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qhull.c

/*<html><pre>  -<a                             href="qh-qhull.htm"
  >-------------------------------</a><a name="TOP">-</a>

   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() =======*/

/*-<a                             href="qh-qhull.htm#TOC"
  >-------------------------------</a><a name="qhull">-</a>
  
  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 */

/*-<a                             href="qh-qhull.htm#TOC"
  >-------------------------------</a><a name="addpoint">-</a>
  
  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 */

/*-<a                             href="qh-qhull.htm#TOC"
  >-------------------------------</a><a name="build_withrestart">-</a>
  
  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 */

/*-<a                             href="qh-qhull.htm#TOC"
  >-------------------------------</a><a name="buildhull">-</a>
  
  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 */
  

/*-<a                             href="qh-qhull.htm#TOC"
  >-------------------------------</a><a name="buildtracing">-</a>
  
  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 */

/*-<a                             href="qh-qhull.htm#TOC"
  >-------------------------------</a><a name="errexit2">-</a>
  
  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 */


/*-<a                             href="qh-qhull.htm#TOC"
  >-------------------------------</a><a name="findhorizon">-</a>
  
  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 */

/*-<a                             href="qh-qhull.htm#TOC"
  >-------------------------------</a><a name="nextfurthest">-</a>
  
  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 */

/*-<a                             href="qh-qhull.htm#TOC"
  >-------------------------------</a><a name="partitionall">-</a>
  
  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 */


/*-<a                             href="qh-qhull.htm#TOC"
  >-------------------------------</a><a name="partitioncoplanar">-</a>
  
  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 */

/*-<a                             href="qh-qhull.htm#TOC"
  >-------------------------------</a><a name="partitionpoint">-</a>
  
  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 */

/*-<a                             href="qh-qhull.htm#TOC"
  >-------------------------------</a><a name="partitionvisible">-</a>
  
  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 */



/*-<a                             href="qh-qhull.htm#TOC"
  >-------------------------------</a><a name="precision">-</a>
  
  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 */

/*-<a                             href="qh-qhull.htm#TOC"
  >-------------------------------</a><a name="printsummary">-</a>
  
  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 */



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