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

	qhull.h
	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 ==============/

	#include <setjmp.h>
	#include <float.h>
	#include <time.h>

	#if __MWERKS__ && __POWERPC__
	#include <SIOUX.h>
	#include <Files.h>
	#include <Desk.h>
	#endif

	#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
	#endif
	#endif
	#endif

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

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

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

	qh_VERSION
	version string by year and date

	the revision increases on code changes only

	notes:
	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"

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

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

	notes:
	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 href="qh-geom.htm#TOC"
	>--------------------------------</a><a name="pointT">-</a>

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

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

	flagT
	Boolean flag as a bit
	*/
	#define flagT unsigned int

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

	boolT
	boolean value, either True or False

	notes:
	needed for portability
	*/
	#define boolT unsigned int
	#ifdef False
	#undef False
	#endif
	#ifdef True
	#undef True
	#endif
	#define False 0
	#define True 1

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

	qh_CENTER
	to distinguish facet->center
	*/
	typedef enum
	{
	qh_ASnone = 0, qh_ASvoronoi, qh_AScentrum
	}
	qh_CENTER;

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

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


	notes:
	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,
	qh_PRINTEND} qh_PRINT;

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

	qh_ALL
	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 */

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

	qh_ERR
	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 */
	#endif

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

	facetT
	defines a facet

	notes:
	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 */
	#endif
	#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 +2DISTround /
	#endif
	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 */
	}f;
	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) */
	};


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

	ridgeT
	defines a ridge

	notes:
	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 */
	};

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

	vertexT
	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 ============================/

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

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

	notes:
	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 */
	#else
	#define qh qh_qh.
	extern qhT qh_qh;
	#endif

	struct qhT {

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

	qh constants
	configuration flags and constants for Qhull

	notes:
	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 */

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

	qh precision constants
	precision constants for Qhull

	notes:
	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 */

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

	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) */

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

	qh facet and vertex lists
	defines lists of facets, new facets, visible facets, vertices, and
	new vertices. Includes counts, next ids, and trace ids.
	see:
	qh_resetlists()
	*/
	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() */

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

	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 +2DISTround /
	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) */

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

	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 */

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

	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 */

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

	qh static variables
	defines static variables for individual functions

	notes:
	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- =========================/

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

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

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

	getid_(p)
	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 ===================/

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

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

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

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

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

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

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

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

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

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

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

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

	declare:
	vertexT *vertex;

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

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

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

	declare:
	facetT facet, *facetp;

	see:
	<a href="qset.h#FOREACHsetelement_">FOREACHsetelement_</a>
	*/
	#define FOREACHfacet_(facets) FOREACHsetelement_(facetT, facets, facet)

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

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

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

	declare:
	facetT neighbor, *neighborp;

	see:
	<a href="qset.h#FOREACHsetelement_">FOREACHsetelement_</a>
	*/
	#define FOREACHneighbor_(facet) FOREACHsetelement_(facetT, facet->neighbors, neighbor)

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

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

	declare:
	pointT point, *pointp;

	see:
	<a href="qset.h#FOREACHsetelement_">FOREACHsetelement_</a>
	*/
	#define FOREACHpoint_(points) FOREACHsetelement_(pointT, points, point)

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

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

	declare:
	ridgeT ridge, *ridgep;

	see:
	<a href="qset.h#FOREACHsetelement_">FOREACHsetelement_</a>
	*/
	#define FOREACHridge_(ridges) FOREACHsetelement_(ridgeT, ridges, ridge)

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

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

	declare:
	vertexT vertex, *vertexp;

	see:
	<a href="qset.h#FOREACHsetelement_">FOREACHsetelement_</a>
	*/
	#define FOREACHvertex_(vertices) FOREACHsetelement_(vertexT, vertices,vertex)

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

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

	declare:
	facetT *facet;
	int facet_n, facet_i;

	see:
	<a href="qset.h#FOREACHsetelement_i_">FOREACHsetelement_i_</a>
	*/
	#define FOREACHfacet_i_(facets) FOREACHsetelement_i_(facetT, facets, facet)

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

	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

	declare:
	facetT *neighbor;
	int neighbor_n, neighbor_i;

	see:
	<a href="qset.h#FOREACHsetelement_i_">FOREACHsetelement_i_</a>
	*/
	#define FOREACHneighbor_i_(facet) FOREACHsetelement_i_(facetT, facet->neighbors, neighbor)

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

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

	declare:
	pointT *point;
	int point_n, point_i;

	see:
	<a href="qset.h#FOREACHsetelement_i_">FOREACHsetelement_i_</a>
	*/
	#define FOREACHpoint_i_(points) FOREACHsetelement_i_(pointT, points, point)

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

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

	declare:
	ridgeT *ridge;
	int ridge_n, ridge_i;

	see:
	<a href="qset.h#FOREACHsetelement_i_">FOREACHsetelement_i_</a>
	*/
	#define FOREACHridge_i_(ridges) FOREACHsetelement_i_(ridgeT, ridges, ridge)

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

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

	declare:
	vertexT *vertex;
	int vertex_n, vertex_i;

	see:
	<a href="qset.h#FOREACHsetelement_i_">FOREACHsetelement_i_</a>
	*/
	#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);
	#endif

	/*** -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 */

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