/* ---------------------------------
geom.c
geometric routines of qhull
see qh-geom.htm and geom.h
copyright (c) 1993-2001 The Geometry Center
infrequent code goes into geom2.c
*/
#include "qhull_a.h"
/*---------------------------------
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_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, newfacts, dist, isoutside, numpart )
find facet that is furthest below a point
searches neighbors of coplanar and most flipped facets
for upperDelaunay facets
searches if coplanar (within searchdist)
returns facet only if !noupper and clearly above
input:
starts search at 'startfacet' (can not be flipped)
if !bestoutside, stops at qh.MINoutside (DISTround if precise)
returns:
best facet
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:
uses qh.visit_id, qh.searchset
caller traces the results
after qh_distplane, this and qh_partitionpoint are the most expensive in 3-d
avoid calls to distplane, function calls and real number operations.
when called by qh_partitionvisible():
indicated by newfacets and isoutside defined
qh.newfacet_list is list of simplicial, new facets
qh_findbestnew set if qh_findbestsharp returns True (to use qh_findbestnew)
qh.bestfacet_notsharp set if qh_findbestsharp returns False
searches horizon of best facet unless "exact" and !bestoutside
searchdist is 2 * DISTround
when called by qh_check_maxout()
indicated by bestoutside and !newfacets and isoutside == NULL
startfacet must be closest to 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
when called by findfacet() and check_bestdist()
indicated by !newfacets and isoutside defined
searchdist is same as qh_check_maxout()
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 facets to search (searchset)
for each neighbor of facet
exit if outside facet found
restart searchset if neighbor is much better
append flipped and nearby facets to searchset
if point is inside and partitioning
test for new facets with a "sharp" intersection
if so, future calls go to qh_findbestsharp
if testhorizon
also test facet's horizon facet
*/
facetT *qh_findbest (pointT *point, facetT *startfacet,
boolT bestoutside, boolT newfacets, boolT noupper,
realT *dist, boolT *isoutside, int *numpart) {
realT bestdist= -REALmax/2 /* avoid underflow */, searchdist;
realT cutoff, mincutoff; /* skip facets that are too far from point */
facetT *facet, *neighbor, **neighborp, *bestfacet= NULL;
int oldtrace= qh IStracing;
int searchsize= 0; /* non-zero if searchset defined */
boolT newbest;
boolT ischeckmax= bestoutside && !newfacets && !isoutside;
boolT ispartition= newfacets && isoutside;
boolT isfindfacet= !newfacets && isoutside;
boolT testhorizon = ispartition && (bestoutside || qh APPROXhull || qh MERGING);
if (!ischeckmax && !ispartition && !isfindfacet) {
fprintf (qh ferr, "qhull internal error (qh_findbest): unknown combination of arguments\n");
qh_errexit (qh_ERRqhull, startfacet, NULL);
}
if (qh TRACElevel && qh TRACEpoint >= 0 && qh TRACEpoint == qh_pointid (point)) {
qh IStracing= qh TRACElevel;
fprintf (qh ferr, "qh_findbest: point p%d starting at f%d bestoutside? %d newfacets %d\n",
qh TRACEpoint, startfacet->id, bestoutside, newfacets);
fprintf(qh ferr, " ischeckmax %d ispartition %d isfindfacet %d testhorizon %d\n",
ischeckmax, ispartition, isfindfacet, testhorizon);
fprintf (qh ferr, " Last point added to hull was p%d.", qh furthest_id);
fprintf(qh ferr, " Last merge was #%d.\n", zzval_(Ztotmerge));
}
if (isoutside)
*isoutside= True;
if (!startfacet->flipped) {
*numpart= 1;
qh_distplane (point, startfacet, dist); /* this code is duplicated below */
if (!startfacet->upperdelaunay || (!noupper && *dist >= qh MINoutside)) {
bestdist= *dist;
bestfacet= startfacet;
if (!bestoutside && *dist >= qh MINoutside)
goto LABELreturn_best;
}
#if qh_MAXoutside
if (ischeckmax && (!qh ONLYgood || startfacet->good) && *dist > startfacet->maxoutside)
startfacet->maxoutside= *dist;
#endif
}
if (ispartition)
searchdist= 2 * qh DISTround;
else
searchdist= qh max_outside + 2 * qh DISTround
+ fmax_( qh MINvisible, qh MAXcoplanar);
cutoff= bestdist - searchdist;
mincutoff= 0;
if (ischeckmax) {
mincutoff= -(qh DISTround - fmax_(qh MINvisible, qh MAXcoplanar));
minimize_(cutoff, mincutoff);
}
startfacet->visitid= ++qh visit_id;
facet= startfacet;
do { /* search neighbors of coplanar, upperdelaunay, and flipped facets */
if (True) {
LABELrestart: /* directed search whenever improvement > searchdist */
newbest= False;
trace4((qh ferr, "qh_findbest: neighbors of f%d, bestdist %2.2g cutoff %2.2g searchdist %2.2g\n",
facet->id, bestdist, cutoff, searchdist));
FOREACHneighbor_(facet) {
if (ispartition && !neighbor->newfacet)
continue;
if (!neighbor->flipped) {
if (neighbor->visitid == qh visit_id)
continue;
neighbor->visitid= qh visit_id;
(*numpart)++;
qh_distplane (point, neighbor, dist);
if (!bestoutside && *dist >= qh MINoutside
&& (!noupper || !facet->upperdelaunay)) {
bestfacet= neighbor;
goto LABELreturn_best;
}
#if qh_MAXoutside
if (ischeckmax) {
if ((!qh ONLYgood || neighbor->good)
&& *dist > neighbor->maxoutside)
neighbor->maxoutside= *dist;
else if (bestfacet && *dist < cutoff)
continue;
}else
#endif /* dangling else! */
if (bestfacet && *dist < cutoff)
continue;
if (*dist > bestdist) {
if (!neighbor->upperdelaunay
|| (bestoutside && !noupper && *dist >= qh MINoutside)) {
if (ischeckmax && qh_MAXoutside) {
bestdist= *dist;
bestfacet= neighbor;
cutoff= bestdist - searchdist;
minimize_(cutoff, mincutoff);
}else if (*dist > bestdist + searchdist) {
bestdist= *dist;
bestfacet= neighbor;
cutoff= bestdist - searchdist;
searchsize= 0;
facet= neighbor;
if (newbest) /* newbest may be coplanar with facet */
facet->visitid= ++qh visit_id;
goto LABELrestart; /* repeat with a new facet */
}else {
bestdist= *dist;
bestfacet= neighbor;
cutoff= bestdist - searchdist;
}
newbest= True;
}
}
}
if (!searchsize++) {
SETfirst_(qh searchset) = neighbor;
qh_settruncate (qh searchset, 1);
}else
qh_setappend (&qh searchset, neighbor);
} /* FOREACHneighbor */
} /* while (restart) */
}while
(searchsize && (facet= (facetT*)qh_setdellast (qh searchset)));
if (!ischeckmax) {
if (!bestfacet) {
fprintf (qh ferr, "qh_findbest: point p%d starting at f%d bestoutside? %d newfacets %d\n",
qh TRACEpoint, startfacet->id, bestoutside, newfacets);
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 FORCEoutput= True;
qh_errexit (qh_ERRqhull, startfacet, NULL);
}
if (ispartition && !qh findbest_notsharp && bestdist < - qh DISTround) {
if (qh_findbestsharp ( point, &bestfacet, &bestdist, numpart))
qh findbestnew= True;
else
qh findbest_notsharp= True;
}
if (testhorizon) {
facet= SETfirstt_(bestfacet->neighbors, facetT);
trace4((qh ferr, "qh_findbest: horizon facet f%d\n", facet->id));
(*numpart)++;
qh_distplane (point, facet, dist);
if (*dist > bestdist
&& (!facet->upperdelaunay || (!noupper && *dist >= qh MINoutside))) {
bestdist= *dist;
bestfacet= facet;
}
}
}
*dist= bestdist;
if (isoutside && bestdist < qh MINoutside)
*isoutside= False;
LABELreturn_best:
qh IStracing= oldtrace;
return bestfacet;
} /* findbest */
/*---------------------------------
qh_findbestnew( point, startfacet, dist, isoutside, numpart )
find best newfacet for point
searches new facets starting at startfacet
returns:
dist is distance to facet
isoutside is true if point is outside of facet
numpart is number of distance tests
notes:
if qh.BESToutside or !isoutside
stops at furthest facet
if qh.MERGING
stops when distance > qh_DISToutside (max(4*MINoutside, 2*max_outside))
else
stops when distance > MINoutside (DISTround in precise case)
searches newfacets then searchs neighbors of best facet.
avoids upperdelaunay facet unless none other or (isoutside and outside)
uses visit_id and seen flags
caller traces the results
see also:
qh_partitionall() and qh_findbest()
design:
for each new facet
test distance from point to facet
select best facet
test each horizon facet of best facet
return best facet
*/
facetT *qh_findbestnew (pointT *point, facetT *startfacet,
realT *dist, boolT *isoutside, int *numpart) {
realT bestdist= -REALmax, bestdist2= -REALmax;
facetT *neighbor, **neighborp, *bestfacet= NULL, *newfacet, *facet;
facetT *bestfacet2= NULL;
int oldtrace= qh IStracing, i;
realT distoutside;
if (!startfacet) {
if (qh MERGING)
fprintf(qh ferr, "qhull precision error (qh_findbestnew): merging has formed and deleted an independent cycle of 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);
}
if (qh BESToutside || !isoutside)
distoutside= REALmax;
else if (qh MERGING)
distoutside= qh_DISToutside; /* defined in user.h */
else
distoutside= qh MINoutside;
if (qh TRACElevel && qh TRACEpoint >= 0 && qh TRACEpoint == qh_pointid (point)) {
qh IStracing= qh TRACElevel;
fprintf(qh ferr, "qh_findbestnew: point p%d facet f%d. Stop if dist > %2.2g\n",
qh TRACEpoint, startfacet->id, distoutside);
fprintf(qh ferr, " Last point added to hull was p%d.", qh furthest_id);
fprintf(qh ferr, " Last merge was #%d.\n", zzval_(Ztotmerge));
}
if (isoutside)
*isoutside= True;
*numpart= 0;
/* visit all new facets starting with startfacet */
for (i= 0, facet= startfacet; i < 2; i++, facet= qh newfacet_list) {
FORALLfacet_(facet) {
if (facet == startfacet && i)
break;
qh_distplane (point, facet, dist);
(*numpart)++;
if (facet->upperdelaunay) {
if (*dist > bestdist2) {
bestdist2= *dist;
bestfacet2= facet;
if (*dist >= distoutside) {
bestfacet= facet;
goto LABELreturn_bestnew;
}
}
}else if (*dist > bestdist) {
bestdist= *dist;
bestfacet= facet;
if (*dist >= distoutside)
goto LABELreturn_bestnew;
}
}
}
newfacet= bestfacet ? bestfacet : bestfacet2;
/* !bestfacet only occurs if 'd' creates incorrect upper-delaunay facets */
FOREACHneighbor_(newfacet) {
if (!neighbor->newfacet) {
qh_distplane (point, neighbor, dist);
(*numpart)++;
if (neighbor->upperdelaunay) {
if (*dist > bestdist2) {
bestdist2= *dist;
bestfacet2= neighbor;
}
}else if (*dist > bestdist) {
bestdist= *dist;
bestfacet= neighbor;
}
}
}
if (!bestfacet
|| (isoutside && bestdist2 >= qh MINoutside && bestdist2 > bestdist)) {
*dist= bestdist2;
bestfacet= bestfacet2;
}else
*dist= bestdist;
if (isoutside && *dist < qh MINoutside)
*isoutside= False;
LABELreturn_bestnew:
qh IStracing= oldtrace;
return bestfacet;
} /* findbestnew */
/*---------------------------------
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, DIM3 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.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 */