/*
--------------------------------- geom.c geometric routines of qhull see qh-geom.htm and geom.h copyright (c) 1993-2002 The Geometry Center infrequent code goes into geom2.c */ #include "qhull_a.h" /*--------------------------------- qh_distplane( point, facet, dist ) return distance from point to facet returns: dist if qh.RANDOMdist, joggles result notes: dist > 0 if point is above facet (i.e., outside) does not error (for sortfacets) see: qh_distnorm in geom2.c */ void qh_distplane (pointT *point, facetT *facet, realT *dist) { coordT *normal= facet->normal, *coordp, randr; int k; switch(qh hull_dim){ case 2: *dist= facet->offset + point[0] * normal[0] + point[1] * normal[1]; break; case 3: *dist= facet->offset + point[0] * normal[0] + point[1] * normal[1] + point[2] * normal[2]; break; case 4: *dist= facet->offset+point[0]*normal[0]+point[1]*normal[1]+point[2]*normal[2]+point[3]*normal[3]; break; case 5: *dist= facet->offset+point[0]*normal[0]+point[1]*normal[1]+point[2]*normal[2]+point[3]*normal[3]+point[4]*normal[4]; break; case 6: *dist= facet->offset+point[0]*normal[0]+point[1]*normal[1]+point[2]*normal[2]+point[3]*normal[3]+point[4]*normal[4]+point[5]*normal[5]; break; case 7: *dist= facet->offset+point[0]*normal[0]+point[1]*normal[1]+point[2]*normal[2]+point[3]*normal[3]+point[4]*normal[4]+point[5]*normal[5]+point[6]*normal[6]; break; case 8: *dist= facet->offset+point[0]*normal[0]+point[1]*normal[1]+point[2]*normal[2]+point[3]*normal[3]+point[4]*normal[4]+point[5]*normal[5]+point[6]*normal[6]+point[7]*normal[7]; break; default: *dist= facet->offset; coordp= point; for (k= qh hull_dim; k--; ) *dist += *coordp++ * *normal++; break; } zinc_(Zdistplane); if (!qh RANDOMdist && qh IStracing < 4) return; if (qh RANDOMdist) { randr= qh_RANDOMint; *dist += (2.0 * randr / qh_RANDOMmax - 1.0) * qh RANDOMfactor * qh MAXabs_coord; } if (qh IStracing >= 4) { fprintf (qh ferr, "qh_distplane: "); fprintf (qh ferr, qh_REAL_1, *dist); fprintf (qh ferr, "from p%d to f%d\n", qh_pointid(point), facet->id); } return; } /* distplane */ /*--------------------------------- qh_findbest( point, startfacet, bestoutside, qh_ISnewfacets, qh_NOupper, dist, isoutside, numpart ) find facet that is furthest below a point for upperDelaunay facets returns facet only if !qh_NOupper and clearly above input: starts search at 'startfacet' (can not be flipped) if !bestoutside (qh_ALL), stops at qh.MINoutside returns: best facet (reports error if NULL) early out if isoutside defined and bestdist > qh.MINoutside dist is distance to facet isoutside is true if point is outside of facet numpart counts the number of distance tests see also: qh_findbestnew() notes: If merging (testhorizon), searches horizon facets of coplanar best facets because after qh_distplane, this and qh_partitionpoint are the most expensive in 3-d avoid calls to distplane, function calls, and real number operations. caller traces result Optimized for outside points. Tried recording a search set for qh_findhorizon. Made code more complicated. when called by qh_partitionvisible(): indicated by qh_ISnewfacets qh.newfacet_list is list of simplicial, new facets qh_findbestnew set if qh_sharpnewfacets returns True (to use qh_findbestnew) qh.bestfacet_notsharp set if qh_sharpnewfacets returns False when called by qh_findfacet(), qh_partitionpoint(), qh_partitioncoplanar(), qh_check_bestdist(), qh_addpoint() indicated by !qh_ISnewfacets returns best facet in neighborhood of given facet this is best facet overall if dist > - qh.MAXcoplanar or hull has at least a "spherical" curvature design: initialize and test for early exit repeat while there are better facets for each neighbor of facet exit if outside facet found test for better facet if point is inside and partitioning test for new facets with a "sharp" intersection if so, future calls go to qh_findbestnew() test horizon facets */ facetT *qh_findbest (pointT *point, facetT *startfacet, boolT bestoutside, boolT isnewfacets, boolT noupper, realT *dist, boolT *isoutside, int *numpart) { realT bestdist= -REALmax/2 /* avoid underflow */; facetT *facet, *neighbor, **neighborp, *bestfacet= NULL; /* facetT *bestfacet_all= startfacet; */ int oldtrace= qh IStracing; unsigned int visitid= ++qh visit_id; int numpartnew=0; boolT testhorizon = True; /* needed if precise, e.g., rbox c D6 | qhull Q0 Tv */ zinc_(Zfindbest); if (qh IStracing >= 3 || (qh TRACElevel && qh TRACEpoint >= 0 && qh TRACEpoint == qh_pointid (point))) { if (qh TRACElevel > qh IStracing) qh IStracing= qh TRACElevel; fprintf (qh ferr, "qh_findbest: point p%d starting at f%d isnewfacets? %d, unless %d exit if > %2.2g\n", qh_pointid(point), startfacet->id, isnewfacets, bestoutside, qh MINoutside); fprintf(qh ferr, " testhorizon? %d noupper? %d", testhorizon, noupper); fprintf (qh ferr, " Last point added was p%d.", qh furthest_id); fprintf(qh ferr, " Last merge was #%d. max_outside %2.2g\n", zzval_(Ztotmerge), qh max_outside); } if (isoutside) *isoutside= True; if (!startfacet->flipped) { /* test startfacet */ *numpart= 1; qh_distplane (point, startfacet, dist); /* this code is duplicated below */ if (!bestoutside && *dist >= qh MINoutside && (!startfacet->upperdelaunay || !noupper)) { bestfacet= startfacet; goto LABELreturn_best; } bestdist= *dist; if (!startfacet->upperdelaunay) { bestfacet= startfacet; } }else *numpart= 0; startfacet->visitid= visitid; facet= startfacet; while (facet) { trace4((qh ferr, "qh_findbest: neighbors of f%d, bestdist %2.2g f%d\n", facet->id, bestdist, getid_(bestfacet))); FOREACHneighbor_(facet) { if (!neighbor->newfacet && isnewfacets) continue; if (neighbor->visitid == visitid) continue; neighbor->visitid= visitid; if (!neighbor->flipped) { /* code duplicated above */ (*numpart)++; qh_distplane (point, neighbor, dist); if (*dist > bestdist) { if (!bestoutside && *dist >= qh MINoutside && (!neighbor->upperdelaunay || !noupper)) { bestfacet= neighbor; goto LABELreturn_best; } if (!neighbor->upperdelaunay) { bestfacet= neighbor; bestdist= *dist; } break; /* switch to neighor */ } /* end of *dist>bestdist */ } /* end of !flipped */ } /* end of FOREACHneighbor */ facet= neighbor; /* non-NULL only if *dist>bestdist */ } /* end of while facet (directed search) */ if (isnewfacets) { if (!bestfacet) { bestdist= -REALmax/2; bestfacet= qh_findbestnew (point, startfacet->next, &bestdist, bestoutside, isoutside, &numpartnew); testhorizon= False; /* qh_findbestnew calls qh_findbesthorizon */ }else if (!qh findbest_notsharp && bestdist < - qh DISTround) { if (qh_sharpnewfacets()) { /* seldom used, qh_findbestnew will retest all facets */ zinc_(Zfindnewsharp); bestfacet= qh_findbestnew (point, bestfacet, &bestdist, bestoutside, isoutside, &numpartnew); testhorizon= False; /* qh_findbestnew calls qh_findbesthorizon */ qh findbestnew= True; }else qh findbest_notsharp= True; } } if (!bestfacet) { fprintf(qh ferr, "\n\ qh_findbest: all neighbors of facet %d are flipped or upper Delaunay.\n\ Please report this error to qhull_bug@geom.umn.edu with the input and all of the output.\n", startfacet->id); qh_errexit (qh_ERRqhull, startfacet, NULL); } if (testhorizon) bestfacet= qh_findbesthorizon (!qh_IScheckmax, point, bestfacet, noupper, &bestdist, &numpartnew); *dist= bestdist; if (isoutside && bestdist < qh MINoutside) *isoutside= False; LABELreturn_best: zadd_(Zfindbesttot, *numpart); zmax_(Zfindbestmax, *numpart); (*numpart) += numpartnew; qh IStracing= oldtrace; return bestfacet; } /* findbest */ /*--------------------------------- qh_findbesthorizon( qh_IScheckmax, point, startfacet, qh_NOupper, &bestdist, &numpart ) search coplanar and better horizon facets from startfacet/bestdist ischeckmax turns off statistics and minsearch update all arguments must be initialized returns (ischeckmax): best facet returns (!ischeckmax): best facet that is not upperdelaunay allows upperdelaunay that is clearly outside returns: bestdist is distance to bestfacet numpart -- updates number of distance tests notes: no early out -- use qh_findbest() or qh_findbestnew() Searches coplanar or better horizon facets when called by qh_check_maxout() (qh_IScheckmax) startfacet must be closest to the point Otherwise, if point is beyond and below startfacet, startfacet may be a local minimum even though other facets are below the point. updates facet->maxoutside for good, visited facets may return NULL searchdist is qh.max_outside + 2 * DISTround + max( MINvisible('Vn'), MAXcoplanar('Un')); This setting is a guess. It must be at least max_outside + 2*DISTround because a facet may have a geometric neighbor across a vertex design: for each horizon facet of coplanar best facets continue if clearly inside unless upperdelaunay or clearly outside update best facet */ facetT *qh_findbesthorizon (boolT ischeckmax, pointT* point, facetT *startfacet, boolT noupper, realT *bestdist, int *numpart) { facetT *bestfacet= startfacet; realT dist; facetT *neighbor, **neighborp, *facet; facetT *nextfacet= NULL; /* optimize last facet of coplanarset */ int numpartinit= *numpart, coplanarset_size; unsigned int visitid= ++qh visit_id; boolT newbest= False; /* for tracing */ realT minsearch, searchdist; /* skip facets that are too far from point */ if (!ischeckmax) { zinc_(Zfindhorizon); }else { #if qh_MAXoutside if ((!qh ONLYgood || startfacet->good) && *bestdist > startfacet->maxoutside) startfacet->maxoutside= *bestdist; #endif } searchdist= qh_SEARCHdist; /* multiple of qh.max_outside and precision constants */ minsearch= *bestdist - searchdist; if (ischeckmax) { /* Always check coplanar facets. Needed for RBOX 1000 s Z1 G1e-13 t996564279 | QHULL Tv */ minimize_(minsearch, -searchdist); } coplanarset_size= 0; facet= startfacet; while (True) { trace4((qh ferr, "qh_findbesthorizon: neighbors of f%d bestdist %2.2g f%d ischeckmax? %d noupper? %d minsearch %2.2g searchdist %2.2g\n", facet->id, *bestdist, getid_(bestfacet), ischeckmax, noupper, minsearch, searchdist)); FOREACHneighbor_(facet) { if (neighbor->visitid == visitid) continue; neighbor->visitid= visitid; if (!neighbor->flipped) { qh_distplane (point, neighbor, &dist); (*numpart)++; if (dist > *bestdist) { if (!neighbor->upperdelaunay || ischeckmax || (!noupper && dist >= qh MINoutside)) { bestfacet= neighbor; *bestdist= dist; newbest= True; if (!ischeckmax) { minsearch= dist - searchdist; if (dist > *bestdist + searchdist) { zinc_(Zfindjump); /* everything in qh.coplanarset at least searchdist below */ coplanarset_size= 0; } } } }else if (dist < minsearch) continue; /* if ischeckmax, dist can't be positive */ #if qh_MAXoutside if (ischeckmax && dist > neighbor->maxoutside) neighbor->maxoutside= dist; #endif } /* end of !flipped */ if (nextfacet) { if (!coplanarset_size++) { SETfirst_(qh coplanarset)= nextfacet; SETtruncate_(qh coplanarset, 1); }else qh_setappend (&qh coplanarset, nextfacet); /* Was needed for RBOX 1000 s W1e-13 P0 t996547055 | QHULL d Qbb Qc Tv and RBOX 1000 s Z1 G1e-13 t996564279 | qhull Tv */ } nextfacet= neighbor; } /* end of EACHneighbor */ facet= nextfacet; if (facet) nextfacet= NULL; else if (!coplanarset_size) break; else if (!--coplanarset_size) { facet= SETfirst_(qh coplanarset); SETtruncate_(qh coplanarset, 0); }else facet= (facetT*)qh_setdellast (qh coplanarset); } /* while True, for each facet in qh.coplanarset */ if (!ischeckmax) { zadd_(Zfindhorizontot, *numpart - numpartinit); zmax_(Zfindhorizonmax, *numpart - numpartinit); if (newbest) zinc_(Zparthorizon); } trace4((qh ferr, "qh_findbesthorizon: newbest? %d bestfacet f%d bestdist %2.2g\n", newbest, getid_(bestfacet), *bestdist)); return bestfacet; } /* findbesthorizon */ /*--------------------------------- qh_findbestnew( point, startfacet, dist, isoutside, numpart ) find best newfacet for point searches all of qh.newfacet_list starting at startfacet searches horizon facets of coplanar best newfacets searches all facets if startfacet == qh.facet_list returns: best new or horizon facet that is not upperdelaunay early out if isoutside and not 'Qf' dist is distance to facet isoutside is true if point is outside of facet numpart is number of distance tests notes: Always used for merged new facets (see qh_USEfindbestnew) Avoids upperdelaunay facet unless (isoutside and outside) Uses qh.visit_id, qh.coplanarset. If share visit_id with qh_findbest, coplanarset is incorrect. If merging (testhorizon), searches horizon facets of coplanar best facets because a point maybe coplanar to the bestfacet, below its horizon facet, and above a horizon facet of a coplanar newfacet. For example, rbox 1000 s Z1 G1e-13 | qhull rbox 1000 s W1e-13 P0 t992110337 | QHULL d Qbb Qc qh_findbestnew() used if qh_sharpnewfacets -- newfacets contains a sharp angle if many merges, qh_premerge found a merge, or 'Qf' (qh.findbestnew) see also: qh_partitionall() and qh_findbest() design: for each new facet starting from startfacet test distance from point to facet return facet if clearly outside unless upperdelaunay and a lowerdelaunay exists update best facet test horizon facets */ facetT *qh_findbestnew (pointT *point, facetT *startfacet, realT *dist, boolT bestoutside, boolT *isoutside, int *numpart) { realT bestdist= -REALmax/2; /*, minsearch= -REALmax/2;*/ facetT *bestfacet= NULL, *facet; int oldtrace= qh IStracing, i; unsigned int visitid= ++qh visit_id; realT distoutside= 0.0; boolT isdistoutside; /* True if distoutside is defined */ boolT testhorizon = True; /* needed if precise, e.g., rbox c D6 | qhull Q0 Tv */ if (!startfacet) { if (qh MERGING) fprintf(qh ferr, "qhull precision error (qh_findbestnew): merging has formed and deleted a cone of new facets. Can not continue.\n"); else fprintf(qh ferr, "qhull internal error (qh_findbestnew): no new facets for point p%d\n", qh furthest_id); qh_errexit (qh_ERRqhull, NULL, NULL); } zinc_(Zfindnew); if (qh BESToutside || bestoutside) isdistoutside= False; else { isdistoutside= True; distoutside= qh_DISToutside; /* multiple of qh.MINoutside & qh.max_outside, see user.h */ } if (isoutside) *isoutside= True; *numpart= 0; if (qh IStracing >= 3 || (qh TRACElevel && qh TRACEpoint >= 0 && qh TRACEpoint == qh_pointid (point))) { if (qh TRACElevel > qh IStracing) qh IStracing= qh TRACElevel; fprintf(qh ferr, "qh_findbestnew: point p%d facet f%d. Stop? %d if dist > %2.2g\n", qh_pointid(point), startfacet->id, isdistoutside, distoutside); fprintf(qh ferr, " Last point added p%d visitid %d.", qh furthest_id, visitid); fprintf(qh ferr, " Last merge was #%d.\n", zzval_(Ztotmerge)); } /* visit all new facets starting with startfacet, maybe qh facet_list */ for (i= 0, facet= startfacet; i < 2; i++, facet= qh newfacet_list) { FORALLfacet_(facet) { if (facet == startfacet && i) break; facet->visitid= visitid; if (!facet->flipped) { qh_distplane (point, facet, dist); (*numpart)++; if (*dist > bestdist) { if (!facet->upperdelaunay || *dist >= qh MINoutside) { bestfacet= facet; if (isdistoutside && *dist >= distoutside) goto LABELreturn_bestnew; bestdist= *dist; } } } /* end of !flipped */ } /* FORALLfacet from startfacet or qh newfacet_list */ } if (testhorizon || !bestfacet) bestfacet= qh_findbesthorizon (!qh_IScheckmax, point, bestfacet ? bestfacet : startfacet, !qh_NOupper, &bestdist, numpart); *dist= bestdist; if (isoutside && *dist < qh MINoutside) *isoutside= False; LABELreturn_bestnew: zadd_(Zfindnewtot, *numpart); zmax_(Zfindnewmax, *numpart); trace4((qh ferr, "qh_findbestnew: bestfacet f%d bestdist %2.2g\n", getid_(bestfacet), *dist)); qh IStracing= oldtrace; return bestfacet; } /* findbestnew */ /* ============ hyperplane functions -- keep code together [?] ============ */ /*--------------------------------- qh_backnormal( rows, numrow, numcol, sign, normal, nearzero ) given an upper-triangular rows array and a sign, solve for normal equation x using back substitution over rows U returns: normal= x if will not be able to divzero() when normalized (qh.MINdenom_2 and qh.MINdenom_1_2), if fails on last row this means that the hyperplane intersects [0,..,1] sets last coordinate of normal to sign otherwise sets tail of normal to [...,sign,0,...], i.e., solves for b= [0...0] sets nearzero notes: assumes numrow == numcol-1 see Golub & van Loan 4.4-9 for back substitution solves Ux=b where Ax=b and PA=LU b= [0,...,0,sign or 0] (sign is either -1 or +1) last row of A= [0,...,0,1] 1) Ly=Pb == y=b since P only permutes the 0's of b design: for each row from end perform back substitution if near zero use qh_divzero for division if zero divide and not last row set tail of normal to 0 */ void qh_backnormal (realT **rows, int numrow, int numcol, boolT sign, coordT *normal, boolT *nearzero) { int i, j; coordT *normalp, *normal_tail, *ai, *ak; realT diagonal; boolT waszero; int zerocol= -1; normalp= normal + numcol - 1; *normalp--= (sign ? -1.0 : 1.0); for(i= numrow; i--; ) { *normalp= 0.0; ai= rows[i] + i + 1; ak= normalp+1; for(j= i+1; j < numcol; j++) *normalp -= *ai++ * *ak++; diagonal= (rows[i])[i]; if (fabs_(diagonal) > qh MINdenom_2) *(normalp--) /= diagonal; else { waszero= False; *normalp= qh_divzero (*normalp, diagonal, qh MINdenom_1_2, &waszero); if (waszero) { zerocol= i; *(normalp--)= (sign ? -1.0 : 1.0); for (normal_tail= normalp+2; normal_tail < normal + numcol; normal_tail++) *normal_tail= 0.0; }else normalp--; } } if (zerocol != -1) { zzinc_(Zback0); *nearzero= True; trace4((qh ferr, "qh_backnormal: zero diagonal at column %d.\n", i)); qh_precision ("zero diagonal on back substitution"); } } /* backnormal */ /*--------------------------------- qh_gausselim( rows, numrow, numcol, sign ) Gaussian elimination with partial pivoting returns: rows is upper triangular (includes row exchanges) flips sign for each row exchange sets nearzero if pivot[k] < qh.NEARzero[k], else clears it notes: if nearzero, the determinant's sign may be incorrect. assumes numrow <= numcol design: for each row determine pivot and exchange rows if necessary test for near zero perform gaussian elimination step */ void qh_gausselim(realT **rows, int numrow, int numcol, boolT *sign, boolT *nearzero) { realT *ai, *ak, *rowp, *pivotrow; realT n, pivot, pivot_abs= 0.0, temp; int i, j, k, pivoti, flip=0; *nearzero= False; for(k= 0; k < numrow; k++) { pivot_abs= fabs_((rows[k])[k]); pivoti= k; for(i= k+1; i < numrow; i++) { if ((temp= fabs_((rows[i])[k])) > pivot_abs) { pivot_abs= temp; pivoti= i; } } if (pivoti != k) { rowp= rows[pivoti]; rows[pivoti]= rows[k]; rows[k]= rowp; *sign ^= 1; flip ^= 1; } if (pivot_abs <= qh NEARzero[k]) { *nearzero= True; if (pivot_abs == 0.0) { /* remainder of column == 0 */ if (qh IStracing >= 4) { fprintf (qh ferr, "qh_gausselim: 0 pivot at column %d. (%2.2g < %2.2g)\n", k, pivot_abs, qh DISTround); qh_printmatrix (qh ferr, "Matrix:", rows, numrow, numcol); } zzinc_(Zgauss0); qh_precision ("zero pivot for Gaussian elimination"); goto LABELnextcol; } } pivotrow= rows[k] + k; pivot= *pivotrow++; /* signed value of pivot, and remainder of row */ for(i= k+1; i < numrow; i++) { ai= rows[i] + k; ak= pivotrow; n= (*ai++)/pivot; /* divzero() not needed since |pivot| >= |*ai| */ for(j= numcol - (k+1); j--; ) *ai++ -= n * *ak++; } LABELnextcol: ; } wmin_(Wmindenom, pivot_abs); /* last pivot element */ if (qh IStracing >= 5) qh_printmatrix (qh ferr, "qh_gausselem: result", rows, numrow, numcol); } /* gausselim */ /*--------------------------------- qh_getangle( vect1, vect2 ) returns the dot product of two vectors if qh.RANDOMdist, joggles result notes: the angle may be > 1.0 or < -1.0 because of roundoff errors */ realT qh_getangle(pointT *vect1, pointT *vect2) { realT angle= 0, randr; int k; for(k= qh hull_dim; k--; ) angle += *vect1++ * *vect2++; if (qh RANDOMdist) { randr= qh_RANDOMint; angle += (2.0 * randr / qh_RANDOMmax - 1.0) * qh RANDOMfactor; } trace4((qh ferr, "qh_getangle: %2.2g\n", angle)); return(angle); } /* getangle */ /*--------------------------------- qh_getcenter( vertices ) returns arithmetic center of a set of vertices as a new point notes: allocates point array for center */ pointT *qh_getcenter(setT *vertices) { int k; pointT *center, *coord; vertexT *vertex, **vertexp; int count= qh_setsize(vertices); if (count < 2) { fprintf (qh ferr, "qhull internal error (qh_getcenter): not defined for %d points\n", count); qh_errexit (qh_ERRqhull, NULL, NULL); } center= (pointT *)qh_memalloc(qh normal_size); for (k=0; k < qh hull_dim; k++) { coord= center+k; *coord= 0.0; FOREACHvertex_(vertices) *coord += vertex->point[k]; *coord /= count; } return(center); } /* getcenter */ /*--------------------------------- qh_getcentrum( facet ) returns the centrum for a facet as a new point notes: allocates the centrum */ pointT *qh_getcentrum(facetT *facet) { realT dist; pointT *centrum, *point; point= qh_getcenter(facet->vertices); zzinc_(Zcentrumtests); qh_distplane (point, facet, &dist); centrum= qh_projectpoint(point, facet, dist); qh_memfree(point, qh normal_size); trace4((qh ferr, "qh_getcentrum: for f%d, %d vertices dist= %2.2g\n", facet->id, qh_setsize(facet->vertices), dist)); return centrum; } /* getcentrum */ /*--------------------------------- qh_getdistance( facet, neighbor, mindist, maxdist ) returns the maxdist and mindist distance of any vertex from neighbor returns: the max absolute value design: for each vertex of facet that is not in neighbor test the distance from vertex to neighbor */ realT qh_getdistance(facetT *facet, facetT *neighbor, realT *mindist, realT *maxdist) { vertexT *vertex, **vertexp; realT dist, maxd, mind; FOREACHvertex_(facet->vertices) vertex->seen= False; FOREACHvertex_(neighbor->vertices) vertex->seen= True; mind= 0.0; maxd= 0.0; FOREACHvertex_(facet->vertices) { if (!vertex->seen) { zzinc_(Zbestdist); qh_distplane(vertex->point, neighbor, &dist); if (dist < mind) mind= dist; else if (dist > maxd) maxd= dist; } } *mindist= mind; *maxdist= maxd; mind= -mind; if (maxd > mind) return maxd; else return mind; } /* getdistance */ /*--------------------------------- qh_normalize( normal, dim, toporient ) normalize a vector and report if too small does not use min norm see: qh_normalize2 */ void qh_normalize (coordT *normal, int dim, boolT toporient) { qh_normalize2( normal, dim, toporient, NULL, NULL); } /* normalize */ /*--------------------------------- qh_normalize2( normal, dim, toporient, minnorm, ismin ) normalize a vector and report if too small qh.MINdenom/MINdenom1 are the upper limits for divide overflow returns: normalized vector flips sign if !toporient if minnorm non-NULL, sets ismin if normal < minnorm notes: if zero norm sets all elements to sqrt(1.0/dim) if divide by zero (divzero ()) sets largest element to +/-1 bumps Znearlysingular design: computes norm test for minnorm if not near zero normalizes normal else if zero norm sets normal to standard value else uses qh_divzero to normalize if nearzero sets norm to direction of maximum value */ void qh_normalize2 (coordT *normal, int dim, boolT toporient, realT *minnorm, boolT *ismin) { int k; realT *colp, *maxp, norm= 0, temp, *norm1, *norm2, *norm3; boolT zerodiv; norm1= normal+1; norm2= normal+2; norm3= normal+3; if (dim == 2) norm= sqrt((*normal)*(*normal) + (*norm1)*(*norm1)); else if (dim == 3) norm= sqrt((*normal)*(*normal) + (*norm1)*(*norm1) + (*norm2)*(*norm2)); else if (dim == 4) { norm= sqrt((*normal)*(*normal) + (*norm1)*(*norm1) + (*norm2)*(*norm2) + (*norm3)*(*norm3)); }else if (dim > 4) { norm= (*normal)*(*normal) + (*norm1)*(*norm1) + (*norm2)*(*norm2) + (*norm3)*(*norm3); for (k= dim-4, colp= normal+4; k--; colp++) norm += (*colp) * (*colp); norm= sqrt(norm); } if (minnorm) { if (norm < *minnorm) *ismin= True; else *ismin= False; } wmin_(Wmindenom, norm); if (norm > qh MINdenom) { if (!toporient) norm= -norm; *normal /= norm; *norm1 /= norm; if (dim == 2) ; /* all done */ else if (dim == 3) *norm2 /= norm; else if (dim == 4) { *norm2 /= norm; *norm3 /= norm; }else if (dim >4) { *norm2 /= norm; *norm3 /= norm; for (k= dim-4, colp= normal+4; k--; ) *colp++ /= norm; } }else if (norm == 0.0) { temp= sqrt (1.0/dim); for (k= dim, colp= normal; k--; ) *colp++ = temp; }else { if (!toporient) norm= -norm; for (k= dim, colp= normal; k--; colp++) { /* k used below */ temp= qh_divzero (*colp, norm, qh MINdenom_1, &zerodiv); if (!zerodiv) *colp= temp; else { maxp= qh_maxabsval(normal, dim); temp= ((*maxp * norm >= 0.0) ? 1.0 : -1.0); for (k= dim, colp= normal; k--; colp++) *colp= 0.0; *maxp= temp; zzinc_(Znearlysingular); trace0((qh ferr, "qh_normalize: norm=%2.2g too small during p%d\n", norm, qh furthest_id)); return; } } } } /* normalize */ /*--------------------------------- qh_projectpoint( point, facet, dist ) project point onto a facet by dist returns: returns a new point notes: if dist= distplane(point,facet) this projects point to hyperplane assumes qh_memfree_() is valid for normal_size */ pointT *qh_projectpoint(pointT *point, facetT *facet, realT dist) { pointT *newpoint, *np, *normal; int normsize= qh normal_size,k; void **freelistp; /* used !qh_NOmem */ qh_memalloc_(normsize, freelistp, newpoint, pointT); np= newpoint; normal= facet->normal; for(k= qh hull_dim; k--; ) *(np++)= *point++ - dist * *normal++; return(newpoint); } /* projectpoint */ /*--------------------------------- qh_setfacetplane( facet ) sets the hyperplane for a facet if qh.RANDOMdist, joggles hyperplane notes: uses global buffers qh.gm_matrix and qh.gm_row overwrites facet->normal if already defined updates Wnewvertex if PRINTstatistics sets facet->upperdelaunay if upper envelope of Delaunay triangulation design: copy vertex coordinates to qh.gm_matrix/gm_row compute determinate if nearzero recompute determinate with gaussian elimination if nearzero force outside orientation by testing interior point */ void qh_setfacetplane(facetT *facet) { pointT *point; vertexT *vertex, **vertexp; int k,i, normsize= qh normal_size, oldtrace= 0; realT dist; void **freelistp; /* used !qh_NOmem */ coordT *coord, *gmcoord; pointT *point0= SETfirstt_(facet->vertices, vertexT)->point; boolT nearzero= False; zzinc_(Zsetplane); if (!facet->normal) qh_memalloc_(normsize, freelistp, facet->normal, coordT); if (facet == qh tracefacet) { oldtrace= qh IStracing; qh IStracing= 5; fprintf (qh ferr, "qh_setfacetplane: facet f%d created.\n", facet->id); fprintf (qh ferr, " Last point added to hull was p%d.", qh furthest_id); if (zzval_(Ztotmerge)) fprintf(qh ferr, " Last merge was #%d.", zzval_(Ztotmerge)); fprintf (qh ferr, "\n\nCurrent summary is:\n"); qh_printsummary (qh ferr); } if (qh hull_dim <= 4) { i= 0; if (qh RANDOMdist) { gmcoord= qh gm_matrix; FOREACHvertex_(facet->vertices) { qh gm_row[i++]= gmcoord; coord= vertex->point; for (k= qh hull_dim; k--; ) *(gmcoord++)= *coord++ * qh_randomfactor(); } }else { FOREACHvertex_(facet->vertices) qh gm_row[i++]= vertex->point; } qh_sethyperplane_det(qh hull_dim, qh gm_row, point0, facet->toporient, facet->normal, &facet->offset, &nearzero); } if (qh hull_dim > 4 || nearzero) { i= 0; gmcoord= qh gm_matrix; FOREACHvertex_(facet->vertices) { if (vertex->point != point0) { qh gm_row[i++]= gmcoord; coord= vertex->point; point= point0; for(k= qh hull_dim; k--; ) *(gmcoord++)= *coord++ - *point++; } } qh gm_row[i]= gmcoord; /* for areasimplex */ if (qh RANDOMdist) { gmcoord= qh gm_matrix; for (i= qh hull_dim-1; i--; ) { for (k= qh hull_dim; k--; ) *(gmcoord++) *= qh_randomfactor(); } } qh_sethyperplane_gauss(qh hull_dim, qh gm_row, point0, facet->toporient, facet->normal, &facet->offset, &nearzero); if (nearzero) { if (qh_orientoutside (facet)) { trace0((qh ferr, "qh_setfacetplane: flipped orientation after testing interior_point during p%d\n", qh furthest_id)); /* this is part of using Gaussian Elimination. For example in 5-d 1 1 1 1 0 1 1 1 1 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 norm= 0.38 0.38 -0.76 0.38 0 has a determinate of 1, but g.e. after subtracting pt. 0 has 0's in the diagonal, even with full pivoting. It does work if you subtract pt. 4 instead. */ } } } facet->upperdelaunay= False; if (qh DELAUNAY) { if (qh UPPERdelaunay) { /* matches qh_triangulate_facet and qh.lower_threshold in qh_initbuild */ if (facet->normal[qh hull_dim -1] >= qh ANGLEround * qh_ZEROdelaunay) facet->upperdelaunay= True; }else { if (facet->normal[qh hull_dim -1] > -qh ANGLEround * qh_ZEROdelaunay) facet->upperdelaunay= True; } } if (qh PRINTstatistics || qh IStracing || qh TRACElevel || qh JOGGLEmax < REALmax) { qh old_randomdist= qh RANDOMdist; qh RANDOMdist= False; FOREACHvertex_(facet->vertices) { if (vertex->point != point0) { boolT istrace= False; zinc_(Zdiststat); qh_distplane(vertex->point, facet, &dist); dist= fabs_(dist); zinc_(Znewvertex); wadd_(Wnewvertex, dist); if (dist > wwval_(Wnewvertexmax)) { wwval_(Wnewvertexmax)= dist; if (dist > qh max_outside) { qh max_outside= dist; /* used by qh_maxouter() */ if (dist > qh TRACEdist) istrace= True; } }else if (-dist > qh TRACEdist) istrace= True; if (istrace) { fprintf (qh ferr, "qh_setfacetplane: ====== vertex p%d (v%d) increases max_outside to %2.2g for new facet f%d last p%d\n", qh_pointid(vertex->point), vertex->id, dist, facet->id, qh furthest_id); qh_errprint ("DISTANT", facet, NULL, NULL, NULL); } } } qh RANDOMdist= qh old_randomdist; } if (qh IStracing >= 3) { fprintf (qh ferr, "qh_setfacetplane: f%d offset %2.2g normal: ", facet->id, facet->offset); for (k=0; k < qh hull_dim; k++) fprintf (qh ferr, "%2.2g ", facet->normal[k]); fprintf (qh ferr, "\n"); } if (facet == qh tracefacet) qh IStracing= oldtrace; } /* setfacetplane */ /*--------------------------------- qh_sethyperplane_det( dim, rows, point0, toporient, normal, offset, nearzero ) given dim X dim array indexed by rows[], one row per point, toporient (flips all signs), and point0 (any row) set normalized hyperplane equation from oriented simplex returns: normal (normalized) offset (places point0 on the hyperplane) sets nearzero if hyperplane not through points notes: only defined for dim == 2..4 rows[] is not modified solves det(P-V_0, V_n-V_0, ..., V_1-V_0)=0, i.e. every point is on hyperplane see Bower & Woodworth, A programmer's geometry, Butterworths 1983. derivation of 3-d minnorm Goal: all vertices V_i within qh.one_merge of hyperplane Plan: exactly translate the facet so that V_0 is the origin exactly rotate the facet so that V_1 is on the x-axis and y_2=0. exactly rotate the effective perturbation to only effect n_0 this introduces a factor of sqrt(3) n_0 = ((y_2-y_0)*(z_1-z_0) - (z_2-z_0)*(y_1-y_0)) / norm Let M_d be the max coordinate difference Let M_a be the greater of M_d and the max abs. coordinate Let u be machine roundoff and distround be max error for distance computation The max error for n_0 is sqrt(3) u M_a M_d / norm. n_1 is approx. 1 and n_2 is approx. 0 The max error for distance of V_1 is sqrt(3) u M_a M_d M_d / norm. Offset=0 at origin Then minnorm = 1.8 u M_a M_d M_d / qh.ONEmerge Note that qh.one_merge is approx. 45.5 u M_a and norm is usually about M_d M_d derivation of 4-d minnorm same as above except rotate the facet so that V_1 on x-axis and w_2, y_3, w_3=0 [if two vertices fixed on x-axis, can rotate the other two in yzw.] n_0 = det3_(...) = y_2 det2_(z_1, w_1, z_3, w_3) = - y_2 w_1 z_3 [all other terms contain at least two factors nearly zero.] The max error for n_0 is sqrt(4) u M_a M_d M_d / norm Then minnorm = 2 u M_a M_d M_d M_d / qh.ONEmerge Note that qh.one_merge is approx. 82 u M_a and norm is usually about M_d M_d M_d */ void qh_sethyperplane_det (int dim, coordT **rows, coordT *point0, boolT toporient, coordT *normal, realT *offset, boolT *nearzero) { realT maxround, dist; int i; pointT *point; if (dim == 2) { normal[0]= dY(1,0); normal[1]= dX(0,1); qh_normalize2 (normal, dim, toporient, NULL, NULL); *offset= -(point0[0]*normal[0]+point0[1]*normal[1]); *nearzero= False; /* since nearzero norm => incident points */ }else if (dim == 3) { normal[0]= det2_(dY(2,0), dZ(2,0), dY(1,0), dZ(1,0)); normal[1]= det2_(dX(1,0), dZ(1,0), dX(2,0), dZ(2,0)); normal[2]= det2_(dX(2,0), dY(2,0), dX(1,0), dY(1,0)); qh_normalize2 (normal, dim, toporient, NULL, NULL); *offset= -(point0[0]*normal[0] + point0[1]*normal[1] + point0[2]*normal[2]); maxround= qh DISTround; for (i=dim; i--; ) { point= rows[i]; if (point != point0) { dist= *offset + (point[0]*normal[0] + point[1]*normal[1] + point[2]*normal[2]); if (dist > maxround || dist < -maxround) { *nearzero= True; break; } } } }else if (dim == 4) { normal[0]= - det3_(dY(2,0), dZ(2,0), dW(2,0), dY(1,0), dZ(1,0), dW(1,0), dY(3,0), dZ(3,0), dW(3,0)); normal[1]= det3_(dX(2,0), dZ(2,0), dW(2,0), dX(1,0), dZ(1,0), dW(1,0), dX(3,0), dZ(3,0), dW(3,0)); normal[2]= - det3_(dX(2,0), dY(2,0), dW(2,0), dX(1,0), dY(1,0), dW(1,0), dX(3,0), dY(3,0), dW(3,0)); normal[3]= det3_(dX(2,0), dY(2,0), dZ(2,0), dX(1,0), dY(1,0), dZ(1,0), dX(3,0), dY(3,0), dZ(3,0)); qh_normalize2 (normal, dim, toporient, NULL, NULL); *offset= -(point0[0]*normal[0] + point0[1]*normal[1] + point0[2]*normal[2] + point0[3]*normal[3]); maxround= qh DISTround; for (i=dim; i--; ) { point= rows[i]; if (point != point0) { dist= *offset + (point[0]*normal[0] + point[1]*normal[1] + point[2]*normal[2] + point[3]*normal[3]); if (dist > maxround || dist < -maxround) { *nearzero= True; break; } } } } if (*nearzero) { zzinc_(Zminnorm); trace0((qh ferr, "qh_sethyperplane_det: degenerate norm during p%d.\n", qh furthest_id)); zzinc_(Znearlysingular); } } /* sethyperplane_det */ /*--------------------------------- qh_sethyperplane_gauss( dim, rows, point0, toporient, normal, offset, nearzero ) given (dim-1) X dim array of rows[i]= V_{i+1} - V_0 (point0) set normalized hyperplane equation from oriented simplex returns: normal (normalized) offset (places point0 on the hyperplane) notes: if nearzero orientation may be incorrect because of incorrect sign flips in gausselim solves [V_n-V_0,...,V_1-V_0, 0 .. 0 1] * N == [0 .. 0 1] or [V_n-V_0,...,V_1-V_0, 0 .. 0 1] * N == [0] i.e., N is normal to the hyperplane, and the unnormalized distance to [0 .. 1] is either 1 or 0 design: perform gaussian elimination flip sign for negative values perform back substitution normalize result compute offset */ void qh_sethyperplane_gauss (int dim, coordT **rows, pointT *point0, boolT toporient, coordT *normal, coordT *offset, boolT *nearzero) { coordT *pointcoord, *normalcoef; int k; boolT sign= toporient, nearzero2= False; qh_gausselim(rows, dim-1, dim, &sign, nearzero); for(k= dim-1; k--; ) { if ((rows[k])[k] < 0) sign ^= 1; } if (*nearzero) { zzinc_(Znearlysingular); trace0((qh ferr, "qh_sethyperplane_gauss: nearly singular or axis parallel hyperplane during p%d.\n", qh furthest_id)); qh_backnormal(rows, dim-1, dim, sign, normal, &nearzero2); }else { qh_backnormal(rows, dim-1, dim, sign, normal, &nearzero2); if (nearzero2) { zzinc_(Znearlysingular); trace0((qh ferr, "qh_sethyperplane_gauss: singular or axis parallel hyperplane at normalization during p%d.\n", qh furthest_id)); } } if (nearzero2) *nearzero= True; qh_normalize2(normal, dim, True, NULL, NULL); pointcoord= point0; normalcoef= normal; *offset= -(*pointcoord++ * *normalcoef++); for(k= dim-1; k--; ) *offset -= *pointcoord++ * *normalcoef++; } /* sethyperplane_gauss */