r);
342 greg 2.32 if (hastexture) {
343     FVECT bnorm;
344     bnorm[0] = -nd.pnorm[0];
345     bnorm[1] = -nd.pnorm[1];
346     bnorm[2] = -nd.pnorm[2];
347 greg 2.50 multambient(ctmp, r, bnorm);
348 greg 2.32 } else
349 greg 2.50 multambient(ctmp, r, r->ron);
350 greg 1.1 addcolor(r->rcol, ctmp);
351     flipsurface(r);
352     }
353 greg 1.3 /* add direct component */
354     direct(r, dirnorm, &nd);
355 greg 1.9 /* check distance */
356 greg 2.29 d = bright(r->rcol);
357     if (transtest > d)
358 greg 1.9 r->rt = transdist;
359 greg 2.29 else if (mirtest > d)
360     r->rt = mirdist;
361 greg 2.27
362     return(1);
363 greg 2.2 }
364    
365    
366 greg 2.38 static void
367 greg 2.54 gaussamp( /* sample Gaussian specular */
368 schorsch 2.47 RAY *r,
369     register NORMDAT *np
370     )
371 greg 2.2 {
372     RAY sr;
373     FVECT u, v, h;
374     double rv[2];
375     double d, sinp, cosp;
376 greg 2.55 COLOR scol;
377 greg 2.58 int maxiter, ntrials, nstarget, nstaken;
378 greg 2.2 register int i;
379 greg 2.13 /* quick test */
380     if ((np->specfl & (SP_REFL|SP_RBLT)) != SP_REFL &&
381     (np->specfl & (SP_TRAN|SP_TBLT)) != SP_TRAN)
382     return;
383 greg 2.2 /* set up sample coordinates */
384     v[0] = v[1] = v[2] = 0.0;
385     for (i = 0; i < 3; i++)
386     if (np->pnorm[i] < 0.6 && np->pnorm[i] > -0.6)
387     break;
388     v[i] = 1.0;
389     fcross(u, v, np->pnorm);
390     normalize(u);
391     fcross(v, np->pnorm, u);
392     /* compute reflection */
393 greg 2.5 if ((np->specfl & (SP_REFL|SP_RBLT)) == SP_REFL &&
394 greg 2.50 rayorigin(&sr, SPECULAR, r, np->scolor) == 0) {
395 greg 2.58 nstarget = 1;
396 greg 2.55 if (specjitter > 1.5) { /* multiple samples? */
397 greg 2.58 nstarget = specjitter*r->rweight + .5;
398     if (sr.rweight <= minweight*nstarget)
399     nstarget = sr.rweight/minweight;
400     if (nstarget > 1) {
401     d = 1./nstarget;
402     scalecolor(sr.rcoef, d);
403 greg 2.56 sr.rweight *= d;
404 greg 2.55 } else
405 greg 2.58 nstarget = 1;
406 greg 2.55 }
407 greg 2.58 setcolor(scol, 0., 0., 0.);
408 greg 2.2 dimlist[ndims++] = (int)np->mp;
409 greg 2.58 maxiter = MAXITER*nstarget;
410     for (nstaken = ntrials = 0; nstaken < nstarget &&
411     ntrials < maxiter; ntrials++) {
412     if (ntrials)
413 greg 2.34 d = frandom();
414     else
415     d = urand(ilhash(dimlist,ndims)+samplendx);
416     multisamp(rv, 2, d);
417     d = 2.0*PI * rv[0];
418 gwlarson 2.37 cosp = tcos(d);
419     sinp = tsin(d);
420 greg 2.55 if ((0. <= specjitter) & (specjitter < 1.))
421     rv[1] = 1.0 - specjitter*rv[1];
422 greg 2.34 if (rv[1] <= FTINY)
423     d = 1.0;
424     else
425     d = sqrt( np->alpha2 * -log(rv[1]) );
426     for (i = 0; i < 3; i++)
427     h[i] = np->pnorm[i] + d*(cosp*u[i] + sinp*v[i]);
428     d = -2.0 * DOT(h, r->rdir) / (1.0 + d*d);
429 greg 2.55 VSUM(sr.rdir, r->rdir, h, d);
430 greg 2.58 /* sample rejection test */
431     if ((d = DOT(sr.rdir, r->ron)) <= FTINY)
432 greg 2.55 continue;
433     checknorm(sr.rdir);
434 greg 2.58 if (nstarget > 1) { /* W-G-M-D adjustment */
435 greg 2.55 rayclear(&sr);
436 greg 2.58 rayvalue(&sr);
437     d = 2./(1. + r->rod/d);
438     scalecolor(sr.rcol, d);
439     addcolor(scol, sr.rcol);
440     } else {
441     rayvalue(&sr);
442     multcolor(sr.rcol, sr.rcoef);
443     addcolor(r->rcol, sr.rcol);
444 greg 2.34 }
445 greg 2.58 ++nstaken;
446     }
447     if (nstarget > 1) { /* final W-G-M-D weighting */
448     multcolor(scol, sr.rcoef);
449     d = (double)nstarget/ntrials;
450     scalecolor(scol, d);
451     addcolor(r->rcol, scol);
452 greg 2.34 }
453 greg 2.2 ndims--;
454     }
455     /* compute transmission */
456 greg 2.50 copycolor(sr.rcoef, np->mcolor); /* modified by color */
457     scalecolor(sr.rcoef, np->tspec);
458 greg 2.8 if ((np->specfl & (SP_TRAN|SP_TBLT)) == SP_TRAN &&
459 greg 2.50 rayorigin(&sr, SPECULAR, r, sr.rcoef) == 0) {
460 greg 2.58 nstarget = 1;
461 greg 2.55 if (specjitter > 1.5) { /* multiple samples? */
462 greg 2.58 nstarget = specjitter*r->rweight + .5;
463     if (sr.rweight <= minweight*nstarget)
464     nstarget = sr.rweight/minweight;
465     if (nstarget > 1) {
466     d = 1./nstarget;
467 greg 2.56 scalecolor(sr.rcoef, d);
468     sr.rweight *= d;
469 greg 2.55 } else
470 greg 2.58 nstarget = 1;
471 greg 2.55 }
472 greg 2.8 dimlist[ndims++] = (int)np->mp;
473 greg 2.58 maxiter = MAXITER*nstarget;
474     for (nstaken = ntrials = 0; nstaken < nstarget &&
475     ntrials < maxiter; ntrials++) {
476     if (ntrials)
477 greg 2.34 d = frandom();
478     else
479 greg 2.58 d = urand(ilhash(dimlist,ndims)+samplendx);
480 greg 2.34 multisamp(rv, 2, d);
481     d = 2.0*PI * rv[0];
482 gwlarson 2.37 cosp = tcos(d);
483     sinp = tsin(d);
484 greg 2.55 if ((0. <= specjitter) & (specjitter < 1.))
485     rv[1] = 1.0 - specjitter*rv[1];
486 greg 2.34 if (rv[1] <= FTINY)
487     d = 1.0;
488     else
489 gwlarson 2.37 d = sqrt( np->alpha2 * -log(rv[1]) );
490 greg 2.34 for (i = 0; i < 3; i++)
491     sr.rdir[i] = np->prdir[i] + d*(cosp*u[i] + sinp*v[i]);
492 greg 2.58 /* sample rejection test */
493 greg 2.55 if (DOT(sr.rdir, r->ron) >= -FTINY)
494     continue;
495     normalize(sr.rdir); /* OK, normalize */
496 greg 2.58 if (nstarget > 1) /* multi-sampling */
497 greg 2.55 rayclear(&sr);
498     rayvalue(&sr);
499     multcolor(sr.rcol, sr.rcoef);
500     addcolor(r->rcol, sr.rcol);
501 greg 2.58 ++nstaken;
502 greg 2.34 }
503 greg 2.8 ndims--;
504     }
505 greg 1.1 }