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|
/*<html><pre> -<a href="qh-geom_r.htm"
>-------------------------------</a><a name="TOP">-</a>
geom_r.c
geometric routines of qhull
see qh-geom_r.htm and geom_r.h
Copyright (c) 1993-2020 The Geometry Center.
$Id: //main/2019/qhull/src/libqhull_r/geom_r.c#5 $$Change: 2953 $
$DateTime: 2020/05/21 22:05:32 $$Author: bbarber $
infrequent code goes into geom2_r.c
*/
#include "qhull_ra.h"
/*-<a href="qh-geom_r.htm#TOC"
>-------------------------------</a><a name="distplane">-</a>
qh_distplane(qh, 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 qh_sortfacets, qh_outerinner)
for nearly coplanar points, the returned values may be duplicates
for example pairs of nearly incident points, rbox 175 C1,2e-13 t1538759579 | qhull d T4
622 qh_distplane: e-014 # count of two or more duplicate values for unique calls
258 qh_distplane: e-015
38 qh_distplane: e-016
40 qh_distplane: e-017
6 qh_distplane: e-018
5 qh_distplane: -e-018
33 qh_distplane: -e-017
3153 qh_distplane: -2.775557561562891e-017 # duplicated value for 3153 unique calls
42 qh_distplane: -e-016
307 qh_distplane: -e-015
1271 qh_distplane: -e-014
13 qh_distplane: -e-013
see:
qh_distnorm in geom2_r.c
qh_distplane [geom_r.c], QhullFacet::distance, and QhullHyperplane::distance are copies
*/
void qh_distplane(qhT *qh, 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;
}
zzinc_(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;
}
#ifndef qh_NOtrace
if (qh->IStracing >= 4) {
qh_fprintf(qh, qh->ferr, 8001, "qh_distplane: ");
qh_fprintf(qh, qh->ferr, 8002, qh_REAL_1, *dist);
qh_fprintf(qh, qh->ferr, 8003, "from p%d to f%d\n", qh_pointid(qh, point), facet->id);
}
#endif
return;
} /* distplane */
/*-<a href="qh-geom_r.htm#TOC"
>-------------------------------</a><a name="findbest">-</a>
qh_findbest(qh, 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(qhT *qh, 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;
facetT *bestfacet= NULL, *lastfacet= NULL;
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);
#ifndef qh_NOtrace
if (qh->IStracing >= 4 || (qh->TRACElevel && qh->TRACEpoint >= 0 && qh->TRACEpoint == qh_pointid(qh, point))) {
if (qh->TRACElevel > qh->IStracing)
qh->IStracing= qh->TRACElevel;
qh_fprintf(qh, qh->ferr, 8004, "qh_findbest: point p%d starting at f%d isnewfacets? %d, unless %d exit if > %2.2g,",
qh_pointid(qh, point), startfacet->id, isnewfacets, bestoutside, qh->MINoutside);
qh_fprintf(qh, qh->ferr, 8005, " testhorizon? %d, noupper? %d,", testhorizon, noupper);
qh_fprintf(qh, qh->ferr, 8006, " Last qh_addpoint p%d,", qh->furthest_id);
qh_fprintf(qh, qh->ferr, 8007, " Last merge #%d, max_outside %2.2g\n", zzval_(Ztotmerge), qh->max_outside);
}
#endif
if (isoutside)
*isoutside= True;
if (!startfacet->flipped) { /* test startfacet before testing its neighbors */
*numpart= 1;
qh_distplane(qh, 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, qh->ferr, 4001, "qh_findbest: neighbors of f%d, bestdist %2.2g f%d\n",
facet->id, bestdist, getid_(bestfacet)));
lastfacet= facet;
FOREACHneighbor_(facet) {
if (!neighbor->newfacet && isnewfacets)
continue;
if (neighbor->visitid == visitid)
continue;
neighbor->visitid= visitid;
if (!neighbor->flipped) { /* code duplicated above */
(*numpart)++;
qh_distplane(qh, 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 neighbor */
}else if (!bestfacet) {
bestdist= *dist;
break; /* switch to neighbor */
}
} /* 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) { /* startfacet is upperdelaunay (or flipped) w/o !flipped newfacet neighbors */
bestdist= -REALmax/2;
bestfacet= qh_findbestnew(qh, point, qh->newfacet_list, &bestdist, bestoutside, isoutside, &numpartnew);
testhorizon= False; /* qh_findbestnew calls qh_findbesthorizon */
}else if (!qh->findbest_notsharp && bestdist < -qh->DISTround) {
if (qh_sharpnewfacets(qh)) {
/* seldom used, qh_findbestnew will retest all facets */
zinc_(Zfindnewsharp);
bestfacet= qh_findbestnew(qh, point, bestfacet, &bestdist, bestoutside, isoutside, &numpartnew);
testhorizon= False; /* qh_findbestnew calls qh_findbesthorizon */
qh->findbestnew= True;
}else
qh->findbest_notsharp= True;
}
}
if (!bestfacet)
bestfacet= qh_findbestlower(qh, lastfacet, point, &bestdist, numpart); /* lastfacet is non-NULL because startfacet is non-NULL */
if (testhorizon) /* qh_findbestnew not called */
bestfacet= qh_findbesthorizon(qh, !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 */
/*-<a href="qh-geom_r.htm#TOC"
>-------------------------------</a><a name="findbesthorizon">-</a>
qh_findbesthorizon(qh, 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, including *bestdist and *numpart
qh.coplanarfacetset used to maintain current search set, reset whenever best facet is substantially better
returns(ischeckmax):
best facet
updates f.maxoutside for neighbors of searched facets (if qh_MAXoutside)
returns(!ischeckmax):
best facet that is not upperdelaunay or newfacet (qh.first_newfacet)
allows upperdelaunay that is clearly outside
returns:
bestdist is distance to bestfacet
numpart -- updates number of distance tests
notes:
called by qh_findbest if point is not outside a facet (directed search)
called by qh_findbestnew if point is not outside a new facet
called by qh_check_maxout for each point in hull
called by qh_check_bestdist for each point in hull (rarely used)
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(qhT *qh, 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 coplanarfacetset */
int numpartinit= *numpart, coplanarfacetset_size, numcoplanar= 0, numfacet= 0;
unsigned int visitid= ++qh->visit_id;
boolT newbest= False; /* for tracing */
realT minsearch, searchdist; /* skip facets that are too far from point */
boolT is_5x_minsearch;
if (!ischeckmax) {
zinc_(Zfindhorizon);
}else {
#if qh_MAXoutside
if ((!qh->ONLYgood || startfacet->good) && *bestdist > startfacet->maxoutside)
startfacet->maxoutside= *bestdist;
#endif
}
searchdist= qh_SEARCHdist; /* an expression, a 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);
}
coplanarfacetset_size= 0;
startfacet->visitid= visitid;
facet= startfacet;
while (True) {
numfacet++;
is_5x_minsearch= (ischeckmax && facet->nummerge > 10 && qh_setsize(qh, facet->neighbors) > 100); /* QH11033 FIX: qh_findbesthorizon: many tests for facets with many merges and neighbors. Can hide coplanar facets, e.g., 'rbox 1000 s Z1 G1e-13' with 4400+ neighbors */
trace4((qh, qh->ferr, 4002, "qh_findbesthorizon: test neighbors of f%d bestdist %2.2g f%d ischeckmax? %d noupper? %d minsearch %2.2g is_5x? %d searchdist %2.2g\n",
facet->id, *bestdist, getid_(bestfacet), ischeckmax, noupper,
minsearch, is_5x_minsearch, searchdist));
FOREACHneighbor_(facet) {
if (neighbor->visitid == visitid)
continue;
neighbor->visitid= visitid;
if (!neighbor->flipped) { /* neighbors of flipped facets always searched via nextfacet */
qh_distplane(qh, point, neighbor, &dist); /* duplicate qh_distpane for new facets, they may be coplanar */
(*numpart)++;
if (dist > *bestdist) {
if (!neighbor->upperdelaunay || ischeckmax || (!noupper && dist >= qh->MINoutside)) {
if (!ischeckmax) {
minsearch= dist - searchdist;
if (dist > *bestdist + searchdist) {
zinc_(Zfindjump); /* everything in qh.coplanarfacetset at least searchdist below */
coplanarfacetset_size= 0;
}
}
bestfacet= neighbor;
*bestdist= dist;
newbest= True;
}
}else if (is_5x_minsearch) {
if (dist < 5 * minsearch)
continue; /* skip this neighbor, do not set nextfacet. dist is negative */
}else if (dist < minsearch)
continue; /* skip this neighbor, do not set nextfacet. If ischeckmax, dist can't be positive */
#if qh_MAXoutside
if (ischeckmax && dist > neighbor->maxoutside)
neighbor->maxoutside= dist;
#endif
} /* end of !flipped, need to search neighbor */
if (nextfacet) {
numcoplanar++;
if (!coplanarfacetset_size++) {
SETfirst_(qh->coplanarfacetset)= nextfacet;
SETtruncate_(qh->coplanarfacetset, 1);
}else
qh_setappend(qh, &qh->coplanarfacetset, 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 (!coplanarfacetset_size)
break;
else if (!--coplanarfacetset_size) {
facet= SETfirstt_(qh->coplanarfacetset, facetT);
SETtruncate_(qh->coplanarfacetset, 0);
}else
facet= (facetT *)qh_setdellast(qh->coplanarfacetset);
} /* while True, i.e., "for each facet in qh.coplanarfacetset" */
if (!ischeckmax) {
zadd_(Zfindhorizontot, *numpart - numpartinit);
zmax_(Zfindhorizonmax, *numpart - numpartinit);
if (newbest)
zinc_(Znewbesthorizon);
}
trace4((qh, qh->ferr, 4003, "qh_findbesthorizon: p%d, newbest? %d, bestfacet f%d, bestdist %2.2g, numfacet %d, coplanarfacets %d, numdist %d\n",
qh_pointid(qh, point), newbest, getid_(bestfacet), *bestdist, numfacet, numcoplanar, *numpart - numpartinit));
return bestfacet;
} /* findbesthorizon */
/*-<a href="qh-geom_r.htm#TOC"
>-------------------------------</a><a name="findbestnew">-</a>
qh_findbestnew(qh, 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.coplanarfacetset.
If share visit_id with qh_findbest, coplanarfacetset 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(qhT *qh, pointT *point, facetT *startfacet,
realT *dist, boolT bestoutside, boolT *isoutside, int *numpart) {
realT bestdist= -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 || !startfacet->next) {
if (qh->MERGING) {
qh_fprintf(qh, qh->ferr, 6001, "qhull topology error (qh_findbestnew): merging has formed and deleted a cone of new facets. Can not continue.\n");
qh_errexit(qh, qh_ERRtopology, NULL, NULL);
}else {
qh_fprintf(qh, qh->ferr, 6002, "qhull internal error (qh_findbestnew): no new facets for point p%d\n",
qh->furthest_id);
qh_errexit(qh, 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_r.h */
}
if (isoutside)
*isoutside= True;
*numpart= 0;
#ifndef qh_NOtrace
if (qh->IStracing >= 4 || (qh->TRACElevel && qh->TRACEpoint >= 0 && qh->TRACEpoint == qh_pointid(qh, point))) {
if (qh->TRACElevel > qh->IStracing)
qh->IStracing= qh->TRACElevel;
qh_fprintf(qh, qh->ferr, 8008, "qh_findbestnew: point p%d facet f%d. Stop? %d if dist > %2.2g,",
qh_pointid(qh, point), startfacet->id, isdistoutside, distoutside);
qh_fprintf(qh, qh->ferr, 8009, " Last qh_addpoint p%d, qh.visit_id %d, vertex_visit %d,", qh->furthest_id, visitid, qh->vertex_visit);
qh_fprintf(qh, qh->ferr, 8010, " Last merge #%d\n", zzval_(Ztotmerge));
}
#endif
/* 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(qh, 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) /* testhorizon is always True. Keep the same code as qh_findbest */
bestfacet= qh_findbesthorizon(qh, !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, qh->ferr, 4004, "qh_findbestnew: bestfacet f%d bestdist %2.2g for p%d f%d bestoutside? %d \n",
getid_(bestfacet), *dist, qh_pointid(qh, point), startfacet->id, bestoutside));
qh->IStracing= oldtrace;
return bestfacet;
} /* findbestnew */
/* ============ hyperplane functions -- keep code together [?] ============ */
/*-<a href="qh-geom_r.htm#TOC"
>-------------------------------</a><a name="backnormal">-</a>
qh_backnormal(qh, 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, 1983, Eq. 4.4-9 for "Gaussian elimination with complete pivoting"
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(qhT *qh, 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) {
*nearzero= True;
trace4((qh, qh->ferr, 4005, "qh_backnormal: zero diagonal at column %d.\n", i));
zzinc_(Zback0);
qh_joggle_restart(qh, "zero diagonal on back substitution");
}
} /* backnormal */
/*-<a href="qh-geom_r.htm#TOC"
>-------------------------------</a><a name="gausselim">-</a>
qh_gausselim(qh, 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(qhT *qh, 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 */
#ifndef qh_NOtrace
if (qh->IStracing >= 4) {
qh_fprintf(qh, qh->ferr, 8011, "qh_gausselim: 0 pivot at column %d. (%2.2g < %2.2g)\n", k, pivot_abs, qh->DISTround);
qh_printmatrix(qh, qh->ferr, "Matrix:", rows, numrow, numcol);
}
#endif
zzinc_(Zgauss0);
qh_joggle_restart(qh, "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, qh->ferr, "qh_gausselem: result", rows, numrow, numcol);
} /* gausselim */
/*-<a href="qh-geom_r.htm#TOC"
>-------------------------------</a><a name="getangle">-</a>
qh_getangle(qh, 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(qhT *qh, 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, qh->ferr, 4006, "qh_getangle: %4.4g\n", angle));
return(angle);
} /* getangle */
/*-<a href="qh-geom_r.htm#TOC"
>-------------------------------</a><a name="getcenter">-</a>
qh_getcenter(qh, vertices )
returns arithmetic center of a set of vertices as a new point
notes:
allocates point array for center
*/
pointT *qh_getcenter(qhT *qh, setT *vertices) {
int k;
pointT *center, *coord;
vertexT *vertex, **vertexp;
int count= qh_setsize(qh, vertices);
if (count < 2) {
qh_fprintf(qh, qh->ferr, 6003, "qhull internal error (qh_getcenter): not defined for %d points\n", count);
qh_errexit(qh, qh_ERRqhull, NULL, NULL);
}
center= (pointT *)qh_memalloc(qh, 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; /* count>=2 by QH6003 */
}
return(center);
} /* getcenter */
/*-<a href="qh-geom_r.htm#TOC"
>-------------------------------</a><a name="getcentrum">-</a>
qh_getcentrum(qh, facet )
returns the centrum for a facet as a new point
notes:
allocates the centrum
*/
pointT *qh_getcentrum(qhT *qh, facetT *facet) {
realT dist;
pointT *centrum, *point;
point= qh_getcenter(qh, facet->vertices);
zzinc_(Zcentrumtests);
qh_distplane(qh, point, facet, &dist);
centrum= qh_projectpoint(qh, point, facet, dist);
qh_memfree(qh, point, qh->normal_size);
trace4((qh, qh->ferr, 4007, "qh_getcentrum: for f%d, %d vertices dist= %2.2g\n",
facet->id, qh_setsize(qh, facet->vertices), dist));
return centrum;
} /* getcentrum */
/*-<a href="qh-geom_r.htm#TOC"
>-------------------------------</a><a name="getdistance">-</a>
qh_getdistance(qh, facet, neighbor, mindist, maxdist )
returns the min and max distance to neighbor of non-neighbor vertices in facet
returns:
the max absolute value
design:
for each vertex of facet that is not in neighbor
test the distance from vertex to neighbor
*/
coordT qh_getdistance(qhT *qh, facetT *facet, facetT *neighbor, coordT *mindist, coordT *maxdist) {
vertexT *vertex, **vertexp;
coordT 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(qh, 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 */
/*-<a href="qh-geom_r.htm#TOC"
>-------------------------------</a><a name="normalize">-</a>
qh_normalize(qh, normal, dim, toporient )
normalize a vector and report if too small
does not use min norm
see:
qh_normalize2
*/
void qh_normalize(qhT *qh, coordT *normal, int dim, boolT toporient) {
qh_normalize2(qh, normal, dim, toporient, NULL, NULL);
} /* normalize */
/*-<a href="qh-geom_r.htm#TOC"
>-------------------------------</a><a name="normalize2">-</a>
qh_normalize2(qh, 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(qhT *qh, 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);
/* qh_joggle_restart ignored for Znearlysingular, normal part of qh_sethyperplane_gauss */
trace0((qh, qh->ferr, 1, "qh_normalize: norm=%2.2g too small during p%d\n",
norm, qh->furthest_id));
return;
}
}
}
} /* normalize */
/*-<a href="qh-geom_r.htm#TOC"
>-------------------------------</a><a name="projectpoint">-</a>
qh_projectpoint(qh, 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(qhT *qh, pointT *point, facetT *facet, realT dist) {
pointT *newpoint, *np, *normal;
int normsize= qh->normal_size;
int k;
void **freelistp; /* used if !qh_NOmem by qh_memalloc_() */
qh_memalloc_(qh, normsize, freelistp, newpoint, pointT);
np= newpoint;
normal= facet->normal;
for (k=qh->hull_dim; k--; )
*(np++)= *point++ - dist * *normal++;
return(newpoint);
} /* projectpoint */
/*-<a href="qh-geom_r.htm#TOC"
>-------------------------------</a><a name="setfacetplane">-</a>
qh_setfacetplane(qh, 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(qhT *qh, facetT *facet) {
pointT *point;
vertexT *vertex, **vertexp;
int normsize= qh->normal_size;
int k,i, oldtrace= 0;
realT dist;
void **freelistp; /* used if !qh_NOmem by qh_memalloc_() */
coordT *coord, *gmcoord;
pointT *point0= SETfirstt_(facet->vertices, vertexT)->point;
boolT nearzero= False;
zzinc_(Zsetplane);
if (!facet->normal)
qh_memalloc_(qh, normsize, freelistp, facet->normal, coordT);
#ifndef qh_NOtrace
if (facet == qh->tracefacet) {
oldtrace= qh->IStracing;
qh->IStracing= 5;
qh_fprintf(qh, qh->ferr, 8012, "qh_setfacetplane: facet f%d created.\n", facet->id);
qh_fprintf(qh, qh->ferr, 8013, " Last point added to hull was p%d.", qh->furthest_id);
if (zzval_(Ztotmerge))
qh_fprintf(qh, qh->ferr, 8014, " Last merge was #%d.", zzval_(Ztotmerge));
qh_fprintf(qh, qh->ferr, 8015, "\n\nCurrent summary is:\n");
qh_printsummary(qh, qh->ferr);
}
#endif
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(qh, qh->RANDOMa, qh->RANDOMb);
}
}else {
FOREACHvertex_(facet->vertices)
qh->gm_row[i++]= vertex->point;
}
qh_sethyperplane_det(qh, 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, qh->RANDOMa, qh->RANDOMb);
}
}
qh_sethyperplane_gauss(qh, qh->hull_dim, qh->gm_row, point0, facet->toporient,
facet->normal, &facet->offset, &nearzero);
if (nearzero) {
if (qh_orientoutside(qh, facet)) {
trace0((qh, qh->ferr, 2, "qh_setfacetplane: flipped orientation due to nearzero gauss and interior_point test. 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(qh, 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(qh) */
if (dist > qh->TRACEdist)
istrace= True;
}
}else if (-dist > qh->TRACEdist)
istrace= True;
if (istrace) {
qh_fprintf(qh, qh->ferr, 3060, "qh_setfacetplane: ====== vertex p%d(v%d) increases max_outside to %2.2g for new facet f%d last p%d\n",
qh_pointid(qh, vertex->point), vertex->id, dist, facet->id, qh->furthest_id);
qh_errprint(qh, "DISTANT", facet, NULL, NULL, NULL);
}
}
}
qh->RANDOMdist= qh->old_randomdist;
}
#ifndef qh_NOtrace
if (qh->IStracing >= 4) {
qh_fprintf(qh, qh->ferr, 8017, "qh_setfacetplane: f%d offset %2.2g normal: ",
facet->id, facet->offset);
for (k=0; k < qh->hull_dim; k++)
qh_fprintf(qh, qh->ferr, 8018, "%2.2g ", facet->normal[k]);
qh_fprintf(qh, qh->ferr, 8019, "\n");
}
#endif
qh_checkflipped(qh, facet, NULL, qh_ALL);
if (facet == qh->tracefacet) {
qh->IStracing= oldtrace;
qh_printfacet(qh, qh->ferr, facet);
}
} /* setfacetplane */
/*-<a href="qh-geom_r.htm#TOC"
>-------------------------------</a><a name="sethyperplane_det">-</a>
qh_sethyperplane_det(qh, 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(qhT *qh, 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(qh, 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(qh, 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(qh, 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);
/* qh_joggle_restart not needed, will call qh_sethyperplane_gauss instead */
trace0((qh, qh->ferr, 3, "qh_sethyperplane_det: degenerate norm during p%d, use qh_sethyperplane_gauss instead.\n", qh->furthest_id));
}
} /* sethyperplane_det */
/*-<a href="qh-geom_r.htm#TOC"
>-------------------------------</a><a name="sethyperplane_gauss">-</a>
qh_sethyperplane_gauss(qh, 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(qhT *qh, 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(qh, rows, dim-1, dim, &sign, nearzero);
for (k=dim-1; k--; ) {
if ((rows[k])[k] < 0)
sign ^= 1;
}
if (*nearzero) {
zzinc_(Znearlysingular);
/* qh_joggle_restart ignored for Znearlysingular, normal part of qh_sethyperplane_gauss */
trace0((qh, qh->ferr, 4, "qh_sethyperplane_gauss: nearly singular or axis parallel hyperplane during p%d.\n", qh->furthest_id));
qh_backnormal(qh, rows, dim-1, dim, sign, normal, &nearzero2);
}else {
qh_backnormal(qh, rows, dim-1, dim, sign, normal, &nearzero2);
if (nearzero2) {
zzinc_(Znearlysingular);
trace0((qh, qh->ferr, 5, "qh_sethyperplane_gauss: singular or axis parallel hyperplane at normalization during p%d.\n", qh->furthest_id));
}
}
if (nearzero2)
*nearzero= True;
qh_normalize2(qh, normal, dim, True, NULL, NULL);
pointcoord= point0;
normalcoef= normal;
*offset= -(*pointcoord++ * *normalcoef++);
for (k=dim-1; k--; )
*offset -= *pointcoord++ * *normalcoef++;
} /* sethyperplane_gauss */
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