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geom2.c
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/*<html><pre> -<a href="qh-geom.htm"
>-------------------------------</a><a name="TOP">-</a>
geom2.c
infrequently used geometric routines of qhull
see qh-geom.htm and geom.h
Copyright (c) 1993-2015 The Geometry Center.
$Id: //main/2011/qhull/src/libqhull/geom2.c#5 $$Change: 1810 $
$DateTime: 2015/01/17 18:28:15 $$Author: bbarber $
frequently used code goes into geom.c
*/
#include "qhull_a.h"
/*================== functions in alphabetic order ============*/
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="copypoints">-</a>
qh_copypoints( points, numpoints, dimension)
return qh_malloc'd copy of points
*/
coordT *qh_copypoints(coordT *points, int numpoints, int dimension) {
int size;
coordT *newpoints;
size= numpoints * dimension * (int)sizeof(coordT);
if (!(newpoints=(coordT*)qh_malloc((size_t)size))) {
qh_fprintf(qh ferr, 6004, "qhull error: insufficient memory to copy %d points\n",
numpoints);
qh_errexit(qh_ERRmem, NULL, NULL);
}
memcpy((char *)newpoints, (char *)points, (size_t)size);
return newpoints;
} /* copypoints */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="crossproduct">-</a>
qh_crossproduct( dim, vecA, vecB, vecC )
crossproduct of 2 dim vectors
C= A x B
notes:
from Glasner, Graphics Gems I, p. 639
only defined for dim==3
*/
void qh_crossproduct(int dim, realT vecA[3], realT vecB[3], realT vecC[3]){
if (dim == 3) {
vecC[0]= det2_(vecA[1], vecA[2],
vecB[1], vecB[2]);
vecC[1]= - det2_(vecA[0], vecA[2],
vecB[0], vecB[2]);
vecC[2]= det2_(vecA[0], vecA[1],
vecB[0], vecB[1]);
}
} /* vcross */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="determinant">-</a>
qh_determinant( rows, dim, nearzero )
compute signed determinant of a square matrix
uses qh.NEARzero to test for degenerate matrices
returns:
determinant
overwrites rows and the matrix
if dim == 2 or 3
nearzero iff determinant < qh NEARzero[dim-1]
(!quite correct, not critical)
if dim >= 4
nearzero iff diagonal[k] < qh NEARzero[k]
*/
realT qh_determinant(realT **rows, int dim, boolT *nearzero) {
realT det=0;
int i;
boolT sign= False;
*nearzero= False;
if (dim < 2) {
qh_fprintf(qh ferr, 6005, "qhull internal error (qh_determinate): only implemented for dimension >= 2\n");
qh_errexit(qh_ERRqhull, NULL, NULL);
}else if (dim == 2) {
det= det2_(rows[0][0], rows[0][1],
rows[1][0], rows[1][1]);
if (fabs_(det) < qh NEARzero[1]) /* not really correct, what should this be? */
*nearzero= True;
}else if (dim == 3) {
det= det3_(rows[0][0], rows[0][1], rows[0][2],
rows[1][0], rows[1][1], rows[1][2],
rows[2][0], rows[2][1], rows[2][2]);
if (fabs_(det) < qh NEARzero[2]) /* not really correct, what should this be? */
*nearzero= True;
}else {
qh_gausselim(rows, dim, dim, &sign, nearzero); /* if nearzero, diagonal still ok*/
det= 1.0;
for (i=dim; i--; )
det *= (rows[i])[i];
if (sign)
det= -det;
}
return det;
} /* determinant */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="detjoggle">-</a>
qh_detjoggle( points, numpoints, dimension )
determine default max joggle for point array
as qh_distround * qh_JOGGLEdefault
returns:
initial value for JOGGLEmax from points and REALepsilon
notes:
computes DISTround since qh_maxmin not called yet
if qh SCALElast, last dimension will be scaled later to MAXwidth
loop duplicated from qh_maxmin
*/
realT qh_detjoggle(pointT *points, int numpoints, int dimension) {
realT abscoord, distround, joggle, maxcoord, mincoord;
pointT *point, *pointtemp;
realT maxabs= -REALmax;
realT sumabs= 0;
realT maxwidth= 0;
int k;
for (k=0; k < dimension; k++) {
if (qh SCALElast && k == dimension-1)
abscoord= maxwidth;
else if (qh DELAUNAY && k == dimension-1) /* will qh_setdelaunay() */
abscoord= 2 * maxabs * maxabs; /* may be low by qh hull_dim/2 */
else {
maxcoord= -REALmax;
mincoord= REALmax;
FORALLpoint_(points, numpoints) {
maximize_(maxcoord, point[k]);
minimize_(mincoord, point[k]);
}
maximize_(maxwidth, maxcoord-mincoord);
abscoord= fmax_(maxcoord, -mincoord);
}
sumabs += abscoord;
maximize_(maxabs, abscoord);
} /* for k */
distround= qh_distround(qh hull_dim, maxabs, sumabs);
joggle= distround * qh_JOGGLEdefault;
maximize_(joggle, REALepsilon * qh_JOGGLEdefault);
trace2((qh ferr, 2001, "qh_detjoggle: joggle=%2.2g maxwidth=%2.2g\n", joggle, maxwidth));
return joggle;
} /* detjoggle */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="detroundoff">-</a>
qh_detroundoff()
determine maximum roundoff errors from
REALepsilon, REALmax, REALmin, qh.hull_dim, qh.MAXabs_coord,
qh.MAXsumcoord, qh.MAXwidth, qh.MINdenom_1
accounts for qh.SETroundoff, qh.RANDOMdist, qh MERGEexact
qh.premerge_cos, qh.postmerge_cos, qh.premerge_centrum,
qh.postmerge_centrum, qh.MINoutside,
qh_RATIOnearinside, qh_COPLANARratio, qh_WIDEcoplanar
returns:
sets qh.DISTround, etc. (see below)
appends precision constants to qh.qhull_options
see:
qh_maxmin() for qh.NEARzero
design:
determine qh.DISTround for distance computations
determine minimum denominators for qh_divzero
determine qh.ANGLEround for angle computations
adjust qh.premerge_cos,... for roundoff error
determine qh.ONEmerge for maximum error due to a single merge
determine qh.NEARinside, qh.MAXcoplanar, qh.MINvisible,
qh.MINoutside, qh.WIDEfacet
initialize qh.max_vertex and qh.minvertex
*/
void qh_detroundoff(void) {
qh_option("_max-width", NULL, &qh MAXwidth);
if (!qh SETroundoff) {
qh DISTround= qh_distround(qh hull_dim, qh MAXabs_coord, qh MAXsumcoord);
if (qh RANDOMdist)
qh DISTround += qh RANDOMfactor * qh MAXabs_coord;
qh_option("Error-roundoff", NULL, &qh DISTround);
}
qh MINdenom= qh MINdenom_1 * qh MAXabs_coord;
qh MINdenom_1_2= sqrt(qh MINdenom_1 * qh hull_dim) ; /* if will be normalized */
qh MINdenom_2= qh MINdenom_1_2 * qh MAXabs_coord;
/* for inner product */
qh ANGLEround= 1.01 * qh hull_dim * REALepsilon;
if (qh RANDOMdist)
qh ANGLEround += qh RANDOMfactor;
if (qh premerge_cos < REALmax/2) {
qh premerge_cos -= qh ANGLEround;
if (qh RANDOMdist)
qh_option("Angle-premerge-with-random", NULL, &qh premerge_cos);
}
if (qh postmerge_cos < REALmax/2) {
qh postmerge_cos -= qh ANGLEround;
if (qh RANDOMdist)
qh_option("Angle-postmerge-with-random", NULL, &qh postmerge_cos);
}
qh premerge_centrum += 2 * qh DISTround; /*2 for centrum and distplane()*/
qh postmerge_centrum += 2 * qh DISTround;
if (qh RANDOMdist && (qh MERGEexact || qh PREmerge))
qh_option("Centrum-premerge-with-random", NULL, &qh premerge_centrum);
if (qh RANDOMdist && qh POSTmerge)
qh_option("Centrum-postmerge-with-random", NULL, &qh postmerge_centrum);
{ /* compute ONEmerge, max vertex offset for merging simplicial facets */
realT maxangle= 1.0, maxrho;
minimize_(maxangle, qh premerge_cos);
minimize_(maxangle, qh postmerge_cos);
/* max diameter * sin theta + DISTround for vertex to its hyperplane */
qh ONEmerge= sqrt((realT)qh hull_dim) * qh MAXwidth *
sqrt(1.0 - maxangle * maxangle) + qh DISTround;
maxrho= qh hull_dim * qh premerge_centrum + qh DISTround;
maximize_(qh ONEmerge, maxrho);
maxrho= qh hull_dim * qh postmerge_centrum + qh DISTround;
maximize_(qh ONEmerge, maxrho);
if (qh MERGING)
qh_option("_one-merge", NULL, &qh ONEmerge);
}
qh NEARinside= qh ONEmerge * qh_RATIOnearinside; /* only used if qh KEEPnearinside */
if (qh JOGGLEmax < REALmax/2 && (qh KEEPcoplanar || qh KEEPinside)) {
realT maxdist; /* adjust qh.NEARinside for joggle */
qh KEEPnearinside= True;
maxdist= sqrt((realT)qh hull_dim) * qh JOGGLEmax + qh DISTround;
maxdist= 2*maxdist; /* vertex and coplanar point can joggle in opposite directions */
maximize_(qh NEARinside, maxdist); /* must agree with qh_nearcoplanar() */
}
if (qh KEEPnearinside)
qh_option("_near-inside", NULL, &qh NEARinside);
if (qh JOGGLEmax < qh DISTround) {
qh_fprintf(qh ferr, 6006, "qhull error: the joggle for 'QJn', %.2g, is below roundoff for distance computations, %.2g\n",
qh JOGGLEmax, qh DISTround);
qh_errexit(qh_ERRinput, NULL, NULL);
}
if (qh MINvisible > REALmax/2) {
if (!qh MERGING)
qh MINvisible= qh DISTround;
else if (qh hull_dim <= 3)
qh MINvisible= qh premerge_centrum;
else
qh MINvisible= qh_COPLANARratio * qh premerge_centrum;
if (qh APPROXhull && qh MINvisible > qh MINoutside)
qh MINvisible= qh MINoutside;
qh_option("Visible-distance", NULL, &qh MINvisible);
}
if (qh MAXcoplanar > REALmax/2) {
qh MAXcoplanar= qh MINvisible;
qh_option("U-coplanar-distance", NULL, &qh MAXcoplanar);
}
if (!qh APPROXhull) { /* user may specify qh MINoutside */
qh MINoutside= 2 * qh MINvisible;
if (qh premerge_cos < REALmax/2)
maximize_(qh MINoutside, (1- qh premerge_cos) * qh MAXabs_coord);
qh_option("Width-outside", NULL, &qh MINoutside);
}
qh WIDEfacet= qh MINoutside;
maximize_(qh WIDEfacet, qh_WIDEcoplanar * qh MAXcoplanar);
maximize_(qh WIDEfacet, qh_WIDEcoplanar * qh MINvisible);
qh_option("_wide-facet", NULL, &qh WIDEfacet);
if (qh MINvisible > qh MINoutside + 3 * REALepsilon
&& !qh BESToutside && !qh FORCEoutput)
qh_fprintf(qh ferr, 7001, "qhull input warning: minimum visibility V%.2g is greater than \nminimum outside W%.2g. Flipped facets are likely.\n",
qh MINvisible, qh MINoutside);
qh max_vertex= qh DISTround;
qh min_vertex= -qh DISTround;
/* numeric constants reported in printsummary */
} /* detroundoff */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="detsimplex">-</a>
qh_detsimplex( apex, points, dim, nearzero )
compute determinant of a simplex with point apex and base points
returns:
signed determinant and nearzero from qh_determinant
notes:
uses qh.gm_matrix/qh.gm_row (assumes they're big enough)
design:
construct qm_matrix by subtracting apex from points
compute determinate
*/
realT qh_detsimplex(pointT *apex, setT *points, int dim, boolT *nearzero) {
pointT *coorda, *coordp, *gmcoord, *point, **pointp;
coordT **rows;
int k, i=0;
realT det;
zinc_(Zdetsimplex);
gmcoord= qh gm_matrix;
rows= qh gm_row;
FOREACHpoint_(points) {
if (i == dim)
break;
rows[i++]= gmcoord;
coordp= point;
coorda= apex;
for (k=dim; k--; )
*(gmcoord++)= *coordp++ - *coorda++;
}
if (i < dim) {
qh_fprintf(qh ferr, 6007, "qhull internal error (qh_detsimplex): #points %d < dimension %d\n",
i, dim);
qh_errexit(qh_ERRqhull, NULL, NULL);
}
det= qh_determinant(rows, dim, nearzero);
trace2((qh ferr, 2002, "qh_detsimplex: det=%2.2g for point p%d, dim %d, nearzero? %d\n",
det, qh_pointid(apex), dim, *nearzero));
return det;
} /* detsimplex */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="distnorm">-</a>
qh_distnorm( dim, point, normal, offset )
return distance from point to hyperplane at normal/offset
returns:
dist
notes:
dist > 0 if point is outside of hyperplane
see:
qh_distplane in geom.c
*/
realT qh_distnorm(int dim, pointT *point, pointT *normal, realT *offsetp) {
coordT *normalp= normal, *coordp= point;
realT dist;
int k;
dist= *offsetp;
for (k=dim; k--; )
dist += *(coordp++) * *(normalp++);
return dist;
} /* distnorm */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="distround">-</a>
qh_distround(dimension, maxabs, maxsumabs )
compute maximum round-off error for a distance computation
to a normalized hyperplane
maxabs is the maximum absolute value of a coordinate
maxsumabs is the maximum possible sum of absolute coordinate values
returns:
max dist round for REALepsilon
notes:
calculate roundoff error according to
Lemma 3.2-1 of Golub and van Loan "Matrix Computation"
use sqrt(dim) since one vector is normalized
or use maxsumabs since one vector is < 1
*/
realT qh_distround(int dimension, realT maxabs, realT maxsumabs) {
realT maxdistsum, maxround;
maxdistsum= sqrt((realT)dimension) * maxabs;
minimize_( maxdistsum, maxsumabs);
maxround= REALepsilon * (dimension * maxdistsum * 1.01 + maxabs);
/* adds maxabs for offset */
trace4((qh ferr, 4008, "qh_distround: %2.2g maxabs %2.2g maxsumabs %2.2g maxdistsum %2.2g\n",
maxround, maxabs, maxsumabs, maxdistsum));
return maxround;
} /* distround */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="divzero">-</a>
qh_divzero( numer, denom, mindenom1, zerodiv )
divide by a number that's nearly zero
mindenom1= minimum denominator for dividing into 1.0
returns:
quotient
sets zerodiv and returns 0.0 if it would overflow
design:
if numer is nearly zero and abs(numer) < abs(denom)
return numer/denom
else if numer is nearly zero
return 0 and zerodiv
else if denom/numer non-zero
return numer/denom
else
return 0 and zerodiv
*/
realT qh_divzero(realT numer, realT denom, realT mindenom1, boolT *zerodiv) {
realT temp, numerx, denomx;
if (numer < mindenom1 && numer > -mindenom1) {
numerx= fabs_(numer);
denomx= fabs_(denom);
if (numerx < denomx) {
*zerodiv= False;
return numer/denom;
}else {
*zerodiv= True;
return 0.0;
}
}
temp= denom/numer;
if (temp > mindenom1 || temp < -mindenom1) {
*zerodiv= False;
return numer/denom;
}else {
*zerodiv= True;
return 0.0;
}
} /* divzero */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="facetarea">-</a>
qh_facetarea( facet )
return area for a facet
notes:
if non-simplicial,
uses centrum to triangulate facet and sums the projected areas.
if (qh DELAUNAY),
computes projected area instead for last coordinate
assumes facet->normal exists
projecting tricoplanar facets to the hyperplane does not appear to make a difference
design:
if simplicial
compute area
else
for each ridge
compute area from centrum to ridge
negate area if upper Delaunay facet
*/
realT qh_facetarea(facetT *facet) {
vertexT *apex;
pointT *centrum;
realT area= 0.0;
ridgeT *ridge, **ridgep;
if (facet->simplicial) {
apex= SETfirstt_(facet->vertices, vertexT);
area= qh_facetarea_simplex(qh hull_dim, apex->point, facet->vertices,
apex, facet->toporient, facet->normal, &facet->offset);
}else {
if (qh CENTERtype == qh_AScentrum)
centrum= facet->center;
else
centrum= qh_getcentrum(facet);
FOREACHridge_(facet->ridges)
area += qh_facetarea_simplex(qh hull_dim, centrum, ridge->vertices,
NULL, (boolT)(ridge->top == facet), facet->normal, &facet->offset);
if (qh CENTERtype != qh_AScentrum)
qh_memfree(centrum, qh normal_size);
}
if (facet->upperdelaunay && qh DELAUNAY)
area= -area; /* the normal should be [0,...,1] */
trace4((qh ferr, 4009, "qh_facetarea: f%d area %2.2g\n", facet->id, area));
return area;
} /* facetarea */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="facetarea_simplex">-</a>
qh_facetarea_simplex( dim, apex, vertices, notvertex, toporient, normal, offset )
return area for a simplex defined by
an apex, a base of vertices, an orientation, and a unit normal
if simplicial or tricoplanar facet,
notvertex is defined and it is skipped in vertices
returns:
computes area of simplex projected to plane [normal,offset]
returns 0 if vertex too far below plane (qh WIDEfacet)
vertex can't be apex of tricoplanar facet
notes:
if (qh DELAUNAY),
computes projected area instead for last coordinate
uses qh gm_matrix/gm_row and qh hull_dim
helper function for qh_facetarea
design:
if Notvertex
translate simplex to apex
else
project simplex to normal/offset
translate simplex to apex
if Delaunay
set last row/column to 0 with -1 on diagonal
else
set last row to Normal
compute determinate
scale and flip sign for area
*/
realT qh_facetarea_simplex(int dim, coordT *apex, setT *vertices,
vertexT *notvertex, boolT toporient, coordT *normal, realT *offset) {
pointT *coorda, *coordp, *gmcoord;
coordT **rows, *normalp;
int k, i=0;
realT area, dist;
vertexT *vertex, **vertexp;
boolT nearzero;
gmcoord= qh gm_matrix;
rows= qh gm_row;
FOREACHvertex_(vertices) {
if (vertex == notvertex)
continue;
rows[i++]= gmcoord;
coorda= apex;
coordp= vertex->point;
normalp= normal;
if (notvertex) {
for (k=dim; k--; )
*(gmcoord++)= *coordp++ - *coorda++;
}else {
dist= *offset;
for (k=dim; k--; )
dist += *coordp++ * *normalp++;
if (dist < -qh WIDEfacet) {
zinc_(Znoarea);
return 0.0;
}
coordp= vertex->point;
normalp= normal;
for (k=dim; k--; )
*(gmcoord++)= (*coordp++ - dist * *normalp++) - *coorda++;
}
}
if (i != dim-1) {
qh_fprintf(qh ferr, 6008, "qhull internal error (qh_facetarea_simplex): #points %d != dim %d -1\n",
i, dim);
qh_errexit(qh_ERRqhull, NULL, NULL);
}
rows[i]= gmcoord;
if (qh DELAUNAY) {
for (i=0; i < dim-1; i++)
rows[i][dim-1]= 0.0;
for (k=dim; k--; )
*(gmcoord++)= 0.0;
rows[dim-1][dim-1]= -1.0;
}else {
normalp= normal;
for (k=dim; k--; )
*(gmcoord++)= *normalp++;
}
zinc_(Zdetsimplex);
area= qh_determinant(rows, dim, &nearzero);
if (toporient)
area= -area;
area *= qh AREAfactor;
trace4((qh ferr, 4010, "qh_facetarea_simplex: area=%2.2g for point p%d, toporient %d, nearzero? %d\n",
area, qh_pointid(apex), toporient, nearzero));
return area;
} /* facetarea_simplex */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="facetcenter">-</a>
qh_facetcenter( vertices )
return Voronoi center (Voronoi vertex) for a facet's vertices
returns:
return temporary point equal to the center
see:
qh_voronoi_center()
*/
pointT *qh_facetcenter(setT *vertices) {
setT *points= qh_settemp(qh_setsize(vertices));
vertexT *vertex, **vertexp;
pointT *center;
FOREACHvertex_(vertices)
qh_setappend(&points, vertex->point);
center= qh_voronoi_center(qh hull_dim-1, points);
qh_settempfree(&points);
return center;
} /* facetcenter */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="findgooddist">-</a>
qh_findgooddist( point, facetA, dist, facetlist )
find best good facet visible for point from facetA
assumes facetA is visible from point
returns:
best facet, i.e., good facet that is furthest from point
distance to best facet
NULL if none
moves good, visible facets (and some other visible facets)
to end of qh facet_list
notes:
uses qh visit_id
design:
initialize bestfacet if facetA is good
move facetA to end of facetlist
for each facet on facetlist
for each unvisited neighbor of facet
move visible neighbors to end of facetlist
update best good neighbor
if no good neighbors, update best facet
*/
facetT *qh_findgooddist(pointT *point, facetT *facetA, realT *distp,
facetT **facetlist) {
realT bestdist= -REALmax, dist;
facetT *neighbor, **neighborp, *bestfacet=NULL, *facet;
boolT goodseen= False;
if (facetA->good) {
zzinc_(Zcheckpart); /* calls from check_bestdist occur after print stats */
qh_distplane(point, facetA, &bestdist);
bestfacet= facetA;
goodseen= True;
}
qh_removefacet(facetA);
qh_appendfacet(facetA);
*facetlist= facetA;
facetA->visitid= ++qh visit_id;
FORALLfacet_(*facetlist) {
FOREACHneighbor_(facet) {
if (neighbor->visitid == qh visit_id)
continue;
neighbor->visitid= qh visit_id;
if (goodseen && !neighbor->good)
continue;
zzinc_(Zcheckpart);
qh_distplane(point, neighbor, &dist);
if (dist > 0) {
qh_removefacet(neighbor);
qh_appendfacet(neighbor);
if (neighbor->good) {
goodseen= True;
if (dist > bestdist) {
bestdist= dist;
bestfacet= neighbor;
}
}
}
}
}
if (bestfacet) {
*distp= bestdist;
trace2((qh ferr, 2003, "qh_findgooddist: p%d is %2.2g above good facet f%d\n",
qh_pointid(point), bestdist, bestfacet->id));
return bestfacet;
}
trace4((qh ferr, 4011, "qh_findgooddist: no good facet for p%d above f%d\n",
qh_pointid(point), facetA->id));
return NULL;
} /* findgooddist */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="getarea">-</a>
qh_getarea( facetlist )
set area of all facets in facetlist
collect statistics
nop if hasAreaVolume
returns:
sets qh totarea/totvol to total area and volume of convex hull
for Delaunay triangulation, computes projected area of the lower or upper hull
ignores upper hull if qh ATinfinity
notes:
could compute outer volume by expanding facet area by rays from interior
the following attempt at perpendicular projection underestimated badly:
qh.totoutvol += (-dist + facet->maxoutside + qh DISTround)
* area/ qh hull_dim;
design:
for each facet on facetlist
compute facet->area
update qh.totarea and qh.totvol
*/
void qh_getarea(facetT *facetlist) {
realT area;
realT dist;
facetT *facet;
if (qh hasAreaVolume)
return;
if (qh REPORTfreq)
qh_fprintf(qh ferr, 8020, "computing area of each facet and volume of the convex hull\n");
else
trace1((qh ferr, 1001, "qh_getarea: computing volume and area for each facet\n"));
qh totarea= qh totvol= 0.0;
FORALLfacet_(facetlist) {
if (!facet->normal)
continue;
if (facet->upperdelaunay && qh ATinfinity)
continue;
if (!facet->isarea) {
facet->f.area= qh_facetarea(facet);
facet->isarea= True;
}
area= facet->f.area;
if (qh DELAUNAY) {
if (facet->upperdelaunay == qh UPPERdelaunay)
qh totarea += area;
}else {
qh totarea += area;
qh_distplane(qh interior_point, facet, &dist);
qh totvol += -dist * area/ qh hull_dim;
}
if (qh PRINTstatistics) {
wadd_(Wareatot, area);
wmax_(Wareamax, area);
wmin_(Wareamin, area);
}
}
qh hasAreaVolume= True;
} /* getarea */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="gram_schmidt">-</a>
qh_gram_schmidt( dim, row )
implements Gram-Schmidt orthogonalization by rows
returns:
false if zero norm
overwrites rows[dim][dim]
notes:
see Golub & van Loan Algorithm 6.2-2
overflow due to small divisors not handled
design:
for each row
compute norm for row
if non-zero, normalize row
for each remaining rowA
compute inner product of row and rowA
reduce rowA by row * inner product
*/
boolT qh_gram_schmidt(int dim, realT **row) {
realT *rowi, *rowj, norm;
int i, j, k;
for (i=0; i < dim; i++) {
rowi= row[i];
for (norm= 0.0, k= dim; k--; rowi++)
norm += *rowi * *rowi;
norm= sqrt(norm);
wmin_(Wmindenom, norm);
if (norm == 0.0) /* either 0 or overflow due to sqrt */
return False;
for (k=dim; k--; )
*(--rowi) /= norm;
for (j=i+1; j < dim; j++) {
rowj= row[j];
for (norm= 0.0, k=dim; k--; )
norm += *rowi++ * *rowj++;
for (k=dim; k--; )
*(--rowj) -= *(--rowi) * norm;
}
}
return True;
} /* gram_schmidt */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="inthresholds">-</a>
qh_inthresholds( normal, angle )
return True if normal within qh.lower_/upper_threshold
returns:
estimate of angle by summing of threshold diffs
angle may be NULL
smaller "angle" is better
notes:
invalid if qh.SPLITthresholds
see:
qh.lower_threshold in qh_initbuild()
qh_initthresholds()
design:
for each dimension
test threshold
*/
boolT qh_inthresholds(coordT *normal, realT *angle) {
boolT within= True;
int k;
realT threshold;
if (angle)
*angle= 0.0;
for (k=0; k < qh hull_dim; k++) {
threshold= qh lower_threshold[k];
if (threshold > -REALmax/2) {
if (normal[k] < threshold)
within= False;
if (angle) {
threshold -= normal[k];
*angle += fabs_(threshold);
}
}
if (qh upper_threshold[k] < REALmax/2) {
threshold= qh upper_threshold[k];
if (normal[k] > threshold)
within= False;
if (angle) {
threshold -= normal[k];
*angle += fabs_(threshold);
}
}
}
return within;
} /* inthresholds */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="joggleinput">-</a>
qh_joggleinput()
randomly joggle input to Qhull by qh.JOGGLEmax
initial input is qh.first_point/qh.num_points of qh.hull_dim
repeated calls use qh.input_points/qh.num_points
returns:
joggles points at qh.first_point/qh.num_points
copies data to qh.input_points/qh.input_malloc if first time
determines qh.JOGGLEmax if it was zero
if qh.DELAUNAY
computes the Delaunay projection of the joggled points
notes:
if qh.DELAUNAY, unnecessarily joggles the last coordinate
the initial 'QJn' may be set larger than qh_JOGGLEmaxincrease
design:
if qh.DELAUNAY
set qh.SCALElast for reduced precision errors
if first call
initialize qh.input_points to the original input points
if qh.JOGGLEmax == 0
determine default qh.JOGGLEmax
else
increase qh.JOGGLEmax according to qh.build_cnt
joggle the input by adding a random number in [-qh.JOGGLEmax,qh.JOGGLEmax]
if qh.DELAUNAY
sets the Delaunay projection
*/
void qh_joggleinput(void) {
int i, seed, size;
coordT *coordp, *inputp;
realT randr, randa, randb;
if (!qh input_points) { /* first call */
qh input_points= qh first_point;
qh input_malloc= qh POINTSmalloc;
size= qh num_points * qh hull_dim * sizeof(coordT);
if (!(qh first_point=(coordT*)qh_malloc((size_t)size))) {
qh_fprintf(qh ferr, 6009, "qhull error: insufficient memory to joggle %d points\n",
qh num_points);
qh_errexit(qh_ERRmem, NULL, NULL);
}
qh POINTSmalloc= True;
if (qh JOGGLEmax == 0.0) {
qh JOGGLEmax= qh_detjoggle(qh input_points, qh num_points, qh hull_dim);
qh_option("QJoggle", NULL, &qh JOGGLEmax);
}
}else { /* repeated call */
if (!qh RERUN && qh build_cnt > qh_JOGGLEretry) {
if (((qh build_cnt-qh_JOGGLEretry-1) % qh_JOGGLEagain) == 0) {
realT maxjoggle= qh MAXwidth * qh_JOGGLEmaxincrease;
if (qh JOGGLEmax < maxjoggle) {
qh JOGGLEmax *= qh_JOGGLEincrease;
minimize_(qh JOGGLEmax, maxjoggle);
}
}
}
qh_option("QJoggle", NULL, &qh JOGGLEmax);
}
if (qh build_cnt > 1 && qh JOGGLEmax > fmax_(qh MAXwidth/4, 0.1)) {
qh_fprintf(qh ferr, 6010, "qhull error: the current joggle for 'QJn', %.2g, is too large for the width\nof the input. If possible, recompile Qhull with higher-precision reals.\n",
qh JOGGLEmax);
qh_errexit(qh_ERRqhull, NULL, NULL);
}
/* for some reason, using qh ROTATErandom and qh_RANDOMseed does not repeat the run. Use 'TRn' instead */
seed= qh_RANDOMint;
qh_option("_joggle-seed", &seed, NULL);
trace0((qh ferr, 6, "qh_joggleinput: joggle input by %2.2g with seed %d\n",
qh JOGGLEmax, seed));
inputp= qh input_points;
coordp= qh first_point;
randa= 2.0 * qh JOGGLEmax/qh_RANDOMmax;
randb= -qh JOGGLEmax;
size= qh num_points * qh hull_dim;
for (i=size; i--; ) {
randr= qh_RANDOMint;
*(coordp++)= *(inputp++) + (randr * randa + randb);
}
if (qh DELAUNAY) {
qh last_low= qh last_high= qh last_newhigh= REALmax;
qh_setdelaunay(qh hull_dim, qh num_points, qh first_point);
}
} /* joggleinput */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="maxabsval">-</a>
qh_maxabsval( normal, dim )
return pointer to maximum absolute value of a dim vector
returns NULL if dim=0
*/
realT *qh_maxabsval(realT *normal, int dim) {
realT maxval= -REALmax;
realT *maxp= NULL, *colp, absval;
int k;
for (k=dim, colp= normal; k--; colp++) {
absval= fabs_(*colp);
if (absval > maxval) {
maxval= absval;
maxp= colp;
}
}
return maxp;
} /* maxabsval */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="maxmin">-</a>
qh_maxmin( points, numpoints, dimension )
return max/min points for each dimension
determine max and min coordinates
returns:
returns a temporary set of max and min points
may include duplicate points. Does not include qh.GOODpoint
sets qh.NEARzero, qh.MAXabs_coord, qh.MAXsumcoord, qh.MAXwidth
qh.MAXlastcoord, qh.MINlastcoord
initializes qh.max_outside, qh.min_vertex, qh.WAScoplanar, qh.ZEROall_ok
notes:
loop duplicated in qh_detjoggle()
design:
initialize global precision variables
checks definition of REAL...
for each dimension
for each point
collect maximum and minimum point
collect maximum of maximums and minimum of minimums
determine qh.NEARzero for Gaussian Elimination
*/
setT *qh_maxmin(pointT *points, int numpoints, int dimension) {
int k;
realT maxcoord, temp;
pointT *minimum, *maximum, *point, *pointtemp;
setT *set;
qh max_outside= 0.0;
qh MAXabs_coord= 0.0;
qh MAXwidth= -REALmax;
qh MAXsumcoord= 0.0;
qh min_vertex= 0.0;
qh WAScoplanar= False;
if (qh ZEROcentrum)
qh ZEROall_ok= True;
if (REALmin < REALepsilon && REALmin < REALmax && REALmin > -REALmax
&& REALmax > 0.0 && -REALmax < 0.0)
; /* all ok */
else {
qh_fprintf(qh ferr, 6011, "qhull error: floating point constants in user.h are wrong\n\
REALepsilon %g REALmin %g REALmax %g -REALmax %g\n",
REALepsilon, REALmin, REALmax, -REALmax);
qh_errexit(qh_ERRinput, NULL, NULL);
}
set= qh_settemp(2*dimension);
for (k=0; k < dimension; k++) {
if (points == qh GOODpointp)
minimum= maximum= points + dimension;
else
minimum= maximum= points;
FORALLpoint_(points, numpoints) {
if (point == qh GOODpointp)
continue;
if (maximum[k] < point[k])
maximum= point;
else if (minimum[k] > point[k])
minimum= point;
}
if (k == dimension-1) {
qh MINlastcoord= minimum[k];
qh MAXlastcoord= maximum[k];
}
if (qh SCALElast && k == dimension-1)
maxcoord= qh MAXwidth;
else {
maxcoord= fmax_(maximum[k], -minimum[k]);
if (qh GOODpointp) {
temp= fmax_(qh GOODpointp[k], -qh GOODpointp[k]);
maximize_(maxcoord, temp);
}
temp= maximum[k] - minimum[k];
maximize_(qh MAXwidth, temp);
}
maximize_(qh MAXabs_coord, maxcoord);
qh MAXsumcoord += maxcoord;
qh_setappend(&set, maximum);
qh_setappend(&set, minimum);
/* calculation of qh NEARzero is based on error formula 4.4-13 of
Golub & van Loan, authors say n^3 can be ignored and 10 be used in
place of rho */
qh NEARzero[k]= 80 * qh MAXsumcoord * REALepsilon;
}
if (qh IStracing >=1)
qh_printpoints(qh ferr, "qh_maxmin: found the max and min points(by dim):", set);
return(set);
} /* maxmin */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="maxouter">-</a>
qh_maxouter()
return maximum distance from facet to outer plane
normally this is qh.max_outside+qh.DISTround
does not include qh.JOGGLEmax
see:
qh_outerinner()
notes:
need to add another qh.DISTround if testing actual point with computation
for joggle:
qh_setfacetplane() updated qh.max_outer for Wnewvertexmax (max distance to vertex)
need to use Wnewvertexmax since could have a coplanar point for a high
facet that is replaced by a low facet
need to add qh.JOGGLEmax if testing input points
*/
realT qh_maxouter(void) {
realT dist;
dist= fmax_(qh max_outside, qh DISTround);
dist += qh DISTround;
trace4((qh ferr, 4012, "qh_maxouter: max distance from facet to outer plane is %2.2g max_outside is %2.2g\n", dist, qh max_outside));
return dist;
} /* maxouter */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="maxsimplex">-</a>
qh_maxsimplex( dim, maxpoints, points, numpoints, simplex )
determines maximum simplex for a set of points
starts from points already in simplex
skips qh.GOODpointp (assumes that it isn't in maxpoints)
returns:
simplex with dim+1 points
notes:
assumes at least pointsneeded points in points
maximizes determinate for x,y,z,w, etc.
uses maxpoints as long as determinate is clearly non-zero
design:
initialize simplex with at least two points
(find points with max or min x coordinate)
for each remaining dimension
add point that maximizes the determinate
(use points from maxpoints first)
*/
void qh_maxsimplex(int dim, setT *maxpoints, pointT *points, int numpoints, setT **simplex) {
pointT *point, **pointp, *pointtemp, *maxpoint, *minx=NULL, *maxx=NULL;
boolT nearzero, maxnearzero= False;
int k, sizinit;
realT maxdet= -REALmax, det, mincoord= REALmax, maxcoord= -REALmax;
sizinit= qh_setsize(*simplex);
if (sizinit < 2) {
if (qh_setsize(maxpoints) >= 2) {
FOREACHpoint_(maxpoints) {
if (maxcoord < point[0]) {
maxcoord= point[0];
maxx= point;
}
if (mincoord > point[0]) {
mincoord= point[0];
minx= point;
}
}
}else {
FORALLpoint_(points, numpoints) {
if (point == qh GOODpointp)
continue;
if (maxcoord < point[0]) {
maxcoord= point[0];
maxx= point;
}
if (mincoord > point[0]) {
mincoord= point[0];
minx= point;
}
}
}
qh_setunique(simplex, minx);
if (qh_setsize(*simplex) < 2)
qh_setunique(simplex, maxx);
sizinit= qh_setsize(*simplex);
if (sizinit < 2) {
qh_precision("input has same x coordinate");
if (zzval_(Zsetplane) > qh hull_dim+1) {
qh_fprintf(qh ferr, 6012, "qhull precision error (qh_maxsimplex for voronoi_center):\n%d points with the same x coordinate.\n",
qh_setsize(maxpoints)+numpoints);
qh_errexit(qh_ERRprec, NULL, NULL);
}else {
qh_fprintf(qh ferr, 6013, "qhull input error: input is less than %d-dimensional since it has the same x coordinate\n", qh hull_dim);
qh_errexit(qh_ERRinput, NULL, NULL);
}
}
}
for (k=sizinit; k < dim+1; k++) {
maxpoint= NULL;
maxdet= -REALmax;
FOREACHpoint_(maxpoints) {
if (!qh_setin(*simplex, point)) {
det= qh_detsimplex(point, *simplex, k, &nearzero);
if ((det= fabs_(det)) > maxdet) {
maxdet= det;
maxpoint= point;
maxnearzero= nearzero;
}
}
}
if (!maxpoint || maxnearzero) {
zinc_(Zsearchpoints);
if (!maxpoint) {
trace0((qh ferr, 7, "qh_maxsimplex: searching all points for %d-th initial vertex.\n", k+1));
}else {
trace0((qh ferr, 8, "qh_maxsimplex: searching all points for %d-th initial vertex, better than p%d det %2.2g\n",
k+1, qh_pointid(maxpoint), maxdet));
}
FORALLpoint_(points, numpoints) {
if (point == qh GOODpointp)
continue;
if (!qh_setin(*simplex, point)) {
det= qh_detsimplex(point, *simplex, k, &nearzero);
if ((det= fabs_(det)) > maxdet) {
maxdet= det;
maxpoint= point;
maxnearzero= nearzero;
}
}
}
} /* !maxpoint */
if (!maxpoint) {
qh_fprintf(qh ferr, 6014, "qhull internal error (qh_maxsimplex): not enough points available\n");
qh_errexit(qh_ERRqhull, NULL, NULL);
}
qh_setappend(simplex, maxpoint);
trace1((qh ferr, 1002, "qh_maxsimplex: selected point p%d for %d`th initial vertex, det=%2.2g\n",
qh_pointid(maxpoint), k+1, maxdet));
} /* k */
} /* maxsimplex */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="minabsval">-</a>
qh_minabsval( normal, dim )
return minimum absolute value of a dim vector
*/
realT qh_minabsval(realT *normal, int dim) {
realT minval= 0;
realT maxval= 0;
realT *colp;
int k;
for (k=dim, colp=normal; k--; colp++) {
maximize_(maxval, *colp);
minimize_(minval, *colp);
}
return fmax_(maxval, -minval);
} /* minabsval */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="mindiff">-</a>
qh_mindif( vecA, vecB, dim )
return index of min abs. difference of two vectors
*/
int qh_mindiff(realT *vecA, realT *vecB, int dim) {
realT mindiff= REALmax, diff;
realT *vecAp= vecA, *vecBp= vecB;
int k, mink= 0;
for (k=0; k < dim; k++) {
diff= *vecAp++ - *vecBp++;
diff= fabs_(diff);
if (diff < mindiff) {
mindiff= diff;
mink= k;
}
}
return mink;
} /* mindiff */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="orientoutside">-</a>
qh_orientoutside( facet )
make facet outside oriented via qh.interior_point
returns:
True if facet reversed orientation.
*/
boolT qh_orientoutside(facetT *facet) {
int k;
realT dist;
qh_distplane(qh interior_point, facet, &dist);
if (dist > 0) {
for (k=qh hull_dim; k--; )
facet->normal[k]= -facet->normal[k];
facet->offset= -facet->offset;
return True;
}
return False;
} /* orientoutside */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="outerinner">-</a>
qh_outerinner( facet, outerplane, innerplane )
if facet and qh.maxoutdone (i.e., qh_check_maxout)
returns outer and inner plane for facet
else
returns maximum outer and inner plane
accounts for qh.JOGGLEmax
see:
qh_maxouter(), qh_check_bestdist(), qh_check_points()
notes:
outerplaner or innerplane may be NULL
facet is const
Does not error (QhullFacet)
includes qh.DISTround for actual points
adds another qh.DISTround if testing with floating point arithmetic
*/
void qh_outerinner(facetT *facet, realT *outerplane, realT *innerplane) {
realT dist, mindist;
vertexT *vertex, **vertexp;
if (outerplane) {
if (!qh_MAXoutside || !facet || !qh maxoutdone) {
*outerplane= qh_maxouter(); /* includes qh.DISTround */
}else { /* qh_MAXoutside ... */
#if qh_MAXoutside
*outerplane= facet->maxoutside + qh DISTround;
#endif
}
if (qh JOGGLEmax < REALmax/2)
*outerplane += qh JOGGLEmax * sqrt((realT)qh hull_dim);
}
if (innerplane) {
if (facet) {
mindist= REALmax;
FOREACHvertex_(facet->vertices) {
zinc_(Zdistio);
qh_distplane(vertex->point, facet, &dist);
minimize_(mindist, dist);
}
*innerplane= mindist - qh DISTround;
}else
*innerplane= qh min_vertex - qh DISTround;
if (qh JOGGLEmax < REALmax/2)
*innerplane -= qh JOGGLEmax * sqrt((realT)qh hull_dim);
}
} /* outerinner */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="pointdist">-</a>
qh_pointdist( point1, point2, dim )
return distance between two points
notes:
returns distance squared if 'dim' is negative
*/
coordT qh_pointdist(pointT *point1, pointT *point2, int dim) {
coordT dist, diff;
int k;
dist= 0.0;
for (k= (dim > 0 ? dim : -dim); k--; ) {
diff= *point1++ - *point2++;
dist += diff * diff;
}
if (dim > 0)
return(sqrt(dist));
return dist;
} /* pointdist */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="printmatrix">-</a>
qh_printmatrix( fp, string, rows, numrow, numcol )
print matrix to fp given by row vectors
print string as header
notes:
print a vector by qh_printmatrix(fp, "", &vect, 1, len)
*/
void qh_printmatrix(FILE *fp, const char *string, realT **rows, int numrow, int numcol) {
realT *rowp;
realT r; /*bug fix*/
int i,k;
qh_fprintf(fp, 9001, "%s\n", string);
for (i=0; i < numrow; i++) {
rowp= rows[i];
for (k=0; k < numcol; k++) {
r= *rowp++;
qh_fprintf(fp, 9002, "%6.3g ", r);
}
qh_fprintf(fp, 9003, "\n");
}
} /* printmatrix */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="printpoints">-</a>
qh_printpoints( fp, string, points )
print pointids to fp for a set of points
if string, prints string and 'p' point ids
*/
void qh_printpoints(FILE *fp, const char *string, setT *points) {
pointT *point, **pointp;
if (string) {
qh_fprintf(fp, 9004, "%s", string);
FOREACHpoint_(points)
qh_fprintf(fp, 9005, " p%d", qh_pointid(point));
qh_fprintf(fp, 9006, "\n");
}else {
FOREACHpoint_(points)
qh_fprintf(fp, 9007, " %d", qh_pointid(point));
qh_fprintf(fp, 9008, "\n");
}
} /* printpoints */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="projectinput">-</a>
qh_projectinput()
project input points using qh.lower_bound/upper_bound and qh DELAUNAY
if qh.lower_bound[k]=qh.upper_bound[k]= 0,
removes dimension k
if halfspace intersection
removes dimension k from qh.feasible_point
input points in qh first_point, num_points, input_dim
returns:
new point array in qh first_point of qh hull_dim coordinates
sets qh POINTSmalloc
if qh DELAUNAY
projects points to paraboloid
lowbound/highbound is also projected
if qh ATinfinity
adds point "at-infinity"
if qh POINTSmalloc
frees old point array
notes:
checks that qh.hull_dim agrees with qh.input_dim, PROJECTinput, and DELAUNAY
design:
sets project[k] to -1 (delete), 0 (keep), 1 (add for Delaunay)
determines newdim and newnum for qh hull_dim and qh num_points
projects points to newpoints
projects qh.lower_bound to itself
projects qh.upper_bound to itself
if qh DELAUNAY
if qh ATINFINITY
projects points to paraboloid
computes "infinity" point as vertex average and 10% above all points
else
uses qh_setdelaunay to project points to paraboloid
*/
void qh_projectinput(void) {
int k,i;
int newdim= qh input_dim, newnum= qh num_points;
signed char *project;
int size= (qh input_dim+1)*sizeof(*project);
pointT *newpoints, *coord, *infinity;
realT paraboloid, maxboloid= 0;
project= (signed char*)qh_memalloc(size);
memset((char*)project, 0, (size_t)size);
for (k=0; k < qh input_dim; k++) { /* skip Delaunay bound */
if (qh lower_bound[k] == 0 && qh upper_bound[k] == 0) {
project[k]= -1;
newdim--;
}
}
if (qh DELAUNAY) {
project[k]= 1;
newdim++;
if (qh ATinfinity)
newnum++;
}
if (newdim != qh hull_dim) {
qh_fprintf(qh ferr, 6015, "qhull internal error (qh_projectinput): dimension after projection %d != hull_dim %d\n", newdim, qh hull_dim);
qh_errexit(qh_ERRqhull, NULL, NULL);
}
if (!(newpoints=(coordT*)qh_malloc(newnum*newdim*sizeof(coordT)))){
qh_fprintf(qh ferr, 6016, "qhull error: insufficient memory to project %d points\n",
qh num_points);
qh_errexit(qh_ERRmem, NULL, NULL);
}
qh_projectpoints(project, qh input_dim+1, qh first_point,
qh num_points, qh input_dim, newpoints, newdim);
trace1((qh ferr, 1003, "qh_projectinput: updating lower and upper_bound\n"));
qh_projectpoints(project, qh input_dim+1, qh lower_bound,
1, qh input_dim+1, qh lower_bound, newdim+1);
qh_projectpoints(project, qh input_dim+1, qh upper_bound,
1, qh input_dim+1, qh upper_bound, newdim+1);
if (qh HALFspace) {
if (!qh feasible_point) {
qh_fprintf(qh ferr, 6017, "qhull internal error (qh_projectinput): HALFspace defined without qh.feasible_point\n");
qh_errexit(qh_ERRqhull, NULL, NULL);
}
qh_projectpoints(project, qh input_dim, qh feasible_point,
1, qh input_dim, qh feasible_point, newdim);
}
qh_memfree(project, (qh input_dim+1)*sizeof(*project));
if (qh POINTSmalloc)
qh_free(qh first_point);
qh first_point= newpoints;
qh POINTSmalloc= True;
if (qh DELAUNAY && qh ATinfinity) {
coord= qh first_point;
infinity= qh first_point + qh hull_dim * qh num_points;
for (k=qh hull_dim-1; k--; )
infinity[k]= 0.0;
for (i=qh num_points; i--; ) {
paraboloid= 0.0;
for (k=0; k < qh hull_dim-1; k++) {
paraboloid += *coord * *coord;
infinity[k] += *coord;
coord++;
}
*(coord++)= paraboloid;
maximize_(maxboloid, paraboloid);
}
/* coord == infinity */
for (k=qh hull_dim-1; k--; )
*(coord++) /= qh num_points;
*(coord++)= maxboloid * 1.1;
qh num_points++;
trace0((qh ferr, 9, "qh_projectinput: projected points to paraboloid for Delaunay\n"));
}else if (qh DELAUNAY) /* !qh ATinfinity */
qh_setdelaunay( qh hull_dim, qh num_points, qh first_point);
} /* projectinput */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="projectpoints">-</a>
qh_projectpoints( project, n, points, numpoints, dim, newpoints, newdim )
project points/numpoints/dim to newpoints/newdim
if project[k] == -1
delete dimension k
if project[k] == 1
add dimension k by duplicating previous column
n is size of project
notes:
newpoints may be points if only adding dimension at end
design:
check that 'project' and 'newdim' agree
for each dimension
if project == -1
skip dimension
else
determine start of column in newpoints
determine start of column in points
if project == +1, duplicate previous column
copy dimension (column) from points to newpoints
*/
void qh_projectpoints(signed char *project, int n, realT *points,
int numpoints, int dim, realT *newpoints, int newdim) {
int testdim= dim, oldk=0, newk=0, i,j=0,k;
realT *newp, *oldp;
for (k=0; k < n; k++)
testdim += project[k];
if (testdim != newdim) {
qh_fprintf(qh ferr, 6018, "qhull internal error (qh_projectpoints): newdim %d should be %d after projection\n",
newdim, testdim);
qh_errexit(qh_ERRqhull, NULL, NULL);
}
for (j=0; j<n; j++) {
if (project[j] == -1)
oldk++;
else {
newp= newpoints+newk++;
if (project[j] == +1) {
if (oldk >= dim)
continue;
oldp= points+oldk;
}else
oldp= points+oldk++;
for (i=numpoints; i--; ) {
*newp= *oldp;
newp += newdim;
oldp += dim;
}
}
if (oldk >= dim)
break;
}
trace1((qh ferr, 1004, "qh_projectpoints: projected %d points from dim %d to dim %d\n",
numpoints, dim, newdim));
} /* projectpoints */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="rotateinput">-</a>
qh_rotateinput( rows )
rotate input using row matrix
input points given by qh first_point, num_points, hull_dim
assumes rows[dim] is a scratch buffer
if qh POINTSmalloc, overwrites input points, else mallocs a new array
returns:
rotated input
sets qh POINTSmalloc
design:
see qh_rotatepoints
*/
void qh_rotateinput(realT **rows) {
if (!qh POINTSmalloc) {
qh first_point= qh_copypoints(qh first_point, qh num_points, qh hull_dim);
qh POINTSmalloc= True;
}
qh_rotatepoints(qh first_point, qh num_points, qh hull_dim, rows);
} /* rotateinput */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="rotatepoints">-</a>
qh_rotatepoints( points, numpoints, dim, row )
rotate numpoints points by a d-dim row matrix
assumes rows[dim] is a scratch buffer
returns:
rotated points in place
design:
for each point
for each coordinate
use row[dim] to compute partial inner product
for each coordinate
rotate by partial inner product
*/
void qh_rotatepoints(realT *points, int numpoints, int dim, realT **row) {
realT *point, *rowi, *coord= NULL, sum, *newval;
int i,j,k;
if (qh IStracing >= 1)
qh_printmatrix(qh ferr, "qh_rotatepoints: rotate points by", row, dim, dim);
for (point= points, j= numpoints; j--; point += dim) {
newval= row[dim];
for (i=0; i < dim; i++) {
rowi= row[i];
coord= point;
for (sum= 0.0, k= dim; k--; )
sum += *rowi++ * *coord++;
*(newval++)= sum;
}
for (k=dim; k--; )
*(--coord)= *(--newval);
}
} /* rotatepoints */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="scaleinput">-</a>
qh_scaleinput()
scale input points using qh low_bound/high_bound
input points given by qh first_point, num_points, hull_dim
if qh POINTSmalloc, overwrites input points, else mallocs a new array
returns:
scales coordinates of points to low_bound[k], high_bound[k]
sets qh POINTSmalloc
design:
see qh_scalepoints
*/
void qh_scaleinput(void) {
if (!qh POINTSmalloc) {
qh first_point= qh_copypoints(qh first_point, qh num_points, qh hull_dim);
qh POINTSmalloc= True;
}
qh_scalepoints(qh first_point, qh num_points, qh hull_dim,
qh lower_bound, qh upper_bound);
} /* scaleinput */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="scalelast">-</a>
qh_scalelast( points, numpoints, dim, low, high, newhigh )
scale last coordinate to [0,m] for Delaunay triangulations
input points given by points, numpoints, dim
returns:
changes scale of last coordinate from [low, high] to [0, newhigh]
overwrites last coordinate of each point
saves low/high/newhigh in qh.last_low, etc. for qh_setdelaunay()
notes:
when called by qh_setdelaunay, low/high may not match actual data
design:
compute scale and shift factors
apply to last coordinate of each point
*/
void qh_scalelast(coordT *points, int numpoints, int dim, coordT low,
coordT high, coordT newhigh) {
realT scale, shift;
coordT *coord;
int i;
boolT nearzero= False;
trace4((qh ferr, 4013, "qh_scalelast: scale last coordinate from [%2.2g, %2.2g] to [0,%2.2g]\n",
low, high, newhigh));
qh last_low= low;
qh last_high= high;
qh last_newhigh= newhigh;
scale= qh_divzero(newhigh, high - low,
qh MINdenom_1, &nearzero);
if (nearzero) {
if (qh DELAUNAY)
qh_fprintf(qh ferr, 6019, "qhull input error: can not scale last coordinate. Input is cocircular\n or cospherical. Use option 'Qz' to add a point at infinity.\n");
else
qh_fprintf(qh ferr, 6020, "qhull input error: can not scale last coordinate. New bounds [0, %2.2g] are too wide for\nexisting bounds [%2.2g, %2.2g] (width %2.2g)\n",
newhigh, low, high, high-low);
qh_errexit(qh_ERRinput, NULL, NULL);
}
shift= - low * newhigh / (high-low);
coord= points + dim - 1;
for (i=numpoints; i--; coord += dim)
*coord= *coord * scale + shift;
} /* scalelast */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="scalepoints">-</a>
qh_scalepoints( points, numpoints, dim, newlows, newhighs )
scale points to new lowbound and highbound
retains old bound when newlow= -REALmax or newhigh= +REALmax
returns:
scaled points
overwrites old points
design:
for each coordinate
compute current low and high bound
compute scale and shift factors
scale all points
enforce new low and high bound for all points
*/
void qh_scalepoints(pointT *points, int numpoints, int dim,
realT *newlows, realT *newhighs) {
int i,k;
realT shift, scale, *coord, low, high, newlow, newhigh, mincoord, maxcoord;
boolT nearzero= False;
for (k=0; k < dim; k++) {
newhigh= newhighs[k];
newlow= newlows[k];
if (newhigh > REALmax/2 && newlow < -REALmax/2)
continue;
low= REALmax;
high= -REALmax;
for (i=numpoints, coord=points+k; i--; coord += dim) {
minimize_(low, *coord);
maximize_(high, *coord);
}
if (newhigh > REALmax/2)
newhigh= high;
if (newlow < -REALmax/2)
newlow= low;
if (qh DELAUNAY && k == dim-1 && newhigh < newlow) {
qh_fprintf(qh ferr, 6021, "qhull input error: 'Qb%d' or 'QB%d' inverts paraboloid since high bound %.2g < low bound %.2g\n",
k, k, newhigh, newlow);
qh_errexit(qh_ERRinput, NULL, NULL);
}
scale= qh_divzero(newhigh - newlow, high - low,
qh MINdenom_1, &nearzero);
if (nearzero) {
qh_fprintf(qh ferr, 6022, "qhull input error: %d'th dimension's new bounds [%2.2g, %2.2g] too wide for\nexisting bounds [%2.2g, %2.2g]\n",
k, newlow, newhigh, low, high);
qh_errexit(qh_ERRinput, NULL, NULL);
}
shift= (newlow * high - low * newhigh)/(high-low);
coord= points+k;
for (i=numpoints; i--; coord += dim)
*coord= *coord * scale + shift;
coord= points+k;
if (newlow < newhigh) {
mincoord= newlow;
maxcoord= newhigh;
}else {
mincoord= newhigh;
maxcoord= newlow;
}
for (i=numpoints; i--; coord += dim) {
minimize_(*coord, maxcoord); /* because of roundoff error */
maximize_(*coord, mincoord);
}
trace0((qh ferr, 10, "qh_scalepoints: scaled %d'th coordinate [%2.2g, %2.2g] to [%.2g, %.2g] for %d points by %2.2g and shifted %2.2g\n",
k, low, high, newlow, newhigh, numpoints, scale, shift));
}
} /* scalepoints */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="setdelaunay">-</a>
qh_setdelaunay( dim, count, points )
project count points to dim-d paraboloid for Delaunay triangulation
dim is one more than the dimension of the input set
assumes dim is at least 3 (i.e., at least a 2-d Delaunay triangulation)
points is a dim*count realT array. The first dim-1 coordinates
are the coordinates of the first input point. array[dim] is
the first coordinate of the second input point. array[2*dim] is
the first coordinate of the third input point.
if qh.last_low defined (i.e., 'Qbb' called qh_scalelast)
calls qh_scalelast to scale the last coordinate the same as the other points
returns:
for each point
sets point[dim-1] to sum of squares of coordinates
scale points to 'Qbb' if needed
notes:
to project one point, use
qh_setdelaunay(qh hull_dim, 1, point)
Do not use options 'Qbk', 'QBk', or 'QbB' since they scale
the coordinates after the original projection.
*/
void qh_setdelaunay(int dim, int count, pointT *points) {
int i, k;
coordT *coordp, coord;
realT paraboloid;
trace0((qh ferr, 11, "qh_setdelaunay: project %d points to paraboloid for Delaunay triangulation\n", count));
coordp= points;
for (i=0; i < count; i++) {
coord= *coordp++;
paraboloid= coord*coord;
for (k=dim-2; k--; ) {
coord= *coordp++;
paraboloid += coord*coord;
}
*coordp++ = paraboloid;
}
if (qh last_low < REALmax/2)
qh_scalelast(points, count, dim, qh last_low, qh last_high, qh last_newhigh);
} /* setdelaunay */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="sethalfspace">-</a>
qh_sethalfspace( dim, coords, nextp, normal, offset, feasible )
set point to dual of halfspace relative to feasible point
halfspace is normal coefficients and offset.
returns:
false if feasible point is outside of hull (error message already reported)
overwrites coordinates for point at dim coords
nextp= next point (coords)
design:
compute distance from feasible point to halfspace
divide each normal coefficient by -dist
*/
boolT qh_sethalfspace(int dim, coordT *coords, coordT **nextp,
coordT *normal, coordT *offset, coordT *feasible) {
coordT *normp= normal, *feasiblep= feasible, *coordp= coords;
realT dist;
realT r; /*bug fix*/
int k;
boolT zerodiv;
dist= *offset;
for (k=dim; k--; )
dist += *(normp++) * *(feasiblep++);
if (dist > 0)
goto LABELerroroutside;
normp= normal;
if (dist < -qh MINdenom) {
for (k=dim; k--; )
*(coordp++)= *(normp++) / -dist;
}else {
for (k=dim; k--; ) {
*(coordp++)= qh_divzero(*(normp++), -dist, qh MINdenom_1, &zerodiv);
if (zerodiv)
goto LABELerroroutside;
}
}
*nextp= coordp;
if (qh IStracing >= 4) {
qh_fprintf(qh ferr, 8021, "qh_sethalfspace: halfspace at offset %6.2g to point: ", *offset);
for (k=dim, coordp=coords; k--; ) {
r= *coordp++;
qh_fprintf(qh ferr, 8022, " %6.2g", r);
}
qh_fprintf(qh ferr, 8023, "\n");
}
return True;
LABELerroroutside:
feasiblep= feasible;
normp= normal;
qh_fprintf(qh ferr, 6023, "qhull input error: feasible point is not clearly inside halfspace\nfeasible point: ");
for (k=dim; k--; )
qh_fprintf(qh ferr, 8024, qh_REAL_1, r=*(feasiblep++));
qh_fprintf(qh ferr, 8025, "\n halfspace: ");
for (k=dim; k--; )
qh_fprintf(qh ferr, 8026, qh_REAL_1, r=*(normp++));
qh_fprintf(qh ferr, 8027, "\n at offset: ");
qh_fprintf(qh ferr, 8028, qh_REAL_1, *offset);
qh_fprintf(qh ferr, 8029, " and distance: ");
qh_fprintf(qh ferr, 8030, qh_REAL_1, dist);
qh_fprintf(qh ferr, 8031, "\n");
return False;
} /* sethalfspace */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="sethalfspace_all">-</a>
qh_sethalfspace_all( dim, count, halfspaces, feasible )
generate dual for halfspace intersection with feasible point
array of count halfspaces
each halfspace is normal coefficients followed by offset
the origin is inside the halfspace if the offset is negative
returns:
malloc'd array of count X dim-1 points
notes:
call before qh_init_B or qh_initqhull_globals
unused/untested code: please email bradb@shore.net if this works ok for you
If using option 'Fp', also set qh feasible_point. It is a malloc'd array
that is freed by qh_freebuffers.
design:
see qh_sethalfspace
*/
coordT *qh_sethalfspace_all(int dim, int count, coordT *halfspaces, pointT *feasible) {
int i, newdim;
pointT *newpoints;
coordT *coordp, *normalp, *offsetp;
trace0((qh ferr, 12, "qh_sethalfspace_all: compute dual for halfspace intersection\n"));
newdim= dim - 1;
if (!(newpoints=(coordT*)qh_malloc(count*newdim*sizeof(coordT)))){
qh_fprintf(qh ferr, 6024, "qhull error: insufficient memory to compute dual of %d halfspaces\n",
count);
qh_errexit(qh_ERRmem, NULL, NULL);
}
coordp= newpoints;
normalp= halfspaces;
for (i=0; i < count; i++) {
offsetp= normalp + newdim;
if (!qh_sethalfspace(newdim, coordp, &coordp, normalp, offsetp, feasible)) {
qh_fprintf(qh ferr, 8032, "The halfspace was at index %d\n", i);
qh_errexit(qh_ERRinput, NULL, NULL);
}
normalp= offsetp + 1;
}
return newpoints;
} /* sethalfspace_all */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="sharpnewfacets">-</a>
qh_sharpnewfacets()
returns:
true if could be an acute angle (facets in different quadrants)
notes:
for qh_findbest
design:
for all facets on qh.newfacet_list
if two facets are in different quadrants
set issharp
*/
boolT qh_sharpnewfacets() {
facetT *facet;
boolT issharp = False;
int *quadrant, k;
quadrant= (int*)qh_memalloc(qh hull_dim * sizeof(int));
FORALLfacet_(qh newfacet_list) {
if (facet == qh newfacet_list) {
for (k=qh hull_dim; k--; )
quadrant[ k]= (facet->normal[ k] > 0);
}else {
for (k=qh hull_dim; k--; ) {
if (quadrant[ k] != (facet->normal[ k] > 0)) {
issharp= True;
break;
}
}
}
if (issharp)
break;
}
qh_memfree( quadrant, qh hull_dim * sizeof(int));
trace3((qh ferr, 3001, "qh_sharpnewfacets: %d\n", issharp));
return issharp;
} /* sharpnewfacets */
/*-<a href="qh-geom.htm#TOC"
>-------------------------------</a><a name="voronoi_center">-</a>
qh_voronoi_center( dim, points )
return Voronoi center for a set of points
dim is the orginal dimension of the points
gh.gm_matrix/qh.gm_row are scratch buffers
returns:
center as a temporary point
if non-simplicial,
returns center for max simplex of points
notes:
from Bowyer & Woodwark, A Programmer's Geometry, 1983, p. 65
design:
if non-simplicial
determine max simplex for points
translate point0 of simplex to origin
compute sum of squares of diagonal
compute determinate
compute Voronoi center (see Bowyer & Woodwark)
*/
pointT *qh_voronoi_center(int dim, setT *points) {
pointT *point, **pointp, *point0;
pointT *center= (pointT*)qh_memalloc(qh center_size);
setT *simplex;
int i, j, k, size= qh_setsize(points);
coordT *gmcoord;
realT *diffp, sum2, *sum2row, *sum2p, det, factor;
boolT nearzero, infinite;
if (size == dim+1)
simplex= points;
else if (size < dim+1) {
qh_fprintf(qh ferr, 6025, "qhull internal error (qh_voronoi_center):\n need at least %d points to construct a Voronoi center\n",
dim+1);
qh_errexit(qh_ERRqhull, NULL, NULL);
simplex= points; /* never executed -- avoids warning */
}else {
simplex= qh_settemp(dim+1);
qh_maxsimplex(dim, points, NULL, 0, &simplex);
}
point0= SETfirstt_(simplex, pointT);
gmcoord= qh gm_matrix;
for (k=0; k < dim; k++) {
qh gm_row[k]= gmcoord;
FOREACHpoint_(simplex) {
if (point != point0)
*(gmcoord++)= point[k] - point0[k];
}
}
sum2row= gmcoord;
for (i=0; i < dim; i++) {
sum2= 0.0;
for (k=0; k < dim; k++) {
diffp= qh gm_row[k] + i;
sum2 += *diffp * *diffp;
}
*(gmcoord++)= sum2;
}
det= qh_determinant(qh gm_row, dim, &nearzero);
factor= qh_divzero(0.5, det, qh MINdenom, &infinite);
if (infinite) {
for (k=dim; k--; )
center[k]= qh_INFINITE;
if (qh IStracing)
qh_printpoints(qh ferr, "qh_voronoi_center: at infinity for ", simplex);
}else {
for (i=0; i < dim; i++) {
gmcoord= qh gm_matrix;
sum2p= sum2row;
for (k=0; k < dim; k++) {
qh gm_row[k]= gmcoord;
if (k == i) {
for (j=dim; j--; )
*(gmcoord++)= *sum2p++;
}else {
FOREACHpoint_(simplex) {
if (point != point0)
*(gmcoord++)= point[k] - point0[k];
}
}
}
center[i]= qh_determinant(qh gm_row, dim, &nearzero)*factor + point0[i];
}
#ifndef qh_NOtrace
if (qh IStracing >= 3) {
qh_fprintf(qh ferr, 8033, "qh_voronoi_center: det %2.2g factor %2.2g ", det, factor);
qh_printmatrix(qh ferr, "center:", &center, 1, dim);
if (qh IStracing >= 5) {
qh_printpoints(qh ferr, "points", simplex);
FOREACHpoint_(simplex)
qh_fprintf(qh ferr, 8034, "p%d dist %.2g, ", qh_pointid(point),
qh_pointdist(point, center, dim));
qh_fprintf(qh ferr, 8035, "\n");
}
}
#endif
}
if (simplex != points)
qh_settempfree(&simplex);
return center;
} /* voronoi_center */

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