hpp-fcl: stat.c Source File

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 /*<html><pre>  -<a                             href="qh-stat.htm"
   >-------------------------------</a><a name="TOP">-</a>
  
    stat.c
    contains all statistics that are collected for qhull
  
    see qh-stat.htm and stat.h
  
    Copyright (c) 1993-2015 The Geometry Center.
    $Id: //main/2015/qhull/src/libqhull/stat.c#5 $$Change: 2062 $
    $DateTime: 2016/01/17 13:13:18 $$Author: bbarber $
 */
  
 #include "qhull_a.h"
  
 /*============ global data structure ==========*/
  
 #if qh_QHpointer
 qhstatT *qh_qhstat=NULL;  /* global data structure */
 #else
 qhstatT qh_qhstat;   /* add "={0}" if this causes a compiler error */
 #endif
  
 /*========== functions in alphabetic order ================*/
  
 /*-<a                             href="qh-stat.htm#TOC"
   >-------------------------------</a><a name="allstatA">-</a>
  
   qh_allstatA()
     define statistics in groups of 20
  
   notes:
     (otherwise, 'gcc -O2' uses too much memory)
     uses qhstat.next
 */
 void qh_allstatA(void) {
  
    /* zdef_(type,name,doc,average) */
   zzdef_(zdoc, Zdoc2, "precision statistics", -1);
   zdef_(zinc, Znewvertex, NULL, -1);
   zdef_(wadd, Wnewvertex, "ave. distance of a new vertex to a facet(!0s)", Znewvertex);
   zzdef_(wmax, Wnewvertexmax, "max. distance of a new vertex to a facet", -1);
   zdef_(wmax, Wvertexmax, "max. distance of an output vertex to a facet", -1);
   zdef_(wmin, Wvertexmin, "min. distance of an output vertex to a facet", -1);
   zdef_(wmin, Wmindenom, "min. denominator in hyperplane computation", -1);
  
   qhstat precision= qhstat next;  /* call qh_precision for each of these */
   zzdef_(zdoc, Zdoc3, "precision problems (corrected unless 'Q0' or an error)", -1);
   zzdef_(zinc, Zcoplanarridges, "coplanar half ridges in output", -1);
   zzdef_(zinc, Zconcaveridges, "concave half ridges in output", -1);
   zzdef_(zinc, Zflippedfacets, "flipped facets", -1);
   zzdef_(zinc, Zcoplanarhorizon, "coplanar horizon facets for new vertices", -1);
   zzdef_(zinc, Zcoplanarpart, "coplanar points during partitioning", -1);
   zzdef_(zinc, Zminnorm, "degenerate hyperplanes recomputed with gaussian elimination", -1);
   zzdef_(zinc, Znearlysingular, "nearly singular or axis-parallel hyperplanes", -1);
   zzdef_(zinc, Zback0, "zero divisors during back substitute", -1);
   zzdef_(zinc, Zgauss0, "zero divisors during gaussian elimination", -1);
   zzdef_(zinc, Zmultiridge, "ridges with multiple neighbors", -1);
 }
 void qh_allstatB(void) {
   zzdef_(zdoc, Zdoc1, "summary information", -1);
   zdef_(zinc, Zvertices, "number of vertices in output", -1);
   zdef_(zinc, Znumfacets, "number of facets in output", -1);
   zdef_(zinc, Znonsimplicial, "number of non-simplicial facets in output", -1);
   zdef_(zinc, Znowsimplicial, "number of simplicial facets that were merged", -1);
   zdef_(zinc, Znumridges, "number of ridges in output", -1);
   zdef_(zadd, Znumridges, "average number of ridges per facet", Znumfacets);
   zdef_(zmax, Zmaxridges, "maximum number of ridges", -1);
   zdef_(zadd, Znumneighbors, "average number of neighbors per facet", Znumfacets);
   zdef_(zmax, Zmaxneighbors, "maximum number of neighbors", -1);
   zdef_(zadd, Znumvertices, "average number of vertices per facet", Znumfacets);
   zdef_(zmax, Zmaxvertices, "maximum number of vertices", -1);
   zdef_(zadd, Znumvneighbors, "average number of neighbors per vertex", Zvertices);
   zdef_(zmax, Zmaxvneighbors, "maximum number of neighbors", -1);
   zdef_(wadd, Wcpu, "cpu seconds for qhull after input", -1);
   zdef_(zinc, Ztotvertices, "vertices created altogether", -1);
   zzdef_(zinc, Zsetplane, "facets created altogether", -1);
   zdef_(zinc, Ztotridges, "ridges created altogether", -1);
   zdef_(zinc, Zpostfacets, "facets before post merge", -1);
   zdef_(zadd, Znummergetot, "average merges per facet(at most 511)", Znumfacets);
   zdef_(zmax, Znummergemax, "  maximum merges for a facet(at most 511)", -1);
   zdef_(zinc, Zangle, NULL, -1);
   zdef_(wadd, Wangle, "average angle(cosine) of facet normals for all ridges", Zangle);
   zdef_(wmax, Wanglemax, "  maximum angle(cosine) of facet normals across a ridge", -1);
   zdef_(wmin, Wanglemin, "  minimum angle(cosine) of facet normals across a ridge", -1);
   zdef_(wadd, Wareatot, "total area of facets", -1);
   zdef_(wmax, Wareamax, "  maximum facet area", -1);
   zdef_(wmin, Wareamin, "  minimum facet area", -1);
 }
 void qh_allstatC(void) {
   zdef_(zdoc, Zdoc9, "build hull statistics", -1);
   zzdef_(zinc, Zprocessed, "points processed", -1);
   zzdef_(zinc, Zretry, "retries due to precision problems", -1);
   zdef_(wmax, Wretrymax, "  max. random joggle", -1);
   zdef_(zmax, Zmaxvertex, "max. vertices at any one time", -1);
   zdef_(zinc, Ztotvisible, "ave. visible facets per iteration", Zprocessed);
   zdef_(zinc, Zinsidevisible, "  ave. visible facets without an horizon neighbor", Zprocessed);
   zdef_(zadd, Zvisfacettot,  "  ave. facets deleted per iteration", Zprocessed);
   zdef_(zmax, Zvisfacetmax,  "    maximum", -1);
   zdef_(zadd, Zvisvertextot, "ave. visible vertices per iteration", Zprocessed);
   zdef_(zmax, Zvisvertexmax, "    maximum", -1);
   zdef_(zinc, Ztothorizon, "ave. horizon facets per iteration", Zprocessed);
   zdef_(zadd, Znewfacettot,  "ave. new or merged facets per iteration", Zprocessed);
   zdef_(zmax, Znewfacetmax,  "    maximum(includes initial simplex)", -1);
   zdef_(wadd, Wnewbalance, "average new facet balance", Zprocessed);
   zdef_(wadd, Wnewbalance2, "  standard deviation", -1);
   zdef_(wadd, Wpbalance, "average partition balance", Zpbalance);
   zdef_(wadd, Wpbalance2, "  standard deviation", -1);
   zdef_(zinc, Zpbalance, "  number of trials", -1);
   zdef_(zinc, Zsearchpoints, "searches of all points for initial simplex", -1);
   zdef_(zinc, Zdetsimplex, "determinants computed(area & initial hull)", -1);
   zdef_(zinc, Znoarea, "determinants not computed because vertex too low", -1);
   zdef_(zinc, Znotmax, "points ignored(!above max_outside)", -1);
   zdef_(zinc, Znotgood, "points ignored(!above a good facet)", -1);
   zdef_(zinc, Znotgoodnew, "points ignored(didn't create a good new facet)", -1);
   zdef_(zinc, Zgoodfacet, "good facets found", -1);
   zzdef_(zinc, Znumvisibility, "distance tests for facet visibility", -1);
   zdef_(zinc, Zdistvertex, "distance tests to report minimum vertex", -1);
   zzdef_(zinc, Ztotcheck, "points checked for facets' outer planes", -1);
   zzdef_(zinc, Zcheckpart, "  ave. distance tests per check", Ztotcheck);
 }
 void qh_allstatD(void) {
   zdef_(zinc, Zvisit, "resets of visit_id", -1);
   zdef_(zinc, Zvvisit, "  resets of vertex_visit", -1);
   zdef_(zmax, Zvisit2max, "  max visit_id/2", -1);
   zdef_(zmax, Zvvisit2max, "  max vertex_visit/2", -1);
  
   zdef_(zdoc, Zdoc4, "partitioning statistics(see previous for outer planes)", -1);
   zzdef_(zadd, Zdelvertextot, "total vertices deleted", -1);
   zdef_(zmax, Zdelvertexmax, "    maximum vertices deleted per iteration", -1);
   zdef_(zinc, Zfindbest, "calls to findbest", -1);
   zdef_(zadd, Zfindbesttot, " ave. facets tested", Zfindbest);
   zdef_(zmax, Zfindbestmax, " max. facets tested", -1);
   zdef_(zadd, Zfindcoplanar, " ave. coplanar search", Zfindbest);
   zdef_(zinc, Zfindnew, "calls to findbestnew", -1);
   zdef_(zadd, Zfindnewtot, " ave. facets tested", Zfindnew);
   zdef_(zmax, Zfindnewmax, " max. facets tested", -1);
   zdef_(zinc, Zfindnewjump, " ave. clearly better", Zfindnew);
   zdef_(zinc, Zfindnewsharp, " calls due to qh_sharpnewfacets", -1);
   zdef_(zinc, Zfindhorizon, "calls to findhorizon", -1);
   zdef_(zadd, Zfindhorizontot, " ave. facets tested", Zfindhorizon);
   zdef_(zmax, Zfindhorizonmax, " max. facets tested", -1);
   zdef_(zinc, Zfindjump,       " ave. clearly better", Zfindhorizon);
   zdef_(zinc, Zparthorizon, " horizon facets better than bestfacet", -1);
   zdef_(zinc, Zpartangle, "angle tests for repartitioned coplanar points", -1);
   zdef_(zinc, Zpartflip, "  repartitioned coplanar points for flipped orientation", -1);
 }
 void qh_allstatE(void) {
   zdef_(zinc, Zpartinside, "inside points", -1);
   zdef_(zinc, Zpartnear, "  inside points kept with a facet", -1);
   zdef_(zinc, Zcoplanarinside, "  inside points that were coplanar with a facet", -1);
   zdef_(zinc, Zbestlower, "calls to findbestlower", -1);
   zdef_(zinc, Zbestlowerv, "  with search of vertex neighbors", -1);
   zdef_(zinc, Zbestlowerall, "  with rare search of all facets", -1);
   zdef_(zmax, Zbestloweralln, "  facets per search of all facets", -1);
   zdef_(wadd, Wmaxout, "difference in max_outside at final check", -1);
   zzdef_(zinc, Zpartitionall, "distance tests for initial partition", -1);
   zdef_(zinc, Ztotpartition, "partitions of a point", -1);
   zzdef_(zinc, Zpartition, "distance tests for partitioning", -1);
   zzdef_(zinc, Zdistcheck, "distance tests for checking flipped facets", -1);
   zzdef_(zinc, Zdistconvex, "distance tests for checking convexity", -1);
   zdef_(zinc, Zdistgood, "distance tests for checking good point", -1);
   zdef_(zinc, Zdistio, "distance tests for output", -1);
   zdef_(zinc, Zdiststat, "distance tests for statistics", -1);
   zdef_(zinc, Zdistplane, "total number of distance tests", -1);
   zdef_(zinc, Ztotpartcoplanar, "partitions of coplanar points or deleted vertices", -1);
   zzdef_(zinc, Zpartcoplanar, "   distance tests for these partitions", -1);
   zdef_(zinc, Zcomputefurthest, "distance tests for computing furthest", -1);
 }
 void qh_allstatE2(void) {
   zdef_(zdoc, Zdoc5, "statistics for matching ridges", -1);
   zdef_(zinc, Zhashlookup, "total lookups for matching ridges of new facets", -1);
   zdef_(zinc, Zhashtests, "average number of tests to match a ridge", Zhashlookup);
   zdef_(zinc, Zhashridge, "total lookups of subridges(duplicates and boundary)", -1);
   zdef_(zinc, Zhashridgetest, "average number of tests per subridge", Zhashridge);
   zdef_(zinc, Zdupsame, "duplicated ridges in same merge cycle", -1);
   zdef_(zinc, Zdupflip, "duplicated ridges with flipped facets", -1);
  
   zdef_(zdoc, Zdoc6, "statistics for determining merges", -1);
   zdef_(zinc, Zangletests, "angles computed for ridge convexity", -1);
   zdef_(zinc, Zbestcentrum, "best merges used centrum instead of vertices",-1);
   zzdef_(zinc, Zbestdist, "distance tests for best merge", -1);
   zzdef_(zinc, Zcentrumtests, "distance tests for centrum convexity", -1);
   zzdef_(zinc, Zdistzero, "distance tests for checking simplicial convexity", -1);
   zdef_(zinc, Zcoplanarangle, "coplanar angles in getmergeset", -1);
   zdef_(zinc, Zcoplanarcentrum, "coplanar centrums in getmergeset", -1);
   zdef_(zinc, Zconcaveridge, "concave ridges in getmergeset", -1);
 }
 void qh_allstatF(void) {
   zdef_(zdoc, Zdoc7, "statistics for merging", -1);
   zdef_(zinc, Zpremergetot, "merge iterations", -1);
   zdef_(zadd, Zmergeinittot, "ave. initial non-convex ridges per iteration", Zpremergetot);
   zdef_(zadd, Zmergeinitmax, "  maximum", -1);
   zdef_(zadd, Zmergesettot, "  ave. additional non-convex ridges per iteration", Zpremergetot);
   zdef_(zadd, Zmergesetmax, "  maximum additional in one pass", -1);
   zdef_(zadd, Zmergeinittot2, "initial non-convex ridges for post merging", -1);
   zdef_(zadd, Zmergesettot2, "  additional non-convex ridges", -1);
   zdef_(wmax, Wmaxoutside, "max distance of vertex or coplanar point above facet(w/roundoff)", -1);
   zdef_(wmin, Wminvertex, "max distance of merged vertex below facet(or roundoff)", -1);
   zdef_(zinc, Zwidefacet, "centrums frozen due to a wide merge", -1);
   zdef_(zinc, Zwidevertices, "centrums frozen due to extra vertices", -1);
   zzdef_(zinc, Ztotmerge, "total number of facets or cycles of facets merged", -1);
   zdef_(zinc, Zmergesimplex, "merged a simplex", -1);
   zdef_(zinc, Zonehorizon, "simplices merged into coplanar horizon", -1);
   zzdef_(zinc, Zcyclehorizon, "cycles of facets merged into coplanar horizon", -1);
   zzdef_(zadd, Zcyclefacettot, "  ave. facets per cycle", Zcyclehorizon);
   zdef_(zmax, Zcyclefacetmax, "  max. facets", -1);
   zdef_(zinc, Zmergeintohorizon, "new facets merged into horizon", -1);
   zdef_(zinc, Zmergenew, "new facets merged", -1);
   zdef_(zinc, Zmergehorizon, "horizon facets merged into new facets", -1);
   zdef_(zinc, Zmergevertex, "vertices deleted by merging", -1);
   zdef_(zinc, Zcyclevertex, "vertices deleted by merging into coplanar horizon", -1);
   zdef_(zinc, Zdegenvertex, "vertices deleted by degenerate facet", -1);
   zdef_(zinc, Zmergeflipdup, "merges due to flipped facets in duplicated ridge", -1);
   zdef_(zinc, Zneighbor, "merges due to redundant neighbors", -1);
   zdef_(zadd, Ztestvneighbor, "non-convex vertex neighbors", -1);
 }
 void qh_allstatG(void) {
   zdef_(zinc, Zacoplanar, "merges due to angle coplanar facets", -1);
   zdef_(wadd, Wacoplanartot, "  average merge distance", Zacoplanar);
   zdef_(wmax, Wacoplanarmax, "  maximum merge distance", -1);
   zdef_(zinc, Zcoplanar, "merges due to coplanar facets", -1);
   zdef_(wadd, Wcoplanartot, "  average merge distance", Zcoplanar);
   zdef_(wmax, Wcoplanarmax, "  maximum merge distance", -1);
   zdef_(zinc, Zconcave, "merges due to concave facets", -1);
   zdef_(wadd, Wconcavetot, "  average merge distance", Zconcave);
   zdef_(wmax, Wconcavemax, "  maximum merge distance", -1);
   zdef_(zinc, Zavoidold, "coplanar/concave merges due to avoiding old merge", -1);
   zdef_(wadd, Wavoidoldtot, "  average merge distance", Zavoidold);
   zdef_(wmax, Wavoidoldmax, "  maximum merge distance", -1);
   zdef_(zinc, Zdegen, "merges due to degenerate facets", -1);
   zdef_(wadd, Wdegentot, "  average merge distance", Zdegen);
   zdef_(wmax, Wdegenmax, "  maximum merge distance", -1);
   zdef_(zinc, Zflipped, "merges due to removing flipped facets", -1);
   zdef_(wadd, Wflippedtot, "  average merge distance", Zflipped);
   zdef_(wmax, Wflippedmax, "  maximum merge distance", -1);
   zdef_(zinc, Zduplicate, "merges due to duplicated ridges", -1);
   zdef_(wadd, Wduplicatetot, "  average merge distance", Zduplicate);
   zdef_(wmax, Wduplicatemax, "  maximum merge distance", -1);
 }
 void qh_allstatH(void) {
   zdef_(zdoc, Zdoc8, "renamed vertex statistics", -1);
   zdef_(zinc, Zrenameshare, "renamed vertices shared by two facets", -1);
   zdef_(zinc, Zrenamepinch, "renamed vertices in a pinched facet", -1);
   zdef_(zinc, Zrenameall, "renamed vertices shared by multiple facets", -1);
   zdef_(zinc, Zfindfail, "rename failures due to duplicated ridges", -1);
   zdef_(zinc, Zdupridge, "  duplicate ridges detected", -1);
   zdef_(zinc, Zdelridge, "deleted ridges due to renamed vertices", -1);
   zdef_(zinc, Zdropneighbor, "dropped neighbors due to renamed vertices", -1);
   zdef_(zinc, Zdropdegen, "degenerate facets due to dropped neighbors", -1);
   zdef_(zinc, Zdelfacetdup, "  facets deleted because of no neighbors", -1);
   zdef_(zinc, Zremvertex, "vertices removed from facets due to no ridges", -1);
   zdef_(zinc, Zremvertexdel, "  deleted", -1);
   zdef_(zinc, Zintersectnum, "vertex intersections for locating redundant vertices", -1);
   zdef_(zinc, Zintersectfail, "intersections failed to find a redundant vertex", -1);
   zdef_(zinc, Zintersect, "intersections found redundant vertices", -1);
   zdef_(zadd, Zintersecttot, "   ave. number found per vertex", Zintersect);
   zdef_(zmax, Zintersectmax, "   max. found for a vertex", -1);
   zdef_(zinc, Zvertexridge, NULL, -1);
   zdef_(zadd, Zvertexridgetot, "  ave. number of ridges per tested vertex", Zvertexridge);
   zdef_(zmax, Zvertexridgemax, "  max. number of ridges per tested vertex", -1);
  
   zdef_(zdoc, Zdoc10, "memory usage statistics(in bytes)", -1);
   zdef_(zadd, Zmemfacets, "for facets and their normals, neighbor and vertex sets", -1);
   zdef_(zadd, Zmemvertices, "for vertices and their neighbor sets", -1);
   zdef_(zadd, Zmempoints, "for input points and outside and coplanar sets",-1);
   zdef_(zadd, Zmemridges, "for ridges and their vertex sets", -1);
 } /* allstat */
  
 void qh_allstatI(void) {
   qhstat vridges= qhstat next;
   zzdef_(zdoc, Zdoc11, "Voronoi ridge statistics", -1);
   zzdef_(zinc, Zridge, "non-simplicial Voronoi vertices for all ridges", -1);
   zzdef_(wadd, Wridge, "  ave. distance to ridge", Zridge);
   zzdef_(wmax, Wridgemax, "  max. distance to ridge", -1);
   zzdef_(zinc, Zridgemid, "bounded ridges", -1);
   zzdef_(wadd, Wridgemid, "  ave. distance of midpoint to ridge", Zridgemid);
   zzdef_(wmax, Wridgemidmax, "  max. distance of midpoint to ridge", -1);
   zzdef_(zinc, Zridgeok, "bounded ridges with ok normal", -1);
   zzdef_(wadd, Wridgeok, "  ave. angle to ridge", Zridgeok);
   zzdef_(wmax, Wridgeokmax, "  max. angle to ridge", -1);
   zzdef_(zinc, Zridge0, "bounded ridges with near-zero normal", -1);
   zzdef_(wadd, Wridge0, "  ave. angle to ridge", Zridge0);
   zzdef_(wmax, Wridge0max, "  max. angle to ridge", -1);
  
   zdef_(zdoc, Zdoc12, "Triangulation statistics(Qt)", -1);
   zdef_(zinc, Ztricoplanar, "non-simplicial facets triangulated", -1);
   zdef_(zadd, Ztricoplanartot, "  ave. new facets created(may be deleted)", Ztricoplanar);
   zdef_(zmax, Ztricoplanarmax, "  max. new facets created", -1);
   zdef_(zinc, Ztrinull, "null new facets deleted(duplicated vertex)", -1);
   zdef_(zinc, Ztrimirror, "mirrored pairs of new facets deleted(same vertices)", -1);
   zdef_(zinc, Ztridegen, "degenerate new facets in output(same ridge)", -1);
 } /* allstat */
  
 /*-<a                             href="qh-stat.htm#TOC"
   >-------------------------------</a><a name="allstatistics">-</a>
  
   qh_allstatistics()
     reset printed flag for all statistics
 */
 void qh_allstatistics(void) {
   int i;
  
   for(i=ZEND; i--; )
     qhstat printed[i]= False;
 } /* allstatistics */
  
 #if qh_KEEPstatistics
 /*-<a                             href="qh-stat.htm#TOC"
   >-------------------------------</a><a name="collectstatistics">-</a>
  
   qh_collectstatistics()
     collect statistics for qh.facet_list
  
 */
 void qh_collectstatistics(void) {
   facetT *facet, *neighbor, **neighborp;
   vertexT *vertex, **vertexp;
   realT dotproduct, dist;
   int sizneighbors, sizridges, sizvertices, i;
  
   qh old_randomdist= qh RANDOMdist;
   qh RANDOMdist= False;
   zval_(Zmempoints)= qh num_points * qh normal_size +
                              sizeof(qhT) + sizeof(qhstatT);
   zval_(Zmemfacets)= 0;
   zval_(Zmemridges)= 0;
   zval_(Zmemvertices)= 0;
   zval_(Zangle)= 0;
   wval_(Wangle)= 0.0;
   zval_(Znumridges)= 0;
   zval_(Znumfacets)= 0;
   zval_(Znumneighbors)= 0;
   zval_(Znumvertices)= 0;
   zval_(Znumvneighbors)= 0;
   zval_(Znummergetot)= 0;
   zval_(Znummergemax)= 0;
   zval_(Zvertices)= qh num_vertices - qh_setsize(qh del_vertices);
   if (qh MERGING || qh APPROXhull || qh JOGGLEmax < REALmax/2)
     wmax_(Wmaxoutside, qh max_outside);
   if (qh MERGING)
     wmin_(Wminvertex, qh min_vertex);
   FORALLfacets
     facet->seen= False;
   if (qh DELAUNAY) {
     FORALLfacets {
       if (facet->upperdelaunay != qh UPPERdelaunay)
         facet->seen= True; /* remove from angle statistics */
     }
   }
   FORALLfacets {
     if (facet->visible && qh NEWfacets)
       continue;
     sizvertices= qh_setsize(facet->vertices);
     sizneighbors= qh_setsize(facet->neighbors);
     sizridges= qh_setsize(facet->ridges);
     zinc_(Znumfacets);
     zadd_(Znumvertices, sizvertices);
     zmax_(Zmaxvertices, sizvertices);
     zadd_(Znumneighbors, sizneighbors);
     zmax_(Zmaxneighbors, sizneighbors);
     zadd_(Znummergetot, facet->nummerge);
     i= facet->nummerge; /* avoid warnings */
     zmax_(Znummergemax, i);
     if (!facet->simplicial) {
       if (sizvertices == qh hull_dim) {
         zinc_(Znowsimplicial);
       }else {
         zinc_(Znonsimplicial);
       }
     }
     if (sizridges) {
       zadd_(Znumridges, sizridges);
       zmax_(Zmaxridges, sizridges);
     }
     zadd_(Zmemfacets, sizeof(facetT) + qh normal_size + 2*sizeof(setT)
        + SETelemsize * (sizneighbors + sizvertices));
     if (facet->ridges) {
       zadd_(Zmemridges,
          sizeof(setT) + SETelemsize * sizridges + sizridges *
          (sizeof(ridgeT) + sizeof(setT) + SETelemsize * (qh hull_dim-1))/2);
     }
     if (facet->outsideset)
       zadd_(Zmempoints, sizeof(setT) + SETelemsize * qh_setsize(facet->outsideset));
     if (facet->coplanarset)
       zadd_(Zmempoints, sizeof(setT) + SETelemsize * qh_setsize(facet->coplanarset));
     if (facet->seen) /* Delaunay upper envelope */
       continue;
     facet->seen= True;
     FOREACHneighbor_(facet) {
       if (neighbor == qh_DUPLICATEridge || neighbor == qh_MERGEridge
           || neighbor->seen || !facet->normal || !neighbor->normal)
         continue;
       dotproduct= qh_getangle(facet->normal, neighbor->normal);
       zinc_(Zangle);
       wadd_(Wangle, dotproduct);
       wmax_(Wanglemax, dotproduct)
       wmin_(Wanglemin, dotproduct)
     }
     if (facet->normal) {
       FOREACHvertex_(facet->vertices) {
         zinc_(Zdiststat);
         qh_distplane(vertex->point, facet, &dist);
         wmax_(Wvertexmax, dist);
         wmin_(Wvertexmin, dist);
       }
     }
   }
   FORALLvertices {
     if (vertex->deleted)
       continue;
     zadd_(Zmemvertices, sizeof(vertexT));
     if (vertex->neighbors) {
       sizneighbors= qh_setsize(vertex->neighbors);
       zadd_(Znumvneighbors, sizneighbors);
       zmax_(Zmaxvneighbors, sizneighbors);
       zadd_(Zmemvertices, sizeof(vertexT) + SETelemsize * sizneighbors);
     }
   }
   qh RANDOMdist= qh old_randomdist;
 } /* collectstatistics */
 #endif /* qh_KEEPstatistics */
  
 /*-<a                             href="qh-stat.htm#TOC"
   >-------------------------------</a><a name="freestatistics">-</a>
  
   qh_freestatistics(  )
     free memory used for statistics
 */
 void qh_freestatistics(void) {
  
 #if qh_QHpointer
   qh_free(qh_qhstat);
   qh_qhstat= NULL;
 #endif
 } /* freestatistics */
  
 /*-<a                             href="qh-stat.htm#TOC"
   >-------------------------------</a><a name="initstatistics">-</a>
  
   qh_initstatistics(  )
     allocate and initialize statistics
  
   notes:
     uses qh_malloc() instead of qh_memalloc() since mem.c not set up yet
     NOerrors -- qh_initstatistics can not use qh_errexit(), qh_fprintf, or qh.ferr
     On first call, only qhmem.ferr is defined.  qh_memalloc is not setup.
     Also invoked by QhullQh().
 */
 void qh_initstatistics(void) {
   int i;
   realT realx;
   int intx;
  
 #if qh_QHpointer
   if(qh_qhstat){  /* qh_initstatistics may be called from Qhull::resetStatistics() */
       qh_free(qh_qhstat);
       qh_qhstat= 0;
   }
   if (!(qh_qhstat= (qhstatT *)qh_malloc(sizeof(qhstatT)))) {
     qh_fprintf_stderr(6183, "qhull error (qh_initstatistics): insufficient memory\n");
     qh_exit(qh_ERRmem);  /* can not use qh_errexit() */
   }
 #endif
  
   qhstat next= 0;
   qh_allstatA();
   qh_allstatB();
   qh_allstatC();
   qh_allstatD();
   qh_allstatE();
   qh_allstatE2();
   qh_allstatF();
   qh_allstatG();
   qh_allstatH();
   qh_allstatI();
   if (qhstat next > (int)sizeof(qhstat id)) {
     qh_fprintf(qhmem.ferr, 6184, "qhull error (qh_initstatistics): increase size of qhstat.id[].\n\
       qhstat.next %d should be <= sizeof(qhstat id) %d\n", qhstat next, (int)sizeof(qhstat id));
 #if 0 /* for locating error, Znumridges should be duplicated */
     for(i=0; i < ZEND; i++) {
       int j;
       for(j=i+1; j < ZEND; j++) {
         if (qhstat id[i] == qhstat id[j]) {
           qh_fprintf(qhmem.ferr, 6185, "qhull error (qh_initstatistics): duplicated statistic %d at indices %d and %d\n",
               qhstat id[i], i, j);
         }
       }
     }
 #endif
     qh_exit(qh_ERRqhull);  /* can not use qh_errexit() */
   }
   qhstat init[zinc].i= 0;
   qhstat init[zadd].i= 0;
   qhstat init[zmin].i= INT_MAX;
   qhstat init[zmax].i= INT_MIN;
   qhstat init[wadd].r= 0;
   qhstat init[wmin].r= REALmax;
   qhstat init[wmax].r= -REALmax;
   for(i=0; i < ZEND; i++) {
     if (qhstat type[i] > ZTYPEreal) {
       realx= qhstat init[(unsigned char)(qhstat type[i])].r;
       qhstat stats[i].r= realx;
     }else if (qhstat type[i] != zdoc) {
       intx= qhstat init[(unsigned char)(qhstat type[i])].i;
       qhstat stats[i].i= intx;
     }
   }
 } /* initstatistics */
  
 /*-<a                             href="qh-stat.htm#TOC"
   >-------------------------------</a><a name="newstats">-</a>
  
   qh_newstats(  )
     returns True if statistics for zdoc
  
   returns:
     next zdoc
 */
 boolT qh_newstats(int idx, int *nextindex) {
   boolT isnew= False;
   int start, i;
  
   if (qhstat type[qhstat id[idx]] == zdoc)
     start= idx+1;
   else
     start= idx;
   for(i= start; i < qhstat next && qhstat type[qhstat id[i]] != zdoc; i++) {
     if (!qh_nostatistic(qhstat id[i]) && !qhstat printed[qhstat id[i]])
         isnew= True;
   }
   *nextindex= i;
   return isnew;
 } /* newstats */
  
 /*-<a                             href="qh-stat.htm#TOC"
   >-------------------------------</a><a name="nostatistic">-</a>
  
   qh_nostatistic( index )
     true if no statistic to print
 */
 boolT qh_nostatistic(int i) {
  
   if ((qhstat type[i] > ZTYPEreal
        &&qhstat stats[i].r == qhstat init[(unsigned char)(qhstat type[i])].r)
       || (qhstat type[i] < ZTYPEreal
           &&qhstat stats[i].i == qhstat init[(unsigned char)(qhstat type[i])].i))
     return True;
   return False;
 } /* nostatistic */
  
 #if qh_KEEPstatistics
 /*-<a                             href="qh-stat.htm#TOC"
   >-------------------------------</a><a name="printallstatistics">-</a>
  
   qh_printallstatistics( fp, string )
     print all statistics with header 'string'
 */
 void qh_printallstatistics(FILE *fp, const char *string) {
  
   qh_allstatistics();
   qh_collectstatistics();
   qh_printstatistics(fp, string);
   qh_memstatistics(fp);
 }
  
  
 /*-<a                             href="qh-stat.htm#TOC"
   >-------------------------------</a><a name="printstatistics">-</a>
  
   qh_printstatistics( fp, string )
     print statistics to a file with header 'string'
     skips statistics with qhstat.printed[] (reset with qh_allstatistics)
  
   see:
     qh_printallstatistics()
 */
 void qh_printstatistics(FILE *fp, const char *string) {
   int i, k;
   realT ave;
  
   if (qh num_points != qh num_vertices) {
     wval_(Wpbalance)= 0;
     wval_(Wpbalance2)= 0;
   }else
     wval_(Wpbalance2)= qh_stddev(zval_(Zpbalance), wval_(Wpbalance),
                                  wval_(Wpbalance2), &ave);
   wval_(Wnewbalance2)= qh_stddev(zval_(Zprocessed), wval_(Wnewbalance),
                                  wval_(Wnewbalance2), &ave);
   qh_fprintf(fp, 9350, "\n\
 %s\n\
  qhull invoked by: %s | %s\n%s with options:\n%s\n", string, qh rbox_command,
      qh qhull_command, qh_version, qh qhull_options);
   qh_fprintf(fp, 9351, "\nprecision constants:\n\
  %6.2g max. abs. coordinate in the (transformed) input('Qbd:n')\n\
  %6.2g max. roundoff error for distance computation('En')\n\
  %6.2g max. roundoff error for angle computations\n\
  %6.2g min. distance for outside points ('Wn')\n\
  %6.2g min. distance for visible facets ('Vn')\n\
  %6.2g max. distance for coplanar facets ('Un')\n\
  %6.2g max. facet width for recomputing centrum and area\n\
 ",
   qh MAXabs_coord, qh DISTround, qh ANGLEround, qh MINoutside,
         qh MINvisible, qh MAXcoplanar, qh WIDEfacet);
   if (qh KEEPnearinside)
     qh_fprintf(fp, 9352, "\
  %6.2g max. distance for near-inside points\n", qh NEARinside);
   if (qh premerge_cos < REALmax/2) qh_fprintf(fp, 9353, "\
  %6.2g max. cosine for pre-merge angle\n", qh premerge_cos);
   if (qh PREmerge) qh_fprintf(fp, 9354, "\
  %6.2g radius of pre-merge centrum\n", qh premerge_centrum);
   if (qh postmerge_cos < REALmax/2) qh_fprintf(fp, 9355, "\
  %6.2g max. cosine for post-merge angle\n", qh postmerge_cos);
   if (qh POSTmerge) qh_fprintf(fp, 9356, "\
  %6.2g radius of post-merge centrum\n", qh postmerge_centrum);
   qh_fprintf(fp, 9357, "\
  %6.2g max. distance for merging two simplicial facets\n\
  %6.2g max. roundoff error for arithmetic operations\n\
  %6.2g min. denominator for divisions\n\
   zero diagonal for Gauss: ", qh ONEmerge, REALepsilon, qh MINdenom);
   for(k=0; k < qh hull_dim; k++)
     qh_fprintf(fp, 9358, "%6.2e ", qh NEARzero[k]);
   qh_fprintf(fp, 9359, "\n\n");
   for(i=0 ; i < qhstat next; )
     qh_printstats(fp, i, &i);
 } /* printstatistics */
 #endif /* qh_KEEPstatistics */
  
 /*-<a                             href="qh-stat.htm#TOC"
   >-------------------------------</a><a name="printstatlevel">-</a>
  
   qh_printstatlevel( fp, id )
     print level information for a statistic
  
   notes:
     nop if id >= ZEND, printed, or same as initial value
 */
 void qh_printstatlevel(FILE *fp, int id) {
 #define NULLfield "       "
  
   if (id >= ZEND || qhstat printed[id])
     return;
   if (qhstat type[id] == zdoc) {
     qh_fprintf(fp, 9360, "%s\n", qhstat doc[id]);
     return;
   }
   if (qh_nostatistic(id) || !qhstat doc[id])
     return;
   qhstat printed[id]= True;
   if (qhstat count[id] != -1
       && qhstat stats[(unsigned char)(qhstat count[id])].i == 0)
     qh_fprintf(fp, 9361, " *0 cnt*");
   else if (qhstat type[id] >= ZTYPEreal && qhstat count[id] == -1)
     qh_fprintf(fp, 9362, "%7.2g", qhstat stats[id].r);
   else if (qhstat type[id] >= ZTYPEreal && qhstat count[id] != -1)
     qh_fprintf(fp, 9363, "%7.2g", qhstat stats[id].r/ qhstat stats[(unsigned char)(qhstat count[id])].i);
   else if (qhstat type[id] < ZTYPEreal && qhstat count[id] == -1)
     qh_fprintf(fp, 9364, "%7d", qhstat stats[id].i);
   else if (qhstat type[id] < ZTYPEreal && qhstat count[id] != -1)
     qh_fprintf(fp, 9365, "%7.3g", (realT) qhstat stats[id].i / qhstat stats[(unsigned char)(qhstat count[id])].i);
   qh_fprintf(fp, 9366, " %s\n", qhstat doc[id]);
 } /* printstatlevel */
  
  
 /*-<a                             href="qh-stat.htm#TOC"
   >-------------------------------</a><a name="printstats">-</a>
  
   qh_printstats( fp, index, nextindex )
     print statistics for a zdoc group
  
   returns:
     next zdoc if non-null
 */
 void qh_printstats(FILE *fp, int idx, int *nextindex) {
   int j, nexti;
  
   if (qh_newstats(idx, &nexti)) {
     qh_fprintf(fp, 9367, "\n");
     for (j=idx; j<nexti; j++)
       qh_printstatlevel(fp, qhstat id[j]);
   }
   if (nextindex)
     *nextindex= nexti;
 } /* printstats */
  
 #if qh_KEEPstatistics
  
 /*-<a                             href="qh-stat.htm#TOC"
   >-------------------------------</a><a name="stddev">-</a>
  
   qh_stddev( num, tot, tot2, ave )
     compute the standard deviation and average from statistics
  
     tot2 is the sum of the squares
   notes:
     computes r.m.s.:
       (x-ave)^2
       == x^2 - 2x tot/num +   (tot/num)^2
       == tot2 - 2 tot tot/num + tot tot/num
       == tot2 - tot ave
 */
 realT qh_stddev(int num, realT tot, realT tot2, realT *ave) {
   realT stddev;
  
   *ave= tot/num;
   stddev= sqrt(tot2/num - *ave * *ave);
   return stddev;
 } /* stddev */
  
 #endif /* qh_KEEPstatistics */
  
 #if !qh_KEEPstatistics
 void    qh_collectstatistics(void) {}
 void    qh_printallstatistics(FILE *fp, char *string) {};
 void    qh_printstatistics(FILE *fp, char *string) {}
 #endif
  