user-level header file for using qhull.a library

   see qh-qhull.htm, qhull_a.h

   copyright (c) 1993-2002, The Geometry Center

   NOTE: access to qh_qh is via the 'qh' macro.  This allows
   qh_qh to be either a pointer or a structure.  An example
   of using qh is "qh DROPdim" which accesses the DROPdim
   field of qh_qh.  Similarly, access to qh_qhstat is via
   the 'qhstat' macro.

   includes function prototypes for qhull.c, geom.c, global.c, io.c, user.c

   use mem.h for mem.c
   use qset.h for qset.c

   see unix.c for an example of using qhull.h

   recompile qhull if you change this file

#ifndef qhDEFqhull
#define qhDEFqhull 1

/*=========================== -included files ==============*/


#if __MWERKS__ && __POWERPC__

#ifndef __STDC__
#ifndef __cplusplus
#if     !_MSC_VER
#error  Neither __STDC__ nor __cplusplus is defined.  Please use strict ANSI C or C++ to compile
#error  Qhull.  You may need to turn off compiler extensions in your project configuration.  If
#error  your compiler is a standard C compiler, you can delete this warning from qhull.h

#include "user.h"      /* user defineable constants */

/*============ constants and basic types ====================*/


    version string by year and date

    the revision increases on code changes only

    change date:    Changes.txt, Announce.txt, README.txt, qhull.man
                    qhull-news.html, Eudora signatures, 
    change version: README.txt, qhull.html, file_id.diz, Makefile
    change year:    Copying.txt
    check download size
    recompile user_eg.c, rbox.c, qhull.c, qconvex.c, qdelaun.c qvoronoi.c, qhalf.c
    make copy of qhull-news.html as qh-news.htm

#define qh_VERSION "2002.1 2002/8/20"


    coordinates and coefficients are stored as realT (i.e., double)

    could use 'float' for data and 'double' for calculations (realT vs. coordT)
      This requires many type casts, and adjusted error bounds.
      Also C compilers may do expressions in double anyway.
#define coordT realT


    a point is an array of DIM3 coordinates
#define pointT coordT


    Boolean flag as a bit
#define flagT unsigned int


    boolean value, either True or False

    needed for portability
#define boolT unsigned int
#ifdef False
#undef False
#ifdef True
#undef True
#define False 0
#define True 1


    to distinguish facet->center
typedef enum
    qh_ASnone = 0, qh_ASvoronoi, qh_AScentrum


    output formats for printing (qh.PRINTout).
    'Fa' 'FV' 'Fc' 'FC' 

   some of these names are similar to qh names.  The similar names are only
   used in switch statements in qh_printbegin() etc.
typedef enum {qh_PRINTnone= 0, 
  qh_PRINTarea, qh_PRINTaverage,           /* 'Fa' 'FV' 'Fc' 'FC' */
  qh_PRINTcoplanars, qh_PRINTcentrums, 
  qh_PRINTfacets, qh_PRINTfacets_xridge,   /* 'f' 'FF' 'G' 'FI' 'Fi' 'Fn' */
  qh_PRINTgeom, qh_PRINTids, qh_PRINTinner, qh_PRINTneighbors, 
  qh_PRINTnormals, qh_PRINTouter,          /* 'n' 'Fo' 'i' 'm' 'Fm' 'o' */
  qh_PRINTincidences, qh_PRINTmathematica, qh_PRINTmerges, qh_PRINToff, 
  qh_PRINToptions, qh_PRINTpointintersect, /* 'FO' 'Fp' 'FP' 'p' 'FQ' 'FS' */
  qh_PRINTpointnearest, qh_PRINTpoints, qh_PRINTqhull, qh_PRINTsize, 
  qh_PRINTsummary, qh_PRINTtriangles,      /* 'Fs' 'Ft' 'Fv' 'FN' 'Fx' */
  qh_PRINTvertices, qh_PRINTvneighbors, qh_PRINTextremes,


    argument flag for selecting everything
#define qh_ALL      True
#define qh_NOupper  True     /* argument for qh_findbest */
#define qh_IScheckmax  True     /* argument for qh_findbesthorizon */
#define qh_ISnewfacets  True     /* argument for qh_findbest */
#define qh_RESETvisible  True     /* argument for qh_resetlists */


    Qhull exit codes, for indicating errors
#define qh_ERRnone  0    /* no error occurred during qhull */
#define qh_ERRinput 1    /* input inconsistency */
#define qh_ERRsingular 2 /* singular input data */
#define qh_ERRprec  3    /* precision error */
#define qh_ERRmem   4    /* insufficient memory, matches mem.h */
#define qh_ERRqhull 5    /* internal error detected, matches mem.h */

/* ============ -structures- ====================
   each of the following structures is defined by a typedef
   all realT and coordT fields occur at the beginning of a structure
        (otherwise space may be wasted due to alignment)
   define all flags together and pack into 32-bit number

typedef struct vertexT vertexT;
typedef struct ridgeT ridgeT;
typedef struct facetT facetT;
#ifndef DEFsetT
#define DEFsetT 1
typedef struct setT setT;          /* defined in qset.h */


    defines a facet

   qhull() generates the hull as a list of facets.

  topological information:
    f.previous,next     doubly-linked list of facets
    f.vertices          set of vertices
    f.ridges            set of ridges
    f.neighbors         set of neighbors
    f.toporient         True if facet has top-orientation (else bottom)

  geometric information:
    f.offset,normal     hyperplane equation
    f.maxoutside        offset to outer plane -- all points inside
    f.center            centrum for testing convexity
    f.simplicial        True if facet is simplicial
    f.flipped           True if facet does not include qh.interior_point

  for constructing hull:
    f.visible           True if facet on list of visible facets (will be deleted)
    f.newfacet          True if facet on list of newly created facets
    f.coplanarset       set of points coplanar with this facet
                        (includes near-inside points for later testing)
    f.outsideset        set of points outside of this facet
    f.furthestdist      distance to furthest point of outside set
    f.visitid           marks visited facets during a loop
    f.replace           replacement facet for to-be-deleted, visible facets
    f.samecycle,newcycle cycle of facets for merging into horizon facet

  see below for other flags and fields
struct facetT {
#if !qh_COMPUTEfurthest
  coordT   furthestdist;/* distance to furthest point of outsideset */
#if qh_MAXoutside
  coordT   maxoutside;  /* max computed distance of point to facet
  			Before QHULLfinished this is an approximation
  			since maxdist not always set for mergefacet
			Actual outer plane is +DISTround and
			computed outer plane is +2*DISTround */
  coordT   offset;      /* exact offset of hyperplane from origin */
  coordT  *normal;      /* normal of hyperplane, hull_dim coefficients */
			/*   if tricoplanar, shared with a neighbor */
  union {               /* in order of testing */
   realT   area;        /* area of facet, only in io.c if  ->isarea */
   facetT *replace;	/*  replacement facet if ->visible and NEWfacets
  			     is NULL only if qh_mergedegen_redundant or interior */
   facetT *samecycle;   /*  cycle of facets from the same visible/horizon intersection,
   			     if ->newfacet */
   facetT *newcycle;    /*  in horizon facet, current samecycle of new facets */ 
   facetT *trivisible;  /* visible facet for ->tricoplanar facets during qh_triangulate() */
   facetT *triowner;    /* owner facet for ->tricoplanar, !isarea facets w/ ->keepcentrum */
  coordT  *center;      /*  centrum for convexity, qh CENTERtype == qh_AScentrum */
      			/*  Voronoi center, qh CENTERtype == qh_ASvoronoi */
			/*   if tricoplanar, shared with a neighbor */
  facetT  *previous;    /* previous facet in the facet_list */
  facetT  *next;        /* next facet in the facet_list */
  setT    *vertices;    /* vertices for this facet, inverse sorted by ID 
                           if simplicial, 1st vertex was apex/furthest */
  setT    *ridges;      /* explicit ridges for nonsimplicial facets.
  			   for simplicial facets, neighbors defines ridge */
  setT    *neighbors;   /* neighbors of the facet.  If simplicial, the kth
			   neighbor is opposite the kth vertex, and the first
			   neighbor is the horizon facet for the first vertex*/
  setT    *outsideset;  /* set of points outside this facet
		           if non-empty, last point is furthest
			   if NARROWhull, includes coplanars for partitioning*/
  setT    *coplanarset; /* set of points coplanar with this facet
  			   > qh.min_vertex and <= facet->max_outside
                           a point is assigned to the furthest facet
		           if non-empty, last point is furthest away */
  unsigned visitid;     /* visit_id, for visiting all neighbors,
			   all uses are independent */
  unsigned id;	        /* unique identifier from qh facet_id */
  unsigned nummerge:9;  /* number of merges */
#define qh_MAXnummerge 511 /*     2^9-1, 32 flags total, see "flags:" in io.c */
  flagT    tricoplanar:1; /* True if TRIangulate and simplicial and coplanar with a neighbor */
			  /*   all tricoplanars share the same ->center, ->normal, ->offset, ->maxoutside */
			  /*   all tricoplanars share the same apex */
                          /*   if ->degenerate, does not span facet (one logical ridge) */
                          /*   one tricoplanar has ->keepcentrum and ->coplanarset */
                          /*   during qh_triangulate, f.trivisible points to original facet */
  flagT	   newfacet:1;  /* True if facet on qh newfacet_list (new or merged) */
  flagT	   visible:1;   /* True if visible facet (will be deleted) */
  flagT    toporient:1; /* True if created with top orientation
			   after merging, use ridge orientation */
  flagT    simplicial:1;/* True if simplicial facet, ->ridges may be implicit */
  flagT    seen:1;      /* used to perform operations only once, like visitid */
  flagT    seen2:1;     /* used to perform operations only once, like visitid */
  flagT	   flipped:1;   /* True if facet is flipped */
  flagT    upperdelaunay:1; /* True if facet is upper envelope of Delaunay triangulation */
  flagT    notfurthest:1; /* True if last point of outsideset is not furthest*/

/*-------- flags primarily for output ---------*/
  flagT	   good:1;      /* True if a facet marked good for output */
  flagT    isarea:1;    /* True if facet->f.area is defined */

/*-------- flags for merging ------------------*/
  flagT    dupridge:1;  /* True if duplicate ridge in facet */
  flagT    mergeridge:1; /* True if facet or neighbor contains a qh_MERGEridge
                            ->normal defined (also defined for mergeridge2) */
  flagT    mergeridge2:1; /* True if neighbor contains a qh_MERGEridge (mark_dupridges */
  flagT    coplanar:1;  /* True if horizon facet is coplanar at last use */
  flagT     mergehorizon:1; /* True if will merge into horizon (->coplanar) */
  flagT	    cycledone:1;/* True if mergecycle_all already done */
  flagT    tested:1;    /* True if facet convexity has been tested (false after merge */
  flagT    keepcentrum:1; /* True if keep old centrum after a merge, or marks owner for ->tricoplanar */
  flagT	   newmerge:1;  /* True if facet is newly merged for reducevertices */
  flagT	   degenerate:1; /* True if facet is degenerate (degen_mergeset or ->tricoplanar) */
  flagT	   redundant:1;  /* True if facet is redundant (degen_mergeset) */


    defines a ridge

  a ridge is DIM3-1 simplex between two neighboring facets.  If the
  facets are non-simplicial, there may be more than one ridge between
  two facets.  E.G. a 4-d hypercube has two triangles between each pair
  of neighboring facets.

  topological information:
    vertices            a set of vertices
    top,bottom          neighboring facets with orientation

  geometric information:
    tested              True if ridge is clearly convex
    nonconvex           True if ridge is non-convex
struct ridgeT {
  setT    *vertices;    /* vertices belonging to this ridge, inverse sorted by ID 
                           NULL if a degen ridge (matchsame) */
  facetT  *top;         /* top facet this ridge is part of */
  facetT  *bottom;      /* bottom facet this ridge is part of */
  unsigned id:24;       /* unique identifier, =>room for 8 flags */
  flagT    seen:1;      /* used to perform operations only once */
  flagT    tested:1;    /* True when ridge is tested for convexity */
  flagT    nonconvex:1; /* True if getmergeset detected a non-convex neighbor
			   only one ridge between neighbors may have nonconvex */


     defines a vertex

  topological information:
    next,previous       doubly-linked list of all vertices
    neighbors           set of adjacent facets (only if qh.VERTEXneighbors)

  geometric information:
    point               array of DIM3 coordinates
struct vertexT {
  vertexT *next;        /* next vertex in vertex_list */
  vertexT *previous;    /* previous vertex in vertex_list */
  pointT  *point;       /* hull_dim coordinates (coordT) */
  setT    *neighbors;   /* neighboring facets of vertex, qh_vertexneighbors()
			   inits in io.c or after first merge */
  unsigned visitid; /* for use with qh vertex_visit */
  unsigned id:24;   /* unique identifier, =>room for 8 flags */
  flagT    seen:1;      /* used to perform operations only once */
  flagT    seen2:1;     /* another seen flag */
  flagT    delridge:1;  /* vertex was part of a deleted ridge */
  flagT	   deleted:1;   /* true if vertex on qh del_vertices */
  flagT    newlist:1;   /* true if vertex on qh newvertex_list */

/*======= -global variables -qh ============================*/


   all global variables for qhull are in qh, qhmem, and qhstat

   qhmem is defined in mem.h and qhstat is defined in stat.h
   access to qh_qh is via the "qh" macro.  See qh_QHpointer in user.h
typedef struct qhT qhT;
#if qh_QHpointer
#define qh qh_qh->
extern qhT *qh_qh;     /* allocated in global.c */
#define qh qh_qh.
extern qhT qh_qh;

struct qhT {


  qh constants
    configuration flags and constants for Qhull

    The user configures Qhull by defining flags.  They are
    copied into qh by qh_setflags().  qh-quick.htm#options defines the flags.
  boolT ALLpoints;        /* true 'Qs' if search all points for initial simplex */
  boolT ANGLEmerge;	  /* true 'Qa' if sort potential merges by angle */
  boolT APPROXhull;       /* true 'Wn' if MINoutside set */
  realT MINoutside;       /*   'Wn' min. distance for an outside point */
  boolT ATinfinity;       /* true 'Qz' if point num_points-1 is "at-infinity"
                             for improving precision in Delaunay triangulations */
  boolT AVOIDold;         /* true 'Q4' if avoid old->new merges */
  boolT BESToutside;      /* true 'Qf' if partition points into best outsideset */
  boolT CDDinput;         /* true 'Pc' if input uses CDD format (1.0/offset first) */
  boolT CDDoutput;        /* true 'PC' if print normals in CDD format (offset first) */
  boolT CHECKfrequently;  /* true 'Tc' if checking frequently */
  realT premerge_cos;     /*   'A-n'   cos_max when pre merging */
  realT postmerge_cos;    /*   'An'    cos_max when post merging */
  boolT DELAUNAY;         /* true 'd' if computing DELAUNAY triangulation */
  boolT DOintersections;  /* true 'Gh' if print hyperplane intersections */
  int   DROPdim;          /* drops dim 'GDn' for 4-d -> 3-d output */
  boolT FORCEoutput;      /* true 'Po' if forcing output despite degeneracies */
  int   GOODpoint;        /* 1+n for 'QGn', good facet if visible/not(-) from point n*/
  pointT *GOODpointp;     /*   the actual point */
  boolT GOODthreshold;    /* true if qh lower_threshold/upper_threshold defined
  			     false if qh SPLITthreshold */
  int   GOODvertex;       /* 1+n, good facet if vertex for point n */
  pointT *GOODvertexp;     /*   the actual point */
  boolT HALFspace;        /* true 'Hn,n,n' if halfspace intersection */
  int   IStracing;        /* trace execution, 0=none, 1=least, 4=most, -1=events */
  int   KEEParea;         /* 'PAn' number of largest facets to keep */
  boolT KEEPcoplanar;     /* true 'Qc' if keeping nearest facet for coplanar points */
  boolT KEEPinside;       /* true 'Qi' if keeping nearest facet for inside points
			      set automatically if 'd Qc' */
  int   KEEPmerge;        /* 'PMn' number of facets to keep with most merges */
  realT KEEPminArea;      /* 'PFn' minimum facet area to keep */
  realT MAXcoplanar;      /* 'Un' max distance below a facet to be coplanar*/
  boolT MERGEexact;	  /* true 'Qx' if exact merges (coplanar, degen, dupridge, flipped) */
  boolT MERGEindependent; /* true 'Q2' if merging independent sets */
  boolT MERGING;          /* true if exact-, pre- or post-merging, with angle and centrum tests */
  realT   premerge_centrum;  /*   'C-n' centrum_radius when pre merging.  Default is round-off */
  realT   postmerge_centrum; /*   'Cn' centrum_radius when post merging.  Default is round-off */
  boolT MERGEvertices;	  /* true 'Q3' if merging redundant vertices */
  realT MINvisible;       /* 'Vn' min. distance for a facet to be visible */
  boolT NOnarrow;         /* true 'Q10' if no special processing for narrow distributions */
  boolT NOnearinside;     /* true 'Q8' if ignore near-inside points when partitioning */
  boolT NOpremerge;       /* true 'Q0' if no defaults for C-0 or Qx */
  boolT ONLYgood; 	  /* true 'Qg' if process points with good visible or horizon facets */
  boolT ONLYmax; 	  /* true 'Qm' if only process points that increase max_outside */
  boolT PICKfurthest;     /* true 'Q9' if process furthest of furthest points*/
  boolT POSTmerge;        /* true if merging after buildhull (Cn or An) */
  boolT PREmerge;         /* true if merging during buildhull (C-n or A-n) */
  			/* NOTE: some of these names are similar to qh_PRINT names */
  boolT PRINTcentrums;	  /* true 'Gc' if printing centrums */
  boolT PRINTcoplanar;    /* true 'Gp' if printing coplanar points */
  int	PRINTdim;      	  /* print dimension for Geomview output */
  boolT PRINTdots;        /* true 'Ga' if printing all points as dots */
  boolT PRINTgood;        /* true 'Pg' if printing good facets */
  boolT PRINTinner;	  /* true 'Gi' if printing inner planes */
  boolT PRINTneighbors;	  /* true 'PG' if printing neighbors of good facets */
  boolT PRINTnoplanes;	  /* true 'Gn' if printing no planes */
  boolT PRINToptions1st;  /* true 'FO' if printing options to stderr */
  boolT PRINTouter;	  /* true 'Go' if printing outer planes */
  boolT PRINTprecision;   /* false 'Pp' if not reporting precision problems */
  qh_PRINT PRINTout[qh_PRINTEND]; /* list of output formats to print */
  boolT PRINTridges;      /* true 'Gr' if print ridges */
  boolT PRINTspheres;     /* true 'Gv' if print vertices as spheres */
  boolT PRINTstatistics;  /* true 'Ts' if printing statistics to stderr */
  boolT PRINTsummary;     /* true 's' if printing summary to stderr */
  boolT PRINTtransparent; /* true 'Gt' if print transparent outer ridges */
  boolT PROJECTdelaunay;  /* true if DELAUNAY, no readpoints() and
			     need projectinput() for Delaunay in qh_init_B */
  int   PROJECTinput;     /* number of projected dimensions 'bn:0Bn:0' */
  boolT QUICKhelp;	  /* true if quick help message for degen input */
  boolT RANDOMdist;       /* true if randomly change distplane and setfacetplane */
  realT RANDOMfactor;     /*    maximum random perturbation */
  realT RANDOMa;         /*  qh_randomfactor is randr * RANDOMa + RANDOMb */
  realT RANDOMb;
  boolT RANDOMoutside;    /* true if select a random outside point */
  int	REPORTfreq;       /* buildtracing reports every n facets */
  int   REPORTfreq2;	  /* tracemerging reports every REPORTfreq/2 facets */
  int	RERUN;            /* 'TRn' rerun qhull n times (qh.build_cnt) */
  int	ROTATErandom;	  /* 'QRn' seed, 0 time, >= rotate input */
  boolT SCALEinput;       /* true 'Qbk' if scaling input */
  boolT SCALElast;        /* true 'Qbb' if scale last coord to max prev coord */
  boolT SETroundoff;      /* true 'E' if qh DISTround is predefined */
  boolT SKIPcheckmax;	  /* true 'Q5' if skip qh_check_maxout */
  boolT SKIPconvex;       /* true 'Q6' if skip convexity testing during pre-merge */
  boolT SPLITthresholds;  /* true if upper_/lower_threshold defines a region
                               used only for printing (not for qh ONLYgood) */
  int	STOPcone;         /* 'TCn' 1+n for stopping after cone for point n*/
			  /*       also used by qh_build_withresart for err exit*/
  int	STOPpoint;        /* 'TVn' 'TV-n' 1+n for stopping after/before(-)
			                adding point n */
  int	TESTpoints;	  /* 'QTn' num of test points after qh.num_points.  Test points always coplanar. */
  boolT TESTvneighbors;   /*  true 'Qv' if test vertex neighbors at end */
  int   TRACElevel;       /* 'Tn' conditional IStracing level */
  int	TRACElastrun;	  /*  qh.TRACElevel applies to last qh.RERUN */
  int   TRACEpoint;       /* 'TPn' start tracing when point n is a vertex */
  realT TRACEdist;        /* 'TWn' start tracing when merge distance too big */
  int   TRACEmerge;       /* 'TMn' start tracing before this merge */
  boolT TRIangulate;	  /* true 'Qt' if triangulate non-simplicial facets */
  boolT TRInormals;	  /* true 'Q11' if triangulate duplicates normals (sets Qt) */
  boolT UPPERdelaunay;    /* true 'Qu' if computing furthest-site Delaunay */
  boolT VERIFYoutput;     /* true 'Tv' if verify output at end of qhull */
  boolT VIRTUALmemory;    /* true 'Q7' if depth-first processing in buildhull */
  boolT VORONOI;	  /* true 'v' if computing Voronoi diagram */

  /*--------input constants ---------*/
  realT AREAfactor;       /* 1/(hull_dim-1)! for converting det's to area */
  boolT DOcheckmax;       /* true if calling qh_check_maxout (qh_initqhull_globals) */
  char	*feasible_string;  /* feasible point 'Hn,n,n' for halfspace intersection */
  coordT *feasible_point;  /*    as coordinates, both malloc'd */
  boolT GETarea;          /* true 'Fa', 'FA', 'FS', 'PAn', 'PFn' if compute facet area/Voronoi volume in io.c */
  boolT KEEPnearinside;   /* true if near-inside points in coplanarset */
  int 	hull_dim;         /* dimension of hull, set by initbuffers */
  int 	input_dim;	  /* dimension of input, set by initbuffers */
  int 	num_points;       /* number of input points */
  pointT *first_point;    /* array of input points, see POINTSmalloc */
  boolT POINTSmalloc;     /*   true if qh first_point/num_points allocated */
  pointT *input_points;   /* copy of original qh.first_point for input points for qh_joggleinput */
  boolT input_malloc;     /* true if qh input_points malloc'd */
  char 	qhull_command[256];/* command line that invoked this program */
  char 	rbox_command[256]; /* command line that produced the input points */
  char  qhull_options[512];/* descriptive list of options */
  int   qhull_optionlen;  /*    length of last line */
  int   qhull_optionsiz;  /*     size of qhull_options before qh_initbuild */
  boolT VERTEXneighbors;  /* true if maintaining vertex neighbors */
  boolT ZEROcentrum;      /* true if 'C-0' or 'C-0 Qx'.  sets ZEROall_ok */
  realT *upper_threshold; /* don't print if facet->normal[k]>=upper_threshold[k]
                             must set either GOODthreshold or SPLITthreshold
  			     if Delaunay, default is 0.0 for upper envelope */
  realT *lower_threshold; /* don't print if facet->normal[k] <=lower_threshold[k] */
  realT *upper_bound;     /* scale point[k] to new upper bound */
  realT *lower_bound;     /* scale point[k] to new lower bound
  			     project if both upper_ and lower_bound == 0 */


  qh precision constants
    precision constants for Qhull

    qh_detroundoff() computes the maximum roundoff error for distance
    and other computations.  It also sets default values for the
    qh constants above.
  realT ANGLEround;       /* max round off error for angles */
  realT centrum_radius;   /* max centrum radius for convexity (roundoff added) */
  realT cos_max;	  /* max cosine for convexity (roundoff added) */
  realT DISTround;        /* max round off error for distances, 'E' overrides */
  realT MAXabs_coord;     /* max absolute coordinate */
  realT MAXlastcoord;     /* max last coordinate for qh_scalelast */
  realT MAXsumcoord;      /* max sum of coordinates */
  realT MAXwidth;         /* max rectilinear width of point coordinates */
  realT MINdenom_1;       /* min. abs. value for 1/x */
  realT MINdenom;         /*    use divzero if denominator < MINdenom */
  realT MINdenom_1_2;     /* min. abs. val for 1/x that allows normalization */
  realT MINdenom_2;       /*    use divzero if denominator < MINdenom_2 */
  realT MINlastcoord;     /* min. last coordinate for qh_scalelast */
  boolT NARROWhull;       /* set in qh_initialhull if angle < qh_MAXnarrow */
  realT *NEARzero;        /* hull_dim array for near zero in gausselim */
  realT NEARinside;       /* keep points for qh_check_maxout if close to facet */
  realT ONEmerge;         /* max distance for merging simplicial facets */
  realT outside_err;      /* application's epsilon for coplanar points
                             qh_check_bestdist() qh_check_points() reports error if point outside */
  realT WIDEfacet;        /* size of wide facet for skipping ridge in
			     area computation and locking centrum */

  qh internal constants
    internal constants for Qhull
  char qhull[sizeof("qhull")]; /* for checking ownership */
  void *old_stat;         /* pointer to saved qh_qhstat, qh_save_qhull */
  jmp_buf errexit;        /* exit label for qh_errexit, defined by setjmp() */
  char jmpXtra[40];       /* extra bytes in case jmp_buf is defined wrong by compiler */
  jmp_buf restartexit;    /* restart label for qh_errexit, defined by setjmp() */
  char jmpXtra2[40];      /* extra bytes in case jmp_buf is defined wrong by compiler*/
  FILE *fin;              /* pointer to input file, init by qh_meminit */
  FILE *fout;             /* pointer to output file */
  FILE *ferr;             /* pointer to error file */
  pointT *interior_point; /* center point of the initial simplex*/
  int   normal_size;      /* size in bytes for facet normals and point coords*/
  int   center_size;      /* size in bytes for Voronoi centers */
  int   TEMPsize;         /* size for small, temporary sets (in quick mem) */


  qh facet and vertex lists
    defines lists of facets, new facets, visible facets, vertices, and
    new vertices.  Includes counts, next ids, and trace ids.
  facetT *facet_list;     /* first facet */
  facetT  *facet_tail;     /* end of facet_list (dummy facet) */
  facetT *facet_next;     /* next facet for buildhull()
    			     previous facets do not have outside sets
                             NARROWhull: previous facets may have coplanar outside sets for qh_outcoplanar */
  facetT *newfacet_list;  /* list of new facets to end of facet_list */
  facetT *visible_list;   /* list of visible facets preceeding newfacet_list,
                             facet->visible set */
  int       num_visible;  /* current number of visible facets */
  unsigned tracefacet_id;  /* set at init, then can print whenever */
  facetT *tracefacet;     /*   set in newfacet/mergefacet, undone in delfacet*/
  unsigned tracevertex_id;  /* set at buildtracing, can print whenever */
  vertexT *tracevertex;     /*   set in newvertex, undone in delvertex*/
  vertexT *vertex_list;     /* list of all vertices, to vertex_tail */
  vertexT  *vertex_tail;    /*      end of vertex_list (dummy vertex) */
  vertexT *newvertex_list; /* list of vertices in newfacet_list, to vertex_tail
                             all vertices have 'newlist' set */
  int 	num_facets;	  /* number of facets in facet_list
			     includes visble faces (num_visible) */
  int 	num_vertices;     /* number of vertices in facet_list */
  int   num_outside;      /* number of points in outsidesets (for tracing and RANDOMoutside)
                               includes coplanar outsideset points for NARROWhull/qh_outcoplanar() */
  int   num_good;         /* number of good facets (after findgood_all) */
  unsigned facet_id;      /* ID of next, new facet from newfacet() */
  unsigned ridge_id;      /* ID of next, new ridge from newridge() */
  unsigned vertex_id;     /* ID of next, new vertex from newvertex() */


  qh global variables
    defines minimum and maximum distances, next visit ids, several flags,
    and other global variables.
    initialize in qh_initbuild or qh_maxmin if used in qh_buildhull
  unsigned long hulltime; /* ignore time to set up input and randomize */
                          /*   use unsigned to avoid wrap-around errors */
  boolT ALLOWrestart;     /* true if qh_precision can use qh.restartexit */
  int   build_cnt;        /* number of calls to qh_initbuild */
  qh_CENTER CENTERtype;   /* current type of facet->center, qh_CENTER */
  int 	furthest_id;      /* pointid of furthest point, for tracing */
  facetT *GOODclosest;    /* closest facet to GOODthreshold in qh_findgood */
  realT JOGGLEmax;        /* set 'QJn' if randomly joggle input */
  boolT maxoutdone;       /* set qh_check_maxout(), cleared by qh_addpoint() */
  realT max_outside;      /* maximum distance from a point to a facet,
			       before roundoff, not simplicial vertices
			       actual outer plane is +DISTround and
			       computed outer plane is +2*DISTround */
  realT max_vertex;       /* maximum distance (>0) from vertex to a facet,
			       before roundoff, due to a merge */
  realT min_vertex;       /* minimum distance (<0) from vertex to a facet,
			       before roundoff, due to a merge
			       if qh.JOGGLEmax, qh_makenewplanes sets it
  			       recomputed if qh.DOcheckmax, default -qh.DISTround */
  boolT NEWfacets;        /* true while visible facets invalid due to new or merge
			      from makecone/attachnewfacets to deletevisible */
  boolT findbestnew;	  /* true if partitioning calls qh_findbestnew */
  boolT findbest_notsharp; /* true if new facets are at least 90 degrees */
  boolT NOerrexit;        /* true if qh.errexit is not available */
  realT PRINTcradius;     /* radius for printing centrums */
  realT PRINTradius;      /* radius for printing vertex spheres and points */
  boolT POSTmerging;      /* true when post merging */
  int 	printoutvar;	  /* temporary variable for qh_printbegin, etc. */
  int 	printoutnum;	  /* number of facets printed */
  boolT QHULLfinished;    /* True after qhull() is finished */
  realT totarea;          /* 'FA': total facet area computed by qh_getarea */
  realT totvol;           /* 'FA': total volume computed by qh_getarea */
  unsigned int visit_id;  /* unique ID for searching neighborhoods, */
  unsigned int vertex_visit; /* unique ID for searching vertices */
  boolT ZEROall_ok;       /* True if qh_checkzero always succeeds */
  boolT WAScoplanar;      /* True if qh_partitioncoplanar (qh_check_maxout) */

  qh global sets
    defines sets for merging, initial simplex, hashing, extra input points,
    and deleted vertices
  setT *facet_mergeset;   /* temporary set of merges to be done */
  setT *degen_mergeset;   /* temporary set of degenerate and redundant merges */
  setT *hash_table;	  /* hash table for matching ridges in qh_matchfacets
                             size is setsize() */
  setT *other_points;     /* additional points (first is qh interior_point) */
  setT *del_vertices;     /* vertices to partition and delete with visible
                             facets.  Have deleted set for checkfacet */


  qh global buffers
    defines buffers for maxtrix operations, input, and error messages
  coordT *gm_matrix;      /* (dim+1)Xdim matrix for geom.c */
  coordT **gm_row;        /* array of gm_matrix rows */
  char* line;             /* malloc'd input line of maxline+1 chars */
  int maxline;
  coordT *half_space;     /* malloc'd input array for halfspace (qh normal_size+coordT) */
  coordT *temp_malloc;    /* malloc'd input array for points */

  qh static variables
    defines static variables for individual functions

    do not use 'static' within a function.  Multiple instances of qhull
    may exist.

    do not assume zero initialization, 'QPn' may cause a restart
  boolT ERREXITcalled;    /* true during errexit (prevents duplicate calls */
  boolT firstcentrum; 	  /* for qh_printcentrum */
  realT last_low;         /* qh_scalelast parameters for qh_setdelaunay */
  realT last_high;
  realT last_newhigh;
  unsigned lastreport;    /* for qh_buildtracing */
  int mergereport;        /* for qh_tracemerging */
  boolT old_randomdist;   /* save RANDOMdist when io, tracing, or statistics */
  int   ridgeoutnum;      /* number of ridges in 4OFF output */
  void *old_qhstat;       /* for saving qh_qhstat in save_qhull() */
  setT *old_tempstack;     /* for saving qhmem.tempstack in save_qhull */
  setT *coplanarset;      /* set of coplanar facets for searching qh_findbesthorizon() */

/*=========== -macros- =========================*/


  otherfacet_(ridge, facet)
    return neighboring facet for a ridge in facet
#define otherfacet_(ridge, facet) \
                        (((ridge)->top == (facet)) ? (ridge)->bottom : (ridge)->top)


    return ID for facet, ridge, or vertex
    return MAXINT if NULL (-1 causes type conversion error )
#define getid_(p)       ((p) ? (p)->id : -1)

/*============== FORALL macros ===================*/


  FORALLfacets { ... }
    assign 'facet' to each facet in qh.facet_list

    uses 'facetT *facet;'
    assumes last facet is a sentinel

    FORALLfacet_( facetlist )
#define FORALLfacets for (facet=qh facet_list;facet && facet->next;facet=facet->next)


  FORALLpoints { ... }
    assign 'point' to each point in qh.first_point, qh.num_points

    coordT *point, *pointtemp;
#define FORALLpoints FORALLpoint_(qh first_point, qh num_points)


  FORALLpoint_( points, num) { ... }
    assign 'point' to each point in points array of num points

    coordT *point, *pointtemp;
#define FORALLpoint_(points, num) for(point= (points), \
      pointtemp= (points)+qh hull_dim*(num); point < pointtemp; point += qh hull_dim)


  FORALLvertices { ... }
    assign 'vertex' to each vertex in qh.vertex_list

    vertexT *vertex;

    assumes qh.vertex_list terminated with a sentinel
#define FORALLvertices for (vertex=qh vertex_list;vertex && vertex->next;vertex= vertex->next)


  FOREACHfacet_( facets ) { ... }
    assign 'facet' to each facet in facets

    facetT *facet, **facetp;

#define FOREACHfacet_(facets)    FOREACHsetelement_(facetT, facets, facet)


  FOREACHneighbor_( facet ) { ... }
    assign 'neighbor' to each neighbor in facet->neighbors

  FOREACHneighbor_( vertex ) { ... }
    assign 'neighbor' to each neighbor in vertex->neighbors

    facetT *neighbor, **neighborp;

#define FOREACHneighbor_(facet)  FOREACHsetelement_(facetT, facet->neighbors, neighbor)


  FOREACHpoint_( points ) { ... }
    assign 'point' to each point in points set

    pointT *point, **pointp;

#define FOREACHpoint_(points)    FOREACHsetelement_(pointT, points, point)


  FOREACHridge_( ridges ) { ... }
    assign 'ridge' to each ridge in ridges set

    ridgeT *ridge, **ridgep;

#define FOREACHridge_(ridges)    FOREACHsetelement_(ridgeT, ridges, ridge)


  FOREACHvertex_( vertices ) { ... }
    assign 'vertex' to each vertex in vertices set

    vertexT *vertex, **vertexp;

#define FOREACHvertex_(vertices) FOREACHsetelement_(vertexT, vertices,vertex)


  FOREACHfacet_i_( facets ) { ... }
    assign 'facet' and 'facet_i' for each facet in facets set

    facetT *facet;
    int     facet_n, facet_i;

#define FOREACHfacet_i_(facets)    FOREACHsetelement_i_(facetT, facets, facet)


  FOREACHneighbor_i_( facet ) { ... }
    assign 'neighbor' and 'neighbor_i' for each neighbor in facet->neighbors

  FOREACHneighbor_i_( vertex ) { ... }
    assign 'neighbor' and 'neighbor_i' for each neighbor in vertex->neighbors

    facetT *neighbor;
    int     neighbor_n, neighbor_i;

#define FOREACHneighbor_i_(facet)  FOREACHsetelement_i_(facetT, facet->neighbors, neighbor)


  FOREACHpoint_i_( points ) { ... }
    assign 'point' and 'point_i' for each point in points set

    pointT *point;
    int     point_n, point_i;

#define FOREACHpoint_i_(points)    FOREACHsetelement_i_(pointT, points, point)


  FOREACHridge_i_( ridges ) { ... }
    assign 'ridge' and 'ridge_i' for each ridge in ridges set

    ridgeT *ridge;
    int     ridge_n, ridge_i;

#define FOREACHridge_i_(ridges)    FOREACHsetelement_i_(ridgeT, ridges, ridge)


  FOREACHvertex_i_( vertices ) { ... }
    assign 'vertex' and 'vertex_i' for each vertex in vertices set

    vertexT *vertex;
    int     vertex_n, vertex_i;

#define FOREACHvertex_i_(vertices) FOREACHsetelement_i_(vertexT, vertices,vertex)

/********* -qhull.c prototypes (duplicated from qhull_a.h) **********************/

void    qh_qhull (void);
boolT   qh_addpoint (pointT *furthest, facetT *facet, boolT checkdist);
void	qh_printsummary(FILE *fp);

/********* -user.c prototypes (alphabetical) **********************/

void 	qh_errexit(int exitcode, facetT *facet, ridgeT *ridge);
void 	qh_errprint(char* string, facetT *atfacet, facetT *otherfacet, ridgeT *atridge, vertexT *atvertex);
int     qh_new_qhull (int dim, int numpoints, coordT *points, boolT ismalloc,
		char *qhull_cmd, FILE *outfile, FILE *errfile);
void    qh_printfacetlist(facetT *facetlist, setT *facets, boolT printall);
void 	qh_user_memsizes (void);

/***** -geom.c/geom2.c prototypes (duplicated from geom.h) ****************/

facetT *qh_findbest (pointT *point, facetT *startfacet,
		     boolT bestoutside, boolT newfacets, boolT noupper,
		     realT *dist, boolT *isoutside, int *numpart);
facetT *qh_findbestnew (pointT *point, facetT *startfacet,
                     realT *dist, boolT bestoutside, boolT *isoutside, int *numpart);
boolT   qh_gram_schmidt(int dim, realT **rows);
void    qh_outerinner (facetT *facet, realT *outerplane, realT *innerplane);
void	qh_printsummary(FILE *fp);
void    qh_projectinput (void);
void    qh_randommatrix (realT *buffer, int dim, realT **row);
void    qh_rotateinput (realT **rows);
void    qh_scaleinput (void);
void    qh_setdelaunay (int dim, int count, pointT *points);
coordT  *qh_sethalfspace_all (int dim, int count, coordT *halfspaces, pointT *feasible);

/***** -global.c prototypes (alphabetical) ***********************/

unsigned long qh_clock (void);
void 	qh_checkflags (char *command, char *hiddenflags);
void 	qh_freebuffers (void);
void    qh_freeqhull (boolT allmem);
void    qh_init_A (FILE *infile, FILE *outfile, FILE *errfile, int argc, char *argv[]);
void    qh_init_B (coordT *points, int numpoints, int dim, boolT ismalloc);
void 	qh_init_qhull_command (int argc, char *argv[]);
void    qh_initbuffers (coordT *points, int numpoints, int dim, boolT ismalloc);
void 	qh_initflags (char *command);
void 	qh_initqhull_buffers (void);
void 	qh_initqhull_globals (coordT *points, int numpoints, int dim, boolT ismalloc);
void    qh_initqhull_mem (void);
void 	qh_initqhull_start (FILE *infile, FILE *outfile, FILE *errfile);
void 	qh_initthresholds (char *command);
void    qh_option (char *option, int *i, realT *r);
#if qh_QHpointer
void 	qh_restore_qhull (qhT **oldqh);
qhT    *qh_save_qhull (void);

/***** -io.c prototypes (duplicated from io.h) ***********************/

void    dfacet( unsigned id);
void    dvertex( unsigned id);
void	qh_printneighborhood (FILE *fp, int format, facetT *facetA, facetT *facetB, boolT printall);
void	qh_produce_output(void);
coordT *qh_readpoints(int *numpoints, int *dimension, boolT *ismalloc);

/********* -mem.c prototypes (duplicated from mem.h) **********************/

void qh_meminit (FILE *ferr);
void qh_memfreeshort (int *curlong, int *totlong);

/********* -poly.c/poly2.c prototypes (duplicated from poly.h) **********************/

void    qh_check_output (void);
void    qh_check_points (void);
setT   *qh_facetvertices (facetT *facetlist, setT *facets, boolT allfacets);
facetT *qh_findbestfacet (pointT *point, boolT bestoutside,
           realT *bestdist, boolT *isoutside);
vertexT *qh_nearvertex (facetT *facet, pointT *point, realT *bestdistp);
pointT *qh_point (int id);
setT   *qh_pointfacet (void /*qh.facet_list*/);
int     qh_pointid (pointT *point);
setT   *qh_pointvertex (void /*qh.facet_list*/);
void    qh_setvoronoi_all (void);
void	qh_triangulate (void /*qh facet_list*/);

/********* -stat.c prototypes (duplicated from stat.h) **********************/

void    qh_collectstatistics (void);
void    qh_printallstatistics (FILE *fp, char *string);

#endif /* qhDEFqhull */