/* ---------------------------------
qhull.c
Quickhull algorithm for convex hulls
qhull() and top-level routines
see qh-qhull.htm, qhull.h, unix.c
see qhull_a.h for internal functions
copyright (c) 1993-2002 The Geometry Center
*/
#include "qhull_a.h"
/*============= functions in alphabetic order after qhull() =======*/
/*---------------------------------
qh_qhull()
compute DIM3 convex hull of qh.num_points starting at qh.first_point
qh contains all global options and variables
returns:
returns polyhedron
qh.facet_list, qh.num_facets, qh.vertex_list, qh.num_vertices,
returns global variables
qh.hulltime, qh.max_outside, qh.interior_point, qh.max_vertex, qh.min_vertex
returns precision constants
qh.ANGLEround, centrum_radius, cos_max, DISTround, MAXabs_coord, ONEmerge
notes:
unless needed for output
qh.max_vertex and qh.min_vertex are max/min due to merges
see:
to add individual points to either qh.num_points
use qh_addpoint()
if qh.GETarea
qh_produceoutput() returns qh.totarea and qh.totvol via qh_getarea()
design:
record starting time
initialize hull and partition points
build convex hull
unless early termination
update facet->maxoutside for vertices, coplanar, and near-inside points
error if temporary sets exist
record end time
*/
void qh_qhull (void) {
int numoutside;
qh hulltime= qh_CPUclock;
if (qh RERUN || qh JOGGLEmax < REALmax/2)
qh_build_withrestart();
else {
qh_initbuild();
qh_buildhull();
}
if (!qh STOPpoint && !qh STOPcone) {
if (qh ZEROall_ok && !qh TESTvneighbors && qh MERGEexact)
qh_checkzero( qh_ALL);
if (qh ZEROall_ok && !qh TESTvneighbors && !qh WAScoplanar) {
trace2((qh ferr, "qh_qhull: all facets are clearly convex and no coplanar points. Post-merging and check of maxout not needed.\n"));
qh DOcheckmax= False;
}else {
if (qh MERGEexact || (qh hull_dim > qh_DIMreduceBuild && qh PREmerge))
qh_postmerge ("First post-merge", qh premerge_centrum, qh premerge_cos,
(qh POSTmerge ? False : qh TESTvneighbors));
else if (!qh POSTmerge && qh TESTvneighbors)
qh_postmerge ("For testing vertex neighbors", qh premerge_centrum,
qh premerge_cos, True);
if (qh POSTmerge)
qh_postmerge ("For post-merging", qh postmerge_centrum,
qh postmerge_cos, qh TESTvneighbors);
if (qh visible_list == qh facet_list) { /* i.e., merging done */
qh findbestnew= True;
qh_partitionvisible (/*visible_list, newfacet_list*/ !qh_ALL, &numoutside);
qh findbestnew= False;
qh_deletevisible (/*qh visible_list*/);
qh_resetlists (False, qh_RESETvisible /*qh visible_list newvertex_list newfacet_list */);
}
}
if (qh DOcheckmax){
if (qh REPORTfreq) {
qh_buildtracing (NULL, NULL);
fprintf (qh ferr, "\nTesting all coplanar points.\n");
}
qh_check_maxout();
}
if (qh KEEPnearinside && !qh maxoutdone)
qh_nearcoplanar();
}
if (qh_setsize ((setT*)qhmem.tempstack) != 0) {
fprintf (qh ferr, "qhull internal error (qh_qhull): temporary sets not empty (%d)\n",
qh_setsize ((setT*)qhmem.tempstack));
qh_errexit (qh_ERRqhull, NULL, NULL);
}
qh hulltime= qh_CPUclock - qh hulltime;
qh QHULLfinished= True;
trace1((qh ferr, "qh_qhull: algorithm completed\n"));
} /* qhull */
/*---------------------------------
qh_addpoint( furthest, facet, checkdist )
add point (usually furthest point) above facet to hull
if checkdist,
check that point is above facet.
if point is not outside of the hull, uses qh_partitioncoplanar()
assumes that facet is defined by qh_findbestfacet()
else if facet specified,
assumes that point is above facet (major damage if below)
for Delaunay triangulations,
Use qh_setdelaunay() to lift point to paraboloid and scale by 'Qbb' if needed
Do not use options 'Qbk', 'QBk', or 'QbB' since they scale the coordinates.
returns:
returns False if user requested an early termination
qh.visible_list, newfacet_list, delvertex_list, NEWfacets may be defined
updates qh.facet_list, qh.num_facets, qh.vertex_list, qh.num_vertices
clear qh.maxoutdone (will need to call qh_check_maxout() for facet->maxoutside)
if unknown point, adds a pointer to qh.other_points
do not deallocate the point's coordinates
notes:
assumes point is near its best facet and not at a local minimum of a lens
distributions. Use qh_findbestfacet to avoid this case.
uses qh.visible_list, qh.newfacet_list, qh.delvertex_list, qh.NEWfacets
see also:
qh_triangulate() -- triangulate non-simplicial facets
design:
check point in qh.first_point/.num_points
if checkdist
if point not above facet
partition coplanar point
exit
exit if pre STOPpoint requested
find horizon and visible facets for point
make new facets for point to horizon
make hyperplanes for point
compute balance statistics
match neighboring new facets
update vertex neighbors and delete interior vertices
exit if STOPcone requested
merge non-convex new facets
if merge found, many merges, or 'Qf'
use qh_findbestnew() instead of qh_findbest()
partition outside points from visible facets
delete visible facets
check polyhedron if requested
exit if post STOPpoint requested
reset working lists of facets and vertices
*/
boolT qh_addpoint (pointT *furthest, facetT *facet, boolT checkdist) {
int goodvisible, goodhorizon;
vertexT *vertex;
facetT *newfacet;
realT dist, newbalance, pbalance;
boolT isoutside= False;
int numpart, numpoints, numnew, firstnew;
qh maxoutdone= False;
if (qh_pointid (furthest) == -1)
qh_setappend (&qh other_points, furthest);
if (!facet) {
fprintf (qh ferr, "qh_addpoint: NULL facet. Need to call qh_findbestfacet first\n");
qh_errexit (qh_ERRqhull, NULL, NULL);
}
if (checkdist) {
facet= qh_findbest (furthest, facet, !qh_ALL, !qh_ISnewfacets, !qh_NOupper,
&dist, &isoutside, &numpart);
zzadd_(Zpartition, numpart);
if (!isoutside) {
zinc_(Znotmax); /* last point of outsideset is no longer furthest. */
facet->notfurthest= True;
qh_partitioncoplanar (furthest, facet, &dist);
return True;
}
}
qh_buildtracing (furthest, facet);
if (qh STOPpoint < 0 && qh furthest_id == -qh STOPpoint-1) {
facet->notfurthest= True;
return False;
}
qh_findhorizon (furthest, facet, &goodvisible, &goodhorizon);
if (qh ONLYgood && !(goodvisible+goodhorizon) && !qh GOODclosest) {
zinc_(Znotgood);
facet->notfurthest= True;
/* last point of outsideset is no longer furthest. This is ok
since all points of the outside are likely to be bad */
qh_resetlists (False, qh_RESETvisible /*qh visible_list newvertex_list newfacet_list */);
return True;
}
zzinc_(Zprocessed);
firstnew= qh facet_id;
vertex= qh_makenewfacets (furthest /*visible_list, attaches if !ONLYgood */);
qh_makenewplanes (/* newfacet_list */);
numnew= qh facet_id - firstnew;
newbalance= numnew - (realT) (qh num_facets-qh num_visible)
* qh hull_dim/qh num_vertices;
wadd_(Wnewbalance, newbalance);
wadd_(Wnewbalance2, newbalance * newbalance);
if (qh ONLYgood
&& !qh_findgood (qh newfacet_list, goodhorizon) && !qh GOODclosest) {
FORALLnew_facets
qh_delfacet (newfacet);
qh_delvertex (vertex);
qh_resetlists (True, qh_RESETvisible /*qh visible_list newvertex_list newfacet_list */);
zinc_(Znotgoodnew);
facet->notfurthest= True;
return True;
}
if (qh ONLYgood)
qh_attachnewfacets(/*visible_list*/);
qh_matchnewfacets();
qh_updatevertices();
if (qh STOPcone && qh furthest_id == qh STOPcone-1) {
facet->notfurthest= True;
return False; /* visible_list etc. still defined */
}
qh findbestnew= False;
if (qh PREmerge || qh MERGEexact) {
qh_premerge (vertex, qh premerge_centrum, qh premerge_cos);
if (qh_USEfindbestnew)
qh findbestnew= True;
else {
FORALLnew_facets {
if (!newfacet->simplicial) {
qh findbestnew= True; /* use qh_findbestnew instead of qh_findbest*/
break;
}
}
}
}else if (qh BESToutside)
qh findbestnew= True;
qh_partitionvisible (/*visible_list, newfacet_list*/ !qh_ALL, &numpoints);
qh findbestnew= False;
qh findbest_notsharp= False;
zinc_(Zpbalance);
pbalance= numpoints - (realT) qh hull_dim /* assumes all points extreme */
* (qh num_points - qh num_vertices)/qh num_vertices;
wadd_(Wpbalance, pbalance);
wadd_(Wpbalance2, pbalance * pbalance);
qh_deletevisible (/*qh visible_list*/);
zmax_(Zmaxvertex, qh num_vertices);
qh NEWfacets= False;
if (qh IStracing >= 4) {
if (qh num_facets < 2000)
qh_printlists();
qh_printfacetlist (qh newfacet_list, NULL, True);
qh_checkpolygon (qh facet_list);
}else if (qh CHECKfrequently) {
if (qh num_facets < 50)
qh_checkpolygon (qh facet_list);
else
qh_checkpolygon (qh newfacet_list);
}
if (qh STOPpoint > 0 && qh furthest_id == qh STOPpoint-1)
return False;
qh_resetlists (True, qh_RESETvisible /*qh visible_list newvertex_list newfacet_list */);
/* qh_triangulate(); to test qh.TRInormals */
trace2((qh ferr, "qh_addpoint: added p%d new facets %d new balance %2.2g point balance %2.2g\n",
qh_pointid (furthest), numnew, newbalance, pbalance));
return True;
} /* addpoint */
/*---------------------------------
qh_build_withrestart()
allow restarts due to qh.JOGGLEmax while calling qh_buildhull()
qh.FIRSTpoint/qh.NUMpoints is point array
it may be moved by qh_joggleinput()
*/
void qh_build_withrestart (void) {
int restart;
qh ALLOWrestart= True;
while (True) {
restart= setjmp (qh restartexit); /* simple statement for CRAY J916 */
if (restart) { /* only from qh_precision() */
zzinc_(Zretry);
wmax_(Wretrymax, qh JOGGLEmax);
qh ERREXITcalled= False;
qh STOPcone= True; /* if break, prevents normal output */
}
if (!qh RERUN && qh JOGGLEmax < REALmax/2) {
if (qh build_cnt > qh_JOGGLEmaxretry) {
fprintf(qh ferr, "\n\
qhull precision error: %d attempts to construct a convex hull\n\
with joggled input. Increase joggle above 'QJ%2.2g'\n\
or modify qh_JOGGLE... parameters in user.h\n",
qh build_cnt, qh JOGGLEmax);
qh_errexit (qh_ERRqhull, NULL, NULL);
}
if (qh build_cnt && !restart)
break;
}else if (qh build_cnt && qh build_cnt >= qh RERUN)
break;
qh STOPcone= False;
qh_freebuild (True); /* first call is a nop */
qh build_cnt++;
if (!qh qhull_optionsiz)
qh qhull_optionsiz= strlen (qh qhull_options);
else {
qh qhull_options [qh qhull_optionsiz]= '\0';
qh qhull_optionlen= 80;
}
qh_option("_run", &qh build_cnt, NULL);
if (qh build_cnt == qh RERUN) {
qh IStracing= qh TRACElastrun; /* duplicated from qh_initqhull_globals */
if (qh TRACEpoint != -1 || qh TRACEdist < REALmax/2 || qh TRACEmerge) {
qh TRACElevel= (qh IStracing? qh IStracing : 3);
qh IStracing= 0;
}
qhmem.IStracing= qh IStracing;
}
if (qh JOGGLEmax < REALmax/2)
qh_joggleinput();
qh_initbuild();
qh_buildhull();
if (qh JOGGLEmax < REALmax/2 && !qh MERGING)
qh_checkconvex (qh facet_list, qh_ALGORITHMfault);
}
qh ALLOWrestart= False;
} /* qh_build_withrestart */
/*---------------------------------
qh_buildhull()
construct a convex hull by adding outside points one at a time
returns:
notes:
may be called multiple times
checks facet and vertex lists for incorrect flags
to recover from STOPcone, call qh_deletevisible and qh_resetlists
design:
check visible facet and newfacet flags
check newlist vertex flags and qh.STOPcone/STOPpoint
for each facet with a furthest outside point
add point to facet
exit if qh.STOPcone or qh.STOPpoint requested
if qh.NARROWhull for initial simplex
partition remaining outside points to coplanar sets
*/
void qh_buildhull(void) {
facetT *facet;
pointT *furthest;
vertexT *vertex;
int id;
trace1((qh ferr, "qh_buildhull: start build hull\n"));
FORALLfacets {
if (facet->visible || facet->newfacet) {
fprintf (qh ferr, "qhull internal error (qh_buildhull): visible or new facet f%d in facet list\n",
facet->id);
qh_errexit (qh_ERRqhull, facet, NULL);
}
}
FORALLvertices {
if (vertex->newlist) {
fprintf (qh ferr, "qhull internal error (qh_buildhull): new vertex f%d in vertex list\n",
vertex->id);
qh_errprint ("ERRONEOUS", NULL, NULL, NULL, vertex);
qh_errexit (qh_ERRqhull, NULL, NULL);
}
id= qh_pointid (vertex->point);
if ((qh STOPpoint>0 && id == qh STOPpoint-1) ||
(qh STOPpoint<0 && id == -qh STOPpoint-1) ||
(qh STOPcone>0 && id == qh STOPcone-1)) {
trace1((qh ferr,"qh_buildhull: stop point or cone P%d in initial hull\n", id));
return;
}
}
qh facet_next= qh facet_list; /* advance facet when processed */
while ((furthest= qh_nextfurthest (&facet))) {
qh num_outside--; /* if ONLYmax, furthest may not be outside */
if (!qh_addpoint (furthest, facet, qh ONLYmax))
break;
}
if (qh NARROWhull) /* move points from outsideset to coplanarset */
qh_outcoplanar( /* facet_list */ );
if (qh num_outside && !furthest) {
fprintf (qh ferr, "qhull internal error (qh_buildhull): %d outside points were never processed.\n", qh num_outside);
qh_errexit (qh_ERRqhull, NULL, NULL);
}
trace1((qh ferr, "qh_buildhull: completed the hull construction\n"));
} /* buildhull */
/*---------------------------------
qh_buildtracing( furthest, facet )
trace an iteration of qh_buildhull() for furthest point and facet
if !furthest, prints progress message
returns:
tracks progress with qh.lastreport
updates qh.furthest_id (-3 if furthest is NULL)
also resets visit_id, vertext_visit on wrap around
see:
qh_tracemerging()
design:
if !furthest
print progress message
exit
if 'TFn' iteration
print progress message
else if tracing
trace furthest point and facet
reset qh.visit_id and qh.vertex_visit if overflow may occur
set qh.furthest_id for tracing
*/
void qh_buildtracing (pointT *furthest, facetT *facet) {
realT dist= 0;
float cpu;
int total, furthestid;
time_t timedata;
struct tm *tp;
vertexT *vertex;
qh old_randomdist= qh RANDOMdist;
qh RANDOMdist= False;
if (!furthest) {
time (&timedata);
tp= localtime (&timedata);
cpu= qh_CPUclock - qh hulltime;
cpu /= qh_SECticks;
total= zzval_(Ztotmerge) - zzval_(Zcyclehorizon) + zzval_(Zcyclefacettot);
fprintf (qh ferr, "\n\
At %02d:%02d:%02d & %2.5g CPU secs, qhull has created %d facets and merged %d.\n\
The current hull contains %d facets and %d vertices. Last point was p%d\n",
tp->tm_hour, tp->tm_min, tp->tm_sec, cpu, qh facet_id -1,
total, qh num_facets, qh num_vertices, qh furthest_id);
return;
}
furthestid= qh_pointid (furthest);
if (qh TRACEpoint == furthestid) {
qh IStracing= qh TRACElevel;
qhmem.IStracing= qh TRACElevel;
}else if (qh TRACEpoint != -1 && qh TRACEdist < REALmax/2) {
qh IStracing= 0;
qhmem.IStracing= 0;
}
if (qh REPORTfreq && (qh facet_id-1 > qh lastreport+qh REPORTfreq)) {
qh lastreport= qh facet_id-1;
time (&timedata);
tp= localtime (&timedata);
cpu= qh_CPUclock - qh hulltime;
cpu /= qh_SECticks;
total= zzval_(Ztotmerge) - zzval_(Zcyclehorizon) + zzval_(Zcyclefacettot);
zinc_(Zdistio);
qh_distplane (furthest, facet, &dist);
fprintf (qh ferr, "\n\
At %02d:%02d:%02d & %2.5g CPU secs, qhull has created %d facets and merged %d.\n\
The current hull contains %d facets and %d vertices. There are %d\n\
outside points. Next is point p%d (v%d), %2.2g above f%d.\n",
tp->tm_hour, tp->tm_min, tp->tm_sec, cpu, qh facet_id -1,
total, qh num_facets, qh num_vertices, qh num_outside+1,
furthestid, qh vertex_id, dist, getid_(facet));
}else if (qh IStracing >=1) {
cpu= qh_CPUclock - qh hulltime;
cpu /= qh_SECticks;
qh_distplane (furthest, facet, &dist);
fprintf (qh ferr, "qh_addpoint: add p%d (v%d) to hull of %d facets (%2.2g above f%d) and %d outside at %4.4g CPU secs. Previous was p%d.\n",
furthestid, qh vertex_id, qh num_facets, dist,
getid_(facet), qh num_outside+1, cpu, qh furthest_id);
}
if (qh visit_id > (unsigned) INT_MAX) {
qh visit_id= 0;
FORALLfacets
facet->visitid= qh visit_id;
}
if (qh vertex_visit > (unsigned) INT_MAX) {
qh vertex_visit= 0;
FORALLvertices
vertex->visitid= qh vertex_visit;
}
qh furthest_id= furthestid;
qh RANDOMdist= qh old_randomdist;
} /* buildtracing */
/*---------------------------------
qh_errexit2( exitcode, facet, otherfacet )
return exitcode to system after an error
report two facets
returns:
assumes exitcode non-zero
see:
normally use qh_errexit() in user.c (reports a facet and a ridge)
*/
void qh_errexit2(int exitcode, facetT *facet, facetT *otherfacet) {
qh_errprint("ERRONEOUS", facet, otherfacet, NULL, NULL);
qh_errexit (exitcode, NULL, NULL);
} /* errexit2 */
/*---------------------------------
qh_findhorizon( point, facet, goodvisible, goodhorizon )
given a visible facet, find the point's horizon and visible facets
for all facets, !facet-visible
returns:
returns qh.visible_list/num_visible with all visible facets
marks visible facets with ->visible
updates count of good visible and good horizon facets
updates qh.max_outside, qh.max_vertex, facet->maxoutside
see:
similar to qh_delpoint()
design:
move facet to qh.visible_list at end of qh.facet_list
for all visible facets
for each unvisited neighbor of a visible facet
compute distance of point to neighbor
if point above neighbor
move neighbor to end of qh.visible_list
else if point is coplanar with neighbor
update qh.max_outside, qh.max_vertex, neighbor->maxoutside
mark neighbor coplanar (will create a samecycle later)
update horizon statistics
*/
void qh_findhorizon(pointT *point, facetT *facet, int *goodvisible, int *goodhorizon) {
facetT *neighbor, **neighborp, *visible;
int numhorizon= 0, coplanar= 0;
realT dist;
trace1((qh ferr,"qh_findhorizon: find horizon for point p%d facet f%d\n",qh_pointid(point),facet->id));
*goodvisible= *goodhorizon= 0;
zinc_(Ztotvisible);
qh_removefacet(facet); /* visible_list at end of qh facet_list */
qh_appendfacet(facet);
qh num_visible= 1;
if (facet->good)
(*goodvisible)++;
qh visible_list= facet;
facet->visible= True;
facet->f.replace= NULL;
if (qh IStracing >=4)
qh_errprint ("visible", facet, NULL, NULL, NULL);
qh visit_id++;
FORALLvisible_facets {
if (visible->tricoplanar && !qh TRInormals) {
fprintf (qh ferr, "qh_findhorizon: does not work for tricoplanar facets. Use option 'Q11'\n");
qh_errexit (qh_ERRqhull, visible, NULL);
}
visible->visitid= qh visit_id;
FOREACHneighbor_(visible) {
if (neighbor->visitid == qh visit_id)
continue;
neighbor->visitid= qh visit_id;
zzinc_(Znumvisibility);
qh_distplane(point, neighbor, &dist);
if (dist > qh MINvisible) {
zinc_(Ztotvisible);
qh_removefacet(neighbor); /* append to end of qh visible_list */
qh_appendfacet(neighbor);
neighbor->visible= True;
neighbor->f.replace= NULL;
qh num_visible++;
if (neighbor->good)
(*goodvisible)++;
if (qh IStracing >=4)
qh_errprint ("visible", neighbor, NULL, NULL, NULL);
}else {
if (dist > - qh MAXcoplanar) {
neighbor->coplanar= True;
zzinc_(Zcoplanarhorizon);
qh_precision ("coplanar horizon");
coplanar++;
if (qh MERGING) {
if (dist > 0) {
maximize_(qh max_outside, dist);
maximize_(qh max_vertex, dist);
#if qh_MAXoutside
maximize_(neighbor->maxoutside, dist);
#endif
}else
minimize_(qh min_vertex, dist); /* due to merge later */
}
trace2((qh ferr, "qh_findhorizon: point p%d is coplanar to horizon f%d, dist=%2.7g < qh MINvisible (%2.7g)\n",
qh_pointid(point), neighbor->id, dist, qh MINvisible));
}else
neighbor->coplanar= False;
zinc_(Ztothorizon);
numhorizon++;
if (neighbor->good)
(*goodhorizon)++;
if (qh IStracing >=4)
qh_errprint ("horizon", neighbor, NULL, NULL, NULL);
}
}
}
if (!numhorizon) {
qh_precision ("empty horizon");
fprintf(qh ferr, "qhull precision error (qh_findhorizon): empty horizon\n\
Point p%d was above all facets.\n", qh_pointid(point));
qh_printfacetlist (qh facet_list, NULL, True);
qh_errexit(qh_ERRprec, NULL, NULL);
}
trace1((qh ferr, "qh_findhorizon: %d horizon facets (good %d), %d visible (good %d), %d coplanar\n",
numhorizon, *goodhorizon, qh num_visible, *goodvisible, coplanar));
if (qh IStracing >= 4 && qh num_facets < 50)
qh_printlists ();
} /* findhorizon */
/*---------------------------------
qh_nextfurthest( visible )
returns next furthest point and visible facet for qh_addpoint()
starts search at qh.facet_next
returns:
removes furthest point from outside set
NULL if none available
advances qh.facet_next over facets with empty outside sets
design:
for each facet from qh.facet_next
if empty outside set
advance qh.facet_next
else if qh.NARROWhull
determine furthest outside point
if furthest point is not outside
advance qh.facet_next (point will be coplanar)
remove furthest point from outside set
*/
pointT *qh_nextfurthest (facetT **visible) {
facetT *facet;
int size, index;
realT randr, dist;
pointT *furthest;
while ((facet= qh facet_next) != qh facet_tail) {
if (!facet->outsideset) {
qh facet_next= facet->next;
continue;
}
SETreturnsize_(facet->outsideset, size);
if (!size) {
qh_setfree (&facet->outsideset);
qh facet_next= facet->next;
continue;
}
if (qh NARROWhull) {
if (facet->notfurthest)
qh_furthestout (facet);
furthest= (pointT*)qh_setlast (facet->outsideset);
#if qh_COMPUTEfurthest
qh_distplane (furthest, facet, &dist);
zinc_(Zcomputefurthest);
#else
dist= facet->furthestdist;
#endif
if (dist < qh MINoutside) { /* remainder of outside set is coplanar for qh_outcoplanar */
qh facet_next= facet->next;
continue;
}
}
if (!qh RANDOMoutside && !qh VIRTUALmemory) {
if (qh PICKfurthest) {
qh_furthestnext (/* qh facet_list */);
facet= qh facet_next;
}
*visible= facet;
return ((pointT*)qh_setdellast (facet->outsideset));
}
if (qh RANDOMoutside) {
int outcoplanar = 0;
if (qh NARROWhull) {
FORALLfacets {
if (facet == qh facet_next)
break;
if (facet->outsideset)
outcoplanar += qh_setsize( facet->outsideset);
}
}
randr= qh_RANDOMint;
randr= randr/(qh_RANDOMmax+1);
index= (int)floor((qh num_outside - outcoplanar) * randr);
FORALLfacet_(qh facet_next) {
if (facet->outsideset) {
SETreturnsize_(facet->outsideset, size);
if (!size)
qh_setfree (&facet->outsideset);
else if (size > index) {
*visible= facet;
return ((pointT*)qh_setdelnth (facet->outsideset, index));
}else
index -= size;
}
}
fprintf (qh ferr, "qhull internal error (qh_nextfurthest): num_outside %d is too low\nby at least %d, or a random real %g >= 1.0\n",
qh num_outside, index+1, randr);
qh_errexit (qh_ERRqhull, NULL, NULL);
}else { /* VIRTUALmemory */
facet= qh facet_tail->previous;
if (!(furthest= (pointT*)qh_setdellast(facet->outsideset))) {
if (facet->outsideset)
qh_setfree (&facet->outsideset);
qh_removefacet (facet);
qh_prependfacet (facet, &qh facet_list);
continue;
}
*visible= facet;
return furthest;
}
}
return NULL;
} /* nextfurthest */
/*---------------------------------
qh_partitionall( vertices, points, numpoints )
partitions all points in points/numpoints to the outsidesets of facets
vertices= vertices in qh.facet_list (not partitioned)
returns:
builds facet->outsideset
does not partition qh.GOODpoint
if qh.ONLYgood && !qh.MERGING,
does not partition qh.GOODvertex
notes:
faster if qh.facet_list sorted by anticipated size of outside set
design:
initialize pointset with all points
remove vertices from pointset
remove qh.GOODpointp from pointset (unless it's qh.STOPcone or qh.STOPpoint)
for all facets
for all remaining points in pointset
compute distance from point to facet
if point is outside facet
remove point from pointset (by not reappending)
update bestpoint
append point or old bestpoint to facet's outside set
append bestpoint to facet's outside set (furthest)
for all points remaining in pointset
partition point into facets' outside sets and coplanar sets
*/
void qh_partitionall(setT *vertices, pointT *points, int numpoints){
setT *pointset;
vertexT *vertex, **vertexp;
pointT *point, **pointp, *bestpoint;
int size, point_i, point_n, point_end, remaining, i, id;
facetT *facet;
realT bestdist= -REALmax, dist, distoutside;
trace1((qh ferr, "qh_partitionall: partition all points into outside sets\n"));
pointset= qh_settemp (numpoints);
qh num_outside= 0;
pointp= SETaddr_(pointset, pointT);
for (i=numpoints, point= points; i--; point += qh hull_dim)
*(pointp++)= point;
qh_settruncate (pointset, numpoints);
FOREACHvertex_(vertices) {
if ((id= qh_pointid(vertex->point)) >= 0)
SETelem_(pointset, id)= NULL;
}
id= qh_pointid (qh GOODpointp);
if (id >=0 && qh STOPcone-1 != id && -qh STOPpoint-1 != id)
SETelem_(pointset, id)= NULL;
if (qh GOODvertexp && qh ONLYgood && !qh MERGING) { /* matches qhull()*/
if ((id= qh_pointid(qh GOODvertexp)) >= 0)
SETelem_(pointset, id)= NULL;
}
if (!qh BESToutside) { /* matches conditional for qh_partitionpoint below */
distoutside= qh_DISToutside; /* multiple of qh.MINoutside & qh.max_outside, see user.h */
zval_(Ztotpartition)= qh num_points - qh hull_dim - 1; /*misses GOOD... */
remaining= qh num_facets;
point_end= numpoints;
FORALLfacets {
size= point_end/(remaining--) + 100;
facet->outsideset= qh_setnew (size);
bestpoint= NULL;
point_end= 0;
FOREACHpoint_i_(pointset) {
if (point) {
zzinc_(Zpartitionall);
qh_distplane (point, facet, &dist);
if (dist < distoutside)
SETelem_(pointset, point_end++)= point;
else {
qh num_outside++;
if (!bestpoint) {
bestpoint= point;
bestdist= dist;
}else if (dist > bestdist) {
qh_setappend (&facet->outsideset, bestpoint);
bestpoint= point;
bestdist= dist;
}else
qh_setappend (&facet->outsideset, point);
}
}
}
if (bestpoint) {
qh_setappend (&facet->outsideset, bestpoint);
#if !qh_COMPUTEfurthest
facet->furthestdist= bestdist;
#endif
}else
qh_setfree (&facet->outsideset);
qh_settruncate (pointset, point_end);
}
}
/* if !qh BESToutside, pointset contains points not assigned to outsideset */
if (qh BESToutside || qh MERGING || qh KEEPcoplanar || qh KEEPinside) {
qh findbestnew= True;
FOREACHpoint_i_(pointset) {
if (point)
qh_partitionpoint(point, qh facet_list);
}
qh findbestnew= False;
}
zzadd_(Zpartitionall, zzval_(Zpartition));
zzval_(Zpartition)= 0;
qh_settempfree(&pointset);
if (qh IStracing >= 4)
qh_printfacetlist (qh facet_list, NULL, True);
} /* partitionall */
/*---------------------------------
qh_partitioncoplanar( point, facet, dist )
partition coplanar point to a facet
dist is distance from point to facet
if dist NULL,
searches for bestfacet and does nothing if inside
if qh.findbestnew set,
searches new facets instead of using qh_findbest()
returns:
qh.max_ouside updated
if qh.KEEPcoplanar or qh.KEEPinside
point assigned to best coplanarset
notes:
facet->maxoutside is updated at end by qh_check_maxout
design:
if dist undefined
find best facet for point
if point sufficiently below facet (depends on qh.NEARinside and qh.KEEPinside)
exit
if keeping coplanar/nearinside/inside points
if point is above furthest coplanar point
append point to coplanar set (it is the new furthest)
update qh.max_outside
else
append point one before end of coplanar set
else if point is clearly outside of qh.max_outside and bestfacet->coplanarset
and bestfacet is more than perpendicular to facet
repartition the point using qh_findbest() -- it may be put on an outsideset
else
update qh.max_outside
*/
void qh_partitioncoplanar (pointT *point, facetT *facet, realT *dist) {
facetT *bestfacet;
pointT *oldfurthest;
realT bestdist, dist2, angle;
int numpart= 0, oldfindbest;
boolT isoutside;
qh WAScoplanar= True;
if (!dist) {
if (qh findbestnew)
bestfacet= qh_findbestnew (point, facet, &bestdist, qh_ALL, &isoutside, &numpart);
else
bestfacet= qh_findbest (point, facet, qh_ALL, !qh_ISnewfacets, qh DELAUNAY,
&bestdist, &isoutside, &numpart);
zinc_(Ztotpartcoplanar);
zzadd_(Zpartcoplanar, numpart);
if (!qh DELAUNAY && !qh KEEPinside) { /* for 'd', bestdist skips upperDelaunay facets */
if (qh KEEPnearinside) {
if (bestdist < -qh NEARinside) {
zinc_(Zcoplanarinside);
trace4((qh ferr, "qh_partitioncoplanar: point p%d is more than near-inside facet f%d dist %2.2g findbestnew %d\n",
qh_pointid(point), bestfacet->id, bestdist, qh findbestnew));
return;
}
}else if (bestdist < -qh MAXcoplanar) {
trace4((qh ferr, "qh_partitioncoplanar: point p%d is inside facet f%d dist %2.2g findbestnew %d\n",
qh_pointid(point), bestfacet->id, bestdist, qh findbestnew));
zinc_(Zcoplanarinside);
return;
}
}
}else {
bestfacet= facet;
bestdist= *dist;
}
if (bestdist > qh max_outside) {
if (!dist && facet != bestfacet) {
zinc_(Zpartangle);
angle= qh_getangle(facet->normal, bestfacet->normal);
if (angle < 0) {
/* typically due to deleted vertex and coplanar facets, e.g.,
RBOX 1000 s Z1 G1e-13 t1001185205 | QHULL Tv */
zinc_(Zpartflip);
trace2((qh ferr, "qh_partitioncoplanar: repartition point p%d from f%d. It is above flipped facet f%d dist %2.2g\n",
qh_pointid(point), facet->id, bestfacet->id, bestdist));
oldfindbest= qh findbestnew;
qh findbestnew= False;
qh_partitionpoint(point, bestfacet);
qh findbestnew= oldfindbest;
return;
}
}
qh max_outside= bestdist;
if (bestdist > qh TRACEdist) {
fprintf (qh ferr, "qh_partitioncoplanar: ====== p%d from f%d increases max_outside to %2.2g of f%d last p%d\n",
qh_pointid(point), facet->id, bestdist, bestfacet->id, qh furthest_id);
qh_errprint ("DISTANT", facet, bestfacet, NULL, NULL);
}
}
if (qh KEEPcoplanar + qh KEEPinside + qh KEEPnearinside) {
oldfurthest= (pointT*)qh_setlast (bestfacet->coplanarset);
if (oldfurthest) {
zinc_(Zcomputefurthest);
qh_distplane (oldfurthest, bestfacet, &dist2);
}
if (!oldfurthest || dist2 < bestdist)
qh_setappend(&bestfacet->coplanarset, point);
else
qh_setappend2ndlast(&bestfacet->coplanarset, point);
}
trace4((qh ferr, "qh_partitioncoplanar: point p%d is coplanar with facet f%d (or inside) dist %2.2g\n",
qh_pointid(point), bestfacet->id, bestdist));
} /* partitioncoplanar */
/*---------------------------------
qh_partitionpoint( point, facet )
assigns point to an outside set, coplanar set, or inside set (i.e., dropt)
if qh.findbestnew
uses qh_findbestnew() to search all new facets
else
uses qh_findbest()
notes:
after qh_distplane(), this and qh_findbest() are most expensive in 3-d
design:
find best facet for point
(either exhaustive search of new facets or directed search from facet)
if qh.NARROWhull
retain coplanar and nearinside points as outside points
if point is outside bestfacet
if point above furthest point for bestfacet
append point to outside set (it becomes the new furthest)
if outside set was empty
move bestfacet to end of qh.facet_list (i.e., after qh.facet_next)
update bestfacet->furthestdist
else
append point one before end of outside set
else if point is coplanar to bestfacet
if keeping coplanar points or need to update qh.max_outside
partition coplanar point into bestfacet
else if near-inside point
partition as coplanar point into bestfacet
else is an inside point
if keeping inside points
partition as coplanar point into bestfacet
*/
void qh_partitionpoint (pointT *point, facetT *facet) {
realT bestdist;
boolT isoutside;
facetT *bestfacet;
int numpart;
#if qh_COMPUTEfurthest
realT dist;
#endif
if (qh findbestnew)
bestfacet= qh_findbestnew (point, facet, &bestdist, qh BESToutside, &isoutside, &numpart);
else
bestfacet= qh_findbest (point, facet, qh BESToutside, qh_ISnewfacets, !qh_NOupper,
&bestdist, &isoutside, &numpart);
zinc_(Ztotpartition);
zzadd_(Zpartition, numpart);
if (qh NARROWhull) {
if (qh DELAUNAY && !isoutside && bestdist >= -qh MAXcoplanar)
qh_precision ("nearly incident point (narrow hull)");
if (qh KEEPnearinside) {
if (bestdist >= -qh NEARinside)
isoutside= True;
}else if (bestdist >= -qh MAXcoplanar)
isoutside= True;
}
if (isoutside) {
if (!bestfacet->outsideset
|| !qh_setlast (bestfacet->outsideset)) {
qh_setappend(&(bestfacet->outsideset), point);
if (!bestfacet->newfacet) {
qh_removefacet (bestfacet); /* make sure it's after qh facet_next */
qh_appendfacet (bestfacet);
}
#if !qh_COMPUTEfurthest
bestfacet->furthestdist= bestdist;
#endif
}else {
#if qh_COMPUTEfurthest
zinc_(Zcomputefurthest);
qh_distplane (oldfurthest, bestfacet, &dist);
if (dist < bestdist)
qh_setappend(&(bestfacet->outsideset), point);
else
qh_setappend2ndlast(&(bestfacet->outsideset), point);
#else
if (bestfacet->furthestdist < bestdist) {
qh_setappend(&(bestfacet->outsideset), point);
bestfacet->furthestdist= bestdist;
}else
qh_setappend2ndlast(&(bestfacet->outsideset), point);
#endif
}
qh num_outside++;
trace4((qh ferr, "qh_partitionpoint: point p%d is outside facet f%d new? %d(or narrowhull)\n",
qh_pointid(point), bestfacet->id, bestfacet->newfacet));
}else if (qh DELAUNAY || bestdist >= -qh MAXcoplanar) { /* for 'd', bestdist skips upperDelaunay facets */
zzinc_(Zcoplanarpart);
if (qh DELAUNAY)
qh_precision ("nearly incident point");
if ((qh KEEPcoplanar + qh KEEPnearinside) || bestdist > qh max_outside)
qh_partitioncoplanar (point, bestfacet, &bestdist);
else {
trace4((qh ferr, "qh_partitionpoint: point p%d is coplanar to facet f%d (dropped)\n",
qh_pointid(point), bestfacet->id));
}
}else if (qh KEEPnearinside && bestdist > -qh NEARinside) {
zinc_(Zpartnear);
qh_partitioncoplanar (point, bestfacet, &bestdist);
}else {
zinc_(Zpartinside);
trace4((qh ferr, "qh_partitionpoint: point p%d is inside all facets, closest to f%d dist %2.2g\n",
qh_pointid(point), bestfacet->id, bestdist));
if (qh KEEPinside)
qh_partitioncoplanar (point, bestfacet, &bestdist);
}
} /* partitionpoint */
/*---------------------------------
qh_partitionvisible( allpoints, numoutside )
partitions points in visible facets to qh.newfacet_list
qh.visible_list= visible facets
for visible facets
1st neighbor (if any) points to a horizon facet or a new facet
if allpoints (not used),
repartitions coplanar points
returns:
updates outside sets and coplanar sets of qh.newfacet_list
updates qh.num_outside (count of outside points)
notes:
qh.findbest_notsharp should be clear (extra work if set)
design:
for all visible facets with outside set or coplanar set
select a newfacet for visible facet
if outside set
partition outside set into new facets
if coplanar set and keeping coplanar/near-inside/inside points
if allpoints
partition coplanar set into new facets, may be assigned outside
else
partition coplanar set into coplanar sets of new facets
for each deleted vertex
if allpoints
partition vertex into new facets, may be assigned outside
else
partition vertex into coplanar sets of new facets
*/
void qh_partitionvisible(/*visible_list*/ boolT allpoints, int *numoutside) {
facetT *visible, *newfacet;
pointT *point, **pointp;
int coplanar=0, size;
unsigned count;
vertexT *vertex, **vertexp;
if (qh ONLYmax)
maximize_(qh MINoutside, qh max_vertex);
*numoutside= 0;
FORALLvisible_facets {
if (!visible->outsideset && !visible->coplanarset)
continue;
newfacet= visible->f.replace;
count= 0;
while (newfacet && newfacet->visible) {
newfacet= newfacet->f.replace;
if (count++ > qh facet_id)
qh_infiniteloop (visible);
}
if (!newfacet)
newfacet= qh newfacet_list;
if (newfacet == qh facet_tail) {
fprintf (qh ferr, "qhull precision error (qh_partitionvisible): all new facets deleted as\n degenerate facets. Can not continue.\n");
qh_errexit (qh_ERRprec, NULL, NULL);
}
if (visible->outsideset) {
size= qh_setsize (visible->outsideset);
*numoutside += size;
qh num_outside -= size;
FOREACHpoint_(visible->outsideset)
qh_partitionpoint (point, newfacet);
}
if (visible->coplanarset && (qh KEEPcoplanar + qh KEEPinside + qh KEEPnearinside)) {
size= qh_setsize (visible->coplanarset);
coplanar += size;
FOREACHpoint_(visible->coplanarset) {
if (allpoints) /* not used */
qh_partitionpoint (point, newfacet);
else
qh_partitioncoplanar (point, newfacet, NULL);
}
}
}
FOREACHvertex_(qh del_vertices) {
if (vertex->point) {
if (allpoints) /* not used */
qh_partitionpoint (vertex->point, qh newfacet_list);
else
qh_partitioncoplanar (vertex->point, qh newfacet_list, NULL);
}
}
trace1((qh ferr,"qh_partitionvisible: partitioned %d points from outsidesets and %d points from coplanarsets\n", *numoutside, coplanar));
} /* partitionvisible */
/*---------------------------------
qh_precision( reason )
restart on precision errors if not merging and if 'QJn'
*/
void qh_precision (char *reason) {
if (qh ALLOWrestart && !qh PREmerge && !qh MERGEexact) {
if (qh JOGGLEmax < REALmax/2) {
trace0((qh ferr, "qh_precision: qhull restart because of %s\n", reason));
longjmp(qh restartexit, qh_ERRprec);
}
}
} /* qh_precision */
/*---------------------------------
qh_printsummary( fp )
prints summary to fp
notes:
not in io.c so that user_eg.c can prevent io.c from loading
qh_printsummary and qh_countfacets must match counts
design:
determine number of points, vertices, and coplanar points
print summary
*/
void qh_printsummary(FILE *fp) {
realT ratio, outerplane, innerplane;
float cpu;
int size, id, nummerged, numvertices, numcoplanars= 0, nonsimplicial=0;
int goodused;
facetT *facet;
char *s;
int numdel= zzval_(Zdelvertextot);
int numtricoplanars= 0;
size= qh num_points + qh_setsize (qh other_points);
numvertices= qh num_vertices - qh_setsize (qh del_vertices);
id= qh_pointid (qh GOODpointp);
FORALLfacets {
if (facet->coplanarset)
numcoplanars += qh_setsize( facet->coplanarset);
if (facet->good) {
if (facet->simplicial) {
if (facet->keepcentrum && facet->tricoplanar)
numtricoplanars++;
}else if (qh_setsize(facet->vertices) != qh hull_dim)
nonsimplicial++;
}
}
if (id >=0 && qh STOPcone-1 != id && -qh STOPpoint-1 != id)
size--;
if (qh STOPcone || qh STOPpoint)
fprintf (fp, "\nAt a premature exit due to 'TVn', 'TCn', 'TRn', or precision error.");
if (qh UPPERdelaunay)
goodused= qh GOODvertex + qh GOODpoint + qh SPLITthresholds;
else if (qh DELAUNAY)
goodused= qh GOODvertex + qh GOODpoint + qh GOODthreshold;
else
goodused= qh num_good;
nummerged= zzval_(Ztotmerge) - zzval_(Zcyclehorizon) + zzval_(Zcyclefacettot);
if (qh VORONOI) {
if (qh UPPERdelaunay)
fprintf (fp, "\n\
Furthest-site Voronoi vertices by the convex hull of %d points in %d-d:\n\n", size, qh hull_dim);
else
fprintf (fp, "\n\
Voronoi diagram by the convex hull of %d points in %d-d:\n\n", size, qh hull_dim);
fprintf(fp, " Number of Voronoi regions%s: %d\n",
qh ATinfinity ? " and at-infinity" : "", numvertices);
if (numdel)
fprintf(fp, " Total number of deleted points due to merging: %d\n", numdel);
if (numcoplanars - numdel > 0)
fprintf(fp, " Number of nearly incident points: %d\n", numcoplanars - numdel);
else if (size - numvertices - numdel > 0)
fprintf(fp, " Total number of nearly incident points: %d\n", size - numvertices - numdel);
fprintf(fp, " Number of%s Voronoi vertices: %d\n",
goodused ? " 'good'" : "", qh num_good);
if (nonsimplicial)
fprintf(fp, " Number of%s non-simplicial Voronoi vertices: %d\n",
goodused ? " 'good'" : "", nonsimplicial);
}else if (qh DELAUNAY) {
if (qh UPPERdelaunay)
fprintf (fp, "\n\
Furthest-site Delaunay triangulation by the convex hull of %d points in %d-d:\n\n", size, qh hull_dim);
else
fprintf (fp, "\n\
Delaunay triangulation by the convex hull of %d points in %d-d:\n\n", size, qh hull_dim);
fprintf(fp, " Number of input sites%s: %d\n",
qh ATinfinity ? " and at-infinity" : "", numvertices);
if (numdel)
fprintf(fp, " Total number of deleted points due to merging: %d\n", numdel);
if (numcoplanars - numdel > 0)
fprintf(fp, " Number of nearly incident points: %d\n", numcoplanars - numdel);
else if (size - numvertices - numdel > 0)
fprintf(fp, " Total number of nearly incident points: %d\n", size - numvertices - numdel);
fprintf(fp, " Number of%s Delaunay regions: %d\n",
goodused ? " 'good'" : "", qh num_good);
if (nonsimplicial)
fprintf(fp, " Number of%s non-simplicial Delaunay regions: %d\n",
goodused ? " 'good'" : "", nonsimplicial);
}else if (qh HALFspace) {
fprintf (fp, "\n\
Halfspace intersection by the convex hull of %d points in %d-d:\n\n", size, qh hull_dim);
fprintf(fp, " Number of halfspaces: %d\n", size);
fprintf(fp, " Number of non-redundant halfspaces: %d\n", numvertices);
if (numcoplanars) {
if (qh KEEPinside && qh KEEPcoplanar)
s= "similar and redundant";
else if (qh KEEPinside)
s= "redundant";
else
s= "similar";
fprintf(fp, " Number of %s halfspaces: %d\n", s, numcoplanars);
}
fprintf(fp, " Number of intersection points: %d\n", qh num_facets - qh num_visible);
if (goodused)
fprintf(fp, " Number of 'good' intersection points: %d\n", qh num_good);
if (nonsimplicial)
fprintf(fp, " Number of%s non-simplicial intersection points: %d\n",
goodused ? " 'good'" : "", nonsimplicial);
}else {
fprintf (fp, "\n\
Convex hull of %d points in %d-d:\n\n", size, qh hull_dim);
fprintf(fp, " Number of vertices: %d\n", numvertices);
if (numcoplanars) {
if (qh KEEPinside && qh KEEPcoplanar)
s= "coplanar and interior";
else if (qh KEEPinside)
s= "interior";
else
s= "coplanar";
fprintf(fp, " Number of %s points: %d\n", s, numcoplanars);
}
fprintf(fp, " Number of facets: %d\n", qh num_facets - qh num_visible);
if (goodused)
fprintf(fp, " Number of 'good' facets: %d\n", qh num_good);
if (nonsimplicial)
fprintf(fp, " Number of%s non-simplicial facets: %d\n",
goodused ? " 'good'" : "", nonsimplicial);
}
if (numtricoplanars)
fprintf(fp, " Number of triangulated facets: %d\n", numtricoplanars);
fprintf(fp, "\nStatistics for: %s | %s",
qh rbox_command, qh qhull_command);
if (qh ROTATErandom != INT_MIN)
fprintf(fp, " QR%d\n\n", qh ROTATErandom);
else
fprintf(fp, "\n\n");
fprintf(fp, " Number of points processed: %d\n", zzval_(Zprocessed));
fprintf(fp, " Number of hyperplanes created: %d\n", zzval_(Zsetplane));
if (qh DELAUNAY)
fprintf(fp, " Number of facets in hull: %d\n", qh num_facets - qh num_visible);
fprintf(fp, " Number of distance tests for qhull: %d\n", zzval_(Zpartition)+
zzval_(Zpartitionall)+zzval_(Znumvisibility)+zzval_(Zpartcoplanar));
#if 0 /* NOTE: must print before printstatistics() */
{realT stddev, ave;
fprintf(fp, " average new facet balance: %2.2g\n",
wval_(Wnewbalance)/zval_(Zprocessed));
stddev= qh_stddev (zval_(Zprocessed), wval_(Wnewbalance),
wval_(Wnewbalance2), &ave);
fprintf(fp, " new facet standard deviation: %2.2g\n", stddev);
fprintf(fp, " average partition balance: %2.2g\n",
wval_(Wpbalance)/zval_(Zpbalance));
stddev= qh_stddev (zval_(Zpbalance), wval_(Wpbalance),
wval_(Wpbalance2), &ave);
fprintf(fp, " partition standard deviation: %2.2g\n", stddev);
}
#endif
if (nummerged) {
fprintf(fp," Number of distance tests for merging: %d\n",zzval_(Zbestdist)+
zzval_(Zcentrumtests)+zzval_(Zdistconvex)+zzval_(Zdistcheck)+
zzval_(Zdistzero));
fprintf(fp," Number of distance tests for checking: %d\n",zzval_(Zcheckpart));
fprintf(fp," Number of merged facets: %d\n", nummerged);
}
if (!qh RANDOMoutside && qh QHULLfinished) {
cpu= qh hulltime;
cpu /= qh_SECticks;
wval_(Wcpu)= cpu;
fprintf (fp, " CPU seconds to compute hull (after input): %2.4g\n", cpu);
}
if (qh RERUN) {
if (!qh PREmerge && !qh MERGEexact)
fprintf(fp, " Percentage of runs with precision errors: %4.1f\n",
zzval_(Zretry)*100.0/qh build_cnt); /* careful of order */
}else if (qh JOGGLEmax < REALmax/2) {
if (zzval_(Zretry))
fprintf(fp, " After %d retries, input joggled by: %2.2g\n",
zzval_(Zretry), qh JOGGLEmax);
else
fprintf(fp, " Input joggled by: %2.2g\n", qh JOGGLEmax);
}
if (qh totarea != 0.0)
fprintf(fp, " %s facet area: %2.8g\n",
zzval_(Ztotmerge) ? "Approximate" : "Total", qh totarea);
if (qh totvol != 0.0)
fprintf(fp, " %s volume: %2.8g\n",
zzval_(Ztotmerge) ? "Approximate" : "Total", qh totvol);
if (qh MERGING) {
qh_outerinner (NULL, &outerplane, &innerplane);
if (outerplane > 2 * qh DISTround) {
fprintf(fp, " Maximum distance of %spoint above facet: %2.2g",
(qh QHULLfinished ? "" : "merged "), outerplane);
ratio= outerplane/(qh ONEmerge + qh DISTround);
/* don't report ratio if MINoutside is large */
if (ratio > 0.05 && 2* qh ONEmerge > qh MINoutside && qh JOGGLEmax > REALmax/2)
fprintf (fp, " (%.1fx)\n", ratio);
else
fprintf (fp, "\n");
}
if (innerplane < -2 * qh DISTround) {
fprintf(fp, " Maximum distance of %svertex below facet: %2.2g",
(qh QHULLfinished ? "" : "merged "), innerplane);
ratio= -innerplane/(qh ONEmerge+qh DISTround);
if (ratio > 0.05 && qh JOGGLEmax > REALmax/2)
fprintf (fp, " (%.1fx)\n", ratio);
else
fprintf (fp, "\n");
}
}
fprintf(fp, "\n");
} /* printsummary */