


qhull(1)						 qhull(1)


NAME
       qhull  -	convex hull, Delaunay triangulation, Voronoi dia-
       gram, halfspace intersection about a point, hull volume, facet area

SYNOPSIS
       qhull- compute convex hulls and related structures
	   input (stdin): dimension, #points, point coordinates
	   first comment (non-numeric) is listed in the	summary
	   halfspace: use dim plus one with offsets after coefficients

       options (qh-quick.htm):
	   d	  - Delaunay triangulation by lifting points to	a paraboloid
	   v	  - Voronoi diagram via	the Delaunay triangulation
	   H1,1	  - Halfspace intersection about [1,1,0,...]
	   d Qu	  - Furthest-site Delaunay triangulation (upper	convex hull)
	   v Qu	  - Furthest-site Voronoi diagram
	   QJ	  - Joggle the input to	avoid precision	problems
	   .	  - concise list of all	options
	   -	  - one-line description of all	options

       Output options (subset):
	   FA	  - compute total area and volume
	   Fx	  - extreme points (convex hull	vertices)
	   G	  - Geomview output (2-d, 3-d and 4-d)
	   Fp	  - halfspace intersection coordinates
	   m	  - Mathematica	output (2-d and	3-d)
	   n	  - normals with offsets
	   o	  - OFF	file format (if	Voronoi, outputs regions)
	   TO file- output results to file, may	be enclosed in single quotes
	   f	  - print all fields of	all facets
	   s	  - summary of results (default)
	   Tv	  - verify result: structure, convexity, and point inclusion
	   p	  - vertex coordinates
	   i	  - vertices incident to each facet

       example:
	   rbox	1000 s | qhull Tv s FA

	- html manual:	  index.htm
	- installation:	  README.txt
	- see also:	  COPYING.txt, REGISTER.txt, Changes.txt
	- WWW:	<http://www.geom.umn.edu/software/qhull>
        - CVS:  <http://savannah.gnu.org/projects/qhull/>
        - mirror: <http://www6.uniovi.es/ftp/pub/mirrors/geom.umn.edu/software/ghindex.html>
	- news:	<http://www.geom.umn.edu/~bradb/qhull-news.html>
	- Geomview:  <http://www.geomview.org>
	- news group:	  <news:comp.graphics.algorithms>
	- FAQ:	     <http://exaflop.org/docs/cgafaq/cga6.html>
	- email:	  qhull@geom.umn.edu
	- bug reports:	  qhull_bug@geom.umn.edu




Geometry Center		     Aug 20, 2002			1





qhull(1)						 qhull(1)


       The sections are:
	- INTRODUCTION
	- DESCRIPTION, a description of	Qhull
	- IMPRECISION, how Qhull handles imprecision
	- OPTIONS
	-    Input and output options
	-    Additional	input/output formats
	-    Precision options
	-    Geomview options
	-    Print options
	-    Qhull options
	-    Trace options
	- BUGS
	- E-MAIL
	- SEE ALSO
	- AUTHORS
	- ACKNOWLEGEMENTS

       This man	page briefly describes all Qhull options.  Please
       report  any  mismatches	with  Qhull's  html  manual  (qh-
       man.htm).



INTRODUCTION
       Qhull  is  a  general  dimension	code for computing convex
       hulls, Delaunay triangulations, Voronoi diagram,	furthest-
       site  Voronoi  diagram,	furthest-site Delaunay triangula-
       tions, and halfspace  intersections  about  a  point.   It
       implements  the Quickhull algorithm for computing the con-
       vex hull.  Qhull	handles	round-off  errors  from	 floating
       point arithmetic.  It can approximate a convex hull.

       The  program includes options for hull volume, facet area,
       partial hulls, input transformations, randomization, trac-
       ing,  multiple  output  formats,	and execution statistics.
       The program can be called from  within  your  application.
       You  can	 view  the  results  in	 2-d,  3-d  and	 4-d with
       Geomview.


DESCRIPTION
       The format of input is the following: first line	 contains
       the  dimension,	second	line contains the number of input
       points, and point coordinates follow.  The  dimension  and
       number  of  points  can	be  reversed.	Comments and line
       breaks are ignored.  A comment starts with  a  non-numeric
       character  and  continues  to  the end of line.	The first
       comment is reported in summaries	 and  statistics.   Error
       reporting is better if there is one point per line.

       The  default printout option is a short summary.	There are
       many other output formats.




Geometry Center		     Aug 20, 2002			2





qhull(1)						 qhull(1)


       Qhull implements	the Quickhull algorithm	for convex  hull.
       This  algorithm	combines the 2-d Quickhull algorithm with
       the n-d beneath-beyond algorithm	[c.f., Preparata & Shamos
       '85].   It  is  similar	to  the	 randomized algorithms of
       Clarkson	and others  [Clarkson  et  al.	'93].	The  main
       advantages  of Quickhull	are output sensitive performance,
       reduced space requirements, and automatic handling of pre-
       cision problems.

       The data	structure produced by Qhull consists of	vertices,
       ridges, and facets.  A vertex is	a point	of the input set.
       A ridge is a set	of d vertices and two neighboring facets.
       For example in 3-d, a ridge is an edge of the  polyhedron.
       A facet is a set	of ridges, a set of neighboring	facets,	a
       set of incident vertices, and a hyperplane equation.   For
       simplicial  facets, the ridges are defined by the vertices
       and neighboring facets.	When Qhull merges two facets,  it
       produces	 a  non-simplicial facet.  A non-simplicial facet
       has more	than d neighbors and  may  share  more	than  one
       ridge with a neighbor.


IMPRECISION
       Since Qhull uses	floating point arithmetic, roundoff error
       may occur for each calculation.	This causes  problems for
       most geometric algorithms.

       Qhull  automatically  sets  option  'C-0' in 2-d, 3-d, and
       4-d, or option 'Qx' in 5-d and higher.  These options han-
       dle  precision problems by merging facets.  Alternatively,
       use option 'QJ' to joggle the input.

       With 'C-0', Qhull  merges  non-convex  facets  while  con-
       structing  the hull. The	remaining facets are clearly con-
       vex. With 'Qx',	Qhull  merges  coplanar	 horizon  facets,
       flipped	facets,	concave	facets and duplicated ridges.  It
       merges coplanar facets after constructing the hull.   With
       'Qx',  coplanar points may be missed, but it appears to be
       unlikely.

       To guarantee triangular	output,	 joggle	 the  input  with
       option 'QJ'.  Facet merging will	not occur.

OPTIONS
       To  get	a  list	 of  the  most important options, execute
       'qhull' by itself.  To get a  complete  list  of	 options,
       execute	'qhull	-'.   To  get a	complete, concise list of
       options,	execute	'qhull .'.

       Options can be in any order.  Capitalized options take  an
       argument	 (except  'PG'	and 'F'	options).  Single letters
       are used	for output formats and precision constants.   The
       other options are grouped into menus for	other output for-
       mats ('F'), Geomview output ('G'), printing  ('P'),  Qhull



Geometry Center		     Aug 20, 2002			3





qhull(1)						 qhull(1)


       control ('Q'), and tracing ('T').

       Main options:

       default
	      Compute  the  convex  hull  of  the  input  points.
	      Report a summary of the result.

       d      Compute the Delaunay triangulation by  lifting  the
	      input  points  to	 a  paraboloid.	  The  'o' option
	      prints the  input	 points	 and  facets.	The  'QJ'
	      option  guarantees  triangular  output.	The  'Ft'
	      option prints a triangulation.  It adds points (the
	      centrums)	to non-simplicial facets.

       v      Compute  the Voronoi diagram from	the Delaunay tri-
	      angulation.  The 'p' option prints the Voronoi ver-
	      tices.   The 'o' option prints the Voronoi vertices
	      and the vertices in each Voronoi region.	It  lists
	      regions  in  site	id order.  The 'Fv' option prints
	      each ridge of the	Voronoi	diagram.   The	first  or
	      zero'th  vertex indicates	the infinity vertex.  Its
	      coordinates are qh_INFINITE  (-10.101).	It  indi-
	      cates   unbounded	 Voronoi  regions  or  degenerate
	      Delaunay triangles.

       Hn,n,...
	      Compute halfspace	intersection  about  [n,n,0,...].
	      The  input  is  a	 set of	halfspaces defined in the
	      same format as 'n', 'Fo',	and 'Fi'.   Use	 'Fp'  to
	      print  the  intersection	points.	 Use 'Fv' to list
	      the intersection points for  each	 halfspace.   The
	      other  output formats display the	dual convex hull.

	      The point	[n,n,n,...] is a feasible point	 for  the
	      halfspaces, i.e.,	a point	that is	inside all of the
	      halfspaces (Hx+b <=  0).	 The  default  coordinate
	      value is 0.

	      The  input may start with	a feasible point.  If so,
	      use 'H' by itself.  The input starts with	a  feasi-
	      ble  point  when the first number	is the dimension,
	      the second number	is "1",	and the	coordinates  com-
	      plete  a line.  The 'FV' option produces a feasible
	      point for	a convex hull.

       d Qu   Compute the  furthest-site  Delaunay  triangulation
	      from  the	upper convex hull.  The	'o' option prints
	      the input	points and facets.  The	'QJ' option guar-
	      antees triangular	otuput.	 You can also use facets.

       v Qu   Compute the furthest-site	Voronoi	diagram.  The 'p'
	      option prints the	Voronoi	vertices.  The 'o' option
	      prints the Voronoi vertices  and	the  vertices  in



Geometry Center		     Aug 20, 2002			4





qhull(1)						 qhull(1)


	      each  Voronoi  region.  The 'Fv' option prints each
	      ridge of the Voronoi diagram.  The first or zero'th
	      vertex  indicates	 the infinity vertex at	infinity.
	      Its  coordinates	are  qh_INFINITE  (-10.101).   It
	      indicates	 unbounded Voronoi regions and degenerate
	      Delaunay triangles.

       Qt     Triangulated output.


       Input/Output options:

       f      Print out	all facets and all fields of each  facet.

       G      Output  the hull in Geomview format.  For	imprecise
	      hulls, Geomview displays the inner and outer  hull.
	      Geomview can also	display	points,	ridges,	vertices,
	      coplanar	points,	 and  facet  intersections.   See
	      below for	a list of options.

	      For  Delaunay triangulations, 'G'	displays the cor-
	      responding paraboloid.  For halfspace intersection,
	      'G' displays the dual polytope.

       i      Output the incident vertices for each facet.  Qhull
	      prints the number	of facets followed  by	the  ver-
	      tices  of	 each  facet.	One  facet is printed per
	      line.  The numbers are the  0-relative  indices  of
	      the  corresponding  input	 points.   The facets are
	      oriented.

	      In 4-d and higher, Qhull	triangulates  non-simpli-
	      cial  facets.   Each  apex  (the first vertex) is	a
	      created point that corresponds to	the facet's  cen-
	      trum.  Its index is greater than the indices of the
	      input points.  Each base corresponds to  a  simpli-
	      cial  ridge  between two facets.	To print the ver-
	      tices without triangulation, use option 'Fv'.

       m      Output  the  hull	 in  Mathematica  format.   Qhull
	      writes  a	 Mathematica  file for 2-d and 3-d convex
	      hulls and	for 2-d	Delaunay triangulations.    Qhull
	      produces a list of objects that you can assign to	a
	      variable in Mathematica,	for  example:  "list=  <<
	      <outputfilename> ". If the object	is 2-d,	it can be
	      visualized  by  "Show[Graphics[list]]  ".	 For  3-d
	      objects the command is "Show[Graphics3D[list]]".

       n      Output  the  normal equation for each facet.  Qhull
	      prints the dimension  (plus  one),  the  number  of
	      facets,  and  the	 normals  for  each  facet.   The
	      facet's offset follows its normal	coefficients.

       o      Output the facets	in OFF file format.  Qhull prints
	      the  dimension, number of	points,	number of facets,
	      and  number  of  ridges.	 Then	it   prints   the



Geometry Center		     Aug 20, 2002			5





qhull(1)						 qhull(1)


	      coordinates  of  the  input points and the vertices
	      for each facet.  Each facet is on	a separate  line.
	      The  first  number  is the number	of vertices.  The
	      remainder	are  the  indices  of  the  corresponding
	      points.  The vertices are	oriented in 2-d, 3-d, and
	      in simplicial facets.

	      For 2-d Voronoi diagrams,	the vertices  are  sorted
	      by adjacency, but	not oriented.  In 3-d and higher,
	      the Voronoi vertices are sorted by index.	 See  the
	      'v' option for more information.

       p      Output the coordinates of	each vertex point.  Qhull
	      prints the dimension, the	number of points, and the
	      coordinates  for	each  vertex.	With the 'Gc' and
	      'Gi' options, it also prints coplanar and	 interior
	      points.	For Voronoi diagrams, it prints	the coor-
	      dinates of each Voronoi vertex.

       s      Print a summary to stderr.  If  no  output  options
	      are  specified  at  all,	a summary goes to stdout.
	      The summary lists	the number of input  points,  the
	      dimension,  the  number  of  vertices in the convex
	      hull, the	number of facets in the	convex hull,  the
	      number of	good facets (if	'Pg'), and statistics.

	      The  last	 two  statistics  (if needed) measure the
	      maximum distance from a point or vertex to a facet.
	      The number in parenthesis	(e.g., 2.1x) is	the ratio
	      between the maximum  distance  and  the  worst-case
	      distance due to merging two simplicial facets.


       Precision options

       An     Maximum  angle  given  as	 a  cosine.  If	the angle
	      between a	pair of	facet normals is greater than n, Qhull
	      merges  one  of the facets into a	neighbor.  If 'n'
	      is negative, Qhull tests angles after  adding  each
	      point  to	 the hull (pre-merging).  If 'n' is posi-
	      tive, Qhull tests	 angles	 after	constructing  the
	      hull  (post-merging).   Both  pre- and post-merging
	      can be defined.

	      Option 'C0' or 'C-0' is set  if  the  corresponding
	      'Cn' or 'C-n' is not set.	 If 'Qx' is set, then 'A-
	      n' and 'C-n' are checked after  the  hull	 is  con-
	      structed and before 'An' and 'Cn'	are checked.

       Cn     Centrum  radius.	If a centrum is	less than n below
	      a	 neighboring  facet,  Qhull  merges  one  of  the
	      facets.	If  'n'	 is negative or	'-0', Qhull tests
	      and merges facets	after adding each  point  to  the
	      hull.   This  is	called	"pre-merging".	If 'n' is



Geometry Center		     Aug 20, 2002			6





qhull(1)						 qhull(1)


	      positive,	Qhull  tests  for  convexity  after  con-
	      structing	the hull ("post-merging").  Both pre- and
	      post-merging can be defined.

	      For 5-d and higher, 'Qx' should be used instead  of
	      'C-n'.  Otherwise, most or all facets may	be merged
	      together.

       En     Maximum roundoff error for distance computations.

       Rn     Randomly perturb distance	computations up	to +/-	n
	      *	 max_coord.  This option perturbs every	distance,
	      hyperplane, and angle computation.  To use time  as
	      the random number	seed, use option 'QR-1'.

       Vn     Minimum  distance	 for  a	 facet	to be visible.	A
	      facet is visible if the distance from the	point  to
	      the facet	is greater than	'Vn'.

	      Without  merging,	the default value for 'Vn' is the
	      round-off	error ('En').  With merging, the  default
	      value  is	 the  pre-merge	centrum	('C-n')	in 2-d or
	      3--d, or three times that	in other dimensions.   If
	      the outside width	is specified ('Wn'), the maximum,
	      default value for	'Vn' is	'Wn'.

       Un     Maximum distance below a facet for a  point  to  be
	      coplanar	to the facet.  The default value is 'Vn'.

       Wn     Minimum outside width  of	 the  hull.   Points  are
	      added  to	 the convex hull only if they are clearly
	      outside of a facet.  A point is outside of a  facet
	      if  its distance to the facet is greater than 'Wn'.
	      The normal value for 'Wn'	is  'En'.   If	the  user
	      specifies	 pre-merging  and does not set 'Wn', than
	      'Wn'  is	set  to	 the  premerge	'Cn'  and  maxco-
	      ord*(1-An).


       Additional input/output formats

       Fa     Print area for each facet.  For Delaunay triangula-
	      tions, the area is the area of the  triangle.   For
	      Voronoi  diagrams, the area is the area of the dual
	      facet.   Use  'PAn'  for	printing  the  n  largest
	      facets, and option 'PFn' for printing facets larger
	      than 'n'.

	      The area for non-simplicial facets is  the  sum  of
	      the areas	for each ridge to the centrum.	 Vertices
	      far below	the facet's hyperplane are ignored.   The
	      reported	area  may  be significantly less than the
	      actual area.




Geometry Center		     Aug 20, 2002			7





qhull(1)						 qhull(1)


       FA     Compute the total	area and volume	for  option  's'.
	      It  is  an  approximation	for non-simplicial facets
	      (see 'Fa').

       Fc     Print coplanar points for	each facet.   The  output
	      starts  with the number of facets.  Then each facet
	      is printed one per line.	Each line is  the  number
	      of  coplanar  points  followed  by  the  point ids.
	      Option 'Qi' includes  the	 interior  points.   Each
	      coplanar	point (interior	point) is assigned to the
	      facet it is furthest above (resp., least below).

       FC     Print centrums for each facet.  The  output  starts
	      with  the	 dimension  followed  by  the  number  of
	      facets.  Then each facet centrum	is  printed,  one
	      per line.

       Fd     Read  input  in cdd format with homogeneous points.
	      The input	starts with comments.  The first  comment
	      is  reported  in	the summary.  Data starts after	a
	      "begin" line.  The  next	line  is  the  number  of
	      points  followed	by  the	dimension+1 and	"real" or
	      "integer".  Then the  points  are	 listed	  with	a
	      leading  "1" or "1.0".  The data ends with an "end"
	      line.

	      For halfspaces ('Fd Hn,n,...'), the input	format is
	      the  same.   Each	halfspace starts with its offset.
	      The sign of the offset is	the opposite  of  Qhull's
	      convention.

       FD     Print  normals ('n', 'Fo', 'Fi') or points ('p') in
	      cdd format.  The first line  is  the  command  line
	      that  invoked  Qhull.   Data  starts with	a "begin"
	      line.  The next line is the number  of  normals  or
	      points  followed	by  the	 dimension+1  and "real".
	      Then the normals or points  are  listed	with  the
	      offset  before  the  coefficients.   The offset for
	      points is	1.0.  The  offset  for	normals	 has  the
	      opposite sign.  The data ends with an "end" line.

       FF     Print  facets  (as  in  'f')  without  printing the
	      ridges.

       Fi     Print inner planes for each facet.  The inner plane
	      is below all vertices.

       Fi     Print  separating	 hyperplanes  for  bounded, inner
	      regions of the Voronoi diagram.  The first line  is
	      the  number  of  ridges.	 Then  each hyperplane is
	      printed, one per line.  A	line starts with the num-
	      ber  of  indices	and floats.  The first pair lists
	      adjacent input sites, the	next  d	 floats	 are  the
	      normalized coefficients for the hyperplane, and the



Geometry Center		     Aug 20, 2002			8





qhull(1)						 qhull(1)


	      last float is the	offset.	 The hyperplane	 is  ori-
	      ented  toward  verify that the hyperplanes are per-
	      pendicular  bisectors.   Use  'Fo'  for	unbounded
	      regions,	and  'Fv'  for	the corresponding Voronoi
	      vertices.

       FI     Print facet identifiers.

       Fm     Print number of merges for each facet.  At most 511
	      merges  are  reported  for  a facet.  See	'PMn' for
	      printing the facets with the most	merges.

       Fn     Print neighbors for each facet.  The output  starts
	      with  the	 number	 of  facets.   Then each facet is
	      printed one per line.  Each line is the  number  of
	      neighbors	 followed  by an index for each	neighbor.
	      The indices match	the other facet	output formats.

	      A	negative index indicates an unprinted  facet  due
	      to  printing  only  good	facets ('Pg').	It is the
	      negation of the  facet's	id  (option  'FI').   For
	      example,	negative  indices are used for facets "at
	      infinity"	in the Delaunay	triangulation.

       FN     Print vertex neighbors or	coplanar facet	for  each
	      point.   The  first  line	 is the	number of points.
	      Then each	point is printed, one per line.	  If  the
	      point  is	coplanar, the line is "1" followed by the
	      facet's id.  If the point	is not a selected vertex,
	      the  line	is "0".	 Otherwise, each line is the num-
	      ber of  neighbors	 followed  by  the  corresponding
	      facet indices (see 'Fn').

       Fo     Print  outer planes for each facet in the	same for-
	      mat as 'n'.  The outer plane is above all	points.

       Fo     Print separating hyperplanes for	unbounded,  outer
	      regions  of the Voronoi diagram.	The first line is
	      the number of  ridges.   Then  each  hyperplane  is
	      printed, one per line.  A	line starts with the num-
	      ber of indices and floats.  The  first  pair  lists
	      adjacent	input  sites,  the  next d floats are the
	      normalized coefficients for the hyperplane, and the
	      last  float  is the offset.  The hyperplane is ori-
	      ented toward verify that the hyperplanes	are  per-
	      pendicular   bisectors.	 Use   'Fi'  for  bounded
	      regions, and 'Fv'	 for  the  corresponding  Voronoi
	      vertices.

       FO     List  all	 options to stderr, including the default
	      values.  Additional 'FO's	are printed to stdout.

       Fp     Print points for	halfspace  intersections  (option
	      'Hn,n,...').   Each  intersection	 corresponds to	a



Geometry Center		     Aug 20, 2002			9





qhull(1)						 qhull(1)


	      facet of the dual	polytope.  The	"infinity"  point
	      [-10.101,-10.101,...]    indicates   an	unbounded
	      intersection.

       FP     For each coplanar	point ('Qc') print the	point  id
	      of  the nearest vertex, the point	id, the	facet id,
	      and the distance.

       FQ     Print command used for qhull and input.

       Fs     Print a summary.	The first line	consists  of  the
	      number  of  integers  ("7"), followed by the dimen-
	      sion, the	number of points, the number of	vertices,
	      the  number  of  facets,	the  number  of	 vertices
	      selected for output, the number of facets	 selected
	      for  output, the number of coplanar points selected
	      for output.

	      The second line consists of  the	number	of  reals
	      ("2"),  followed by the maxmimum offset to an outer
	      plane and	and minimum offset  to	an  inner  plane.
	      Roundoff	is included.  Later versions of	Qhull may
	      produce additional integers or reals.

       FS     Print the	size of	the hull.  The	first  line  con-
	      sists  of	the number of integers ("0").  The second
	      line consists of the number of  reals  ("2"),  fol-
	      lowed  by	 the total facet area, and the total vol-
	      ume.  Later versions of  Qhull  may  produce  addi-
	      tional integers or reals.

	      The  total volume	measures the volume of the inter-
	      section of the halfspaces	defined	 by  each  facet.
	      Both  area  and  volume are approximations for non-
	      simplicial facets.  See option 'Fa'.

       Ft     Print a triangulation with added	points	for  non-
	      simplicial facets.  The first line is the	dimension
	      and the second line is the number	of points and the
	      number of	facets.	 The points follow, one	per line,
	      then the facets follow as	a list of point	 indices.
	      With option points include the point-at-infinity.

       Fv     Print  vertices  for each	facet.	The first line is
	      the number of facets.  Then each facet is	 printed,
	      one  per line.  Each line	is the number of vertices
	      followed by the corresponding point ids.	 Vertices
	      are listed in the	order they were	added to the hull
	      (the last	one is first).

       Fv     Print all	ridges of a Voronoi diagram.   The  first
	      line  is	the number of ridges.  Then each ridge is
	      printed, one per line.  A	line starts with the num-
	      ber  of  indices.	  The  first  pair lists adjacent



Geometry Center		     Aug 20, 2002		       10





qhull(1)						 qhull(1)


	      input sites, the	remaining  indices  list  Voronoi
	      vertices.	  Vertex  '0'  indicates  the  vertex-at-
	      infinity (i.e., an unbounded  ray).   In	3-d,  the
	      vertices	are  listed  in	order.	See 'Fi' and 'Fo'
	      for separating hyperplanes.

       FV     Print average vertex.  The average vertex	is a fea-
	      sible point for halfspace	intersection.

       Fx     List  extreme points (vertices) of the convex hull.
	      The first	line is	the number of points.  The  other
	      lines give the indices of	the corresponding points.
	      The first	point is '0'.  In 2-d, the  points  occur
	      in counter-clockwise order; otherwise they occur in
	      input order.   For  Delaunay  triangulations,  'Fx'
	      lists  the  extreme points of the	input sites.  The
	      points are unordered.


       Geomview	options

       G      Produce a	file for viewing with Geomview.	  Without
	      other  options,  Qhull displays edges in 2-d, outer
	      planes in	3-d, and ridges	in 4-d.	 A ridge  can  be
	      explicit or implicit.  An	explicit ridge is a dim-1
	      dimensional simplex between two  facets.	 In  4-d,
	      the explicit ridges are triangles.  When displaying
	      a	ridge in 4-d, Qhull projects the ridge's vertices
	      to  one  of  its	facets'	hyperplanes.  Use 'Gh' to
	      project ridges to	the intersection of  both  hyper-
	      planes.

       Ga     Display all input	points as dots.

       Gc     Display  the  centrum  for  each facet in	3-d.  The
	      centrum is defined by a green radius sitting  on	a
	      blue  plane.   The plane corresponds to the facet's
	      hyperplane.  The radius  is  defined  by	'C-n'  or
	      'Cn'.

       GDn    Drop  dimension  n  in 3-d or 4-d.  The result is	a
	      2-d or 3-d object.

       Gh     Display hyperplane intersections in  3-d	and  4-d.
	      In  3-d, the intersection	is a black line.  It lies
	      on two  neighboring  hyperplanes	(c.f.,	the  blue
	      squares  associated with centrums	('Gc')).  In 4-d,
	      the ridges are projected	to  the	 intersection  of
	      both hyperplanes.

       Gi     Display  inner  planes  in  2-d and 3-d.	The inner
	      plane of a facet is below	all of its vertices.   It
	      is  parallel  to the facet's hyperplane.	The inner
	      plane's color is the opposite (1-r,1-g,1-b) of  the



Geometry Center		     Aug 20, 2002		       11





qhull(1)						 qhull(1)


	      outer  plane.  Its edges are determined by the ver-
	      tices.

       Gn     Do not display inner or outer planes.  By	 default,
	      Geomview displays	the precise plane (no merging) or
	      both inner  and  output  planes  (merging).   Under
	      merging,	Geomview does not display the inner plane
	      if the the difference between inner  and	outer  is
	      too small.

       Go     Display  outer  planes  in  2-d and 3-d.	The outer
	      plane of a facet is above	all input points.  It  is
	      parallel	to  the	facet's	hyperplane.  Its color is
	      determined by the	facet's	normal,	and its	edges are
	      determined by the	vertices.

       Gp     Display  coplanar	 points	and vertices as	radii.	A
	      radius defines a	ball  which  corresponds  to  the
	      imprecision  of  the point.  The imprecision is the
	      maximum of the roundoff error, the centrum  radius,
	      and  maxcoord  * (1-An).	It is at least 1/20'th of
	      the maximum coordinate, and ignores post-merging if
	      pre-merging is done.

       Gr     Display  ridges  in  3-d.	 A ridge connects the two
	      vertices that are	 shared	 by  neighboring  facets.
	      Ridges are always	displayed in 4-d.

       Gt     A	 3-d  Delaunay	triangulation looks like a convex
	      hull with	interior facets.  Option 'Gt' removes the
	      outside  ridges to reveal	the outermost facets.  It
	      automatically sets options 'Gr' and 'GDn'.

       Gv     Display vertices as spheres.   The  radius  of  the
	      sphere  corresponds to the imprecision of	the data.
	      See 'Gp' for determining the radius.


       Print options

       PAn    Only the n  largest  facets  are	marked	good  for
	      printing.	  Unless  'PG'	is set,	'Pg' is	automati-
	      cally set.

       Pdk:n  Drop facet from output  if  normal[k]  <=	 n.   The
	      option 'Pdk' uses	the default value of 0 for n.

       PDk:n  Drop  facet  from	 output	 if  normal[k] >= n.  The
	      option 'PDk' uses	the default value of 0 for n.

       PFn    Only facets with area at least 'n' are marked  good
	      for printing.  Unless 'PG' is set, 'Pg' is automat-
	      ically set.




Geometry Center		     Aug 20, 2002		       12





qhull(1)						 qhull(1)


       Pg     Print only good facets.  A  good	facet  is  either
	      visible from a point (the	'QGn' option) or includes
	      a	point (the 'QVn'  option).   It	 also  meets  the
	      requirements  of	'Pdk'  and 'PDk' options.  Option
	      'Pg' is automatically set	 for  options  'PAn'  and
	      'PFn'.

       PG     Print neighbors of good facets.

       PMn    Only  the	 n facets with the most	merges are marked
	      good for printing.  Unless 'PG'  is  set,	 'Pg'  is
	      automatically set.

       Po     Force output despite precision problems.	Verify  ('Tv')  does	not check
	      coplanar points.	Flipped	facets are  reported  and
	      concave  facets  are  counted.   If  'Po'	 is used,
	      points are not partitioned into flipped facets  and
	      a	 flipped  facet	 is  always  visible  to a point.
	      Also, if an error	occurs before the  completion  of
	      Qhull  and  tracing  is  not active, 'Po'	outputs	a
	      neighborhood of the erroneous facets (if any).

       Pp     Do not report precision problems.


       Qhull control options

       Qbk:0Bk:0
	      Drop dimension  k	 from  the  input  points.   This
	      allows  the user to take convex hulls of sub-dimen-
	      sional objects.  It happens before the Delaunay and
	      Voronoi transformation.

       QbB    Scale the	input points to	fit the	unit cube.  After
	      scaling, the lower bound will be -0.5 and	the upper
	      bound  +0.5  in  all  dimensions.	 For Delaunay and
	      Voronoi diagrams,	scaling	happens	after  projection
	      to the paraboloid.  Under	precise	arithmetic, scal-
	      ing does not change  the	topology  of  the  convex
	      hull.

       Qbb    Scale  the last coordinate to [0,	m] where m is the
	      maximum absolute value of	 the  other  coordinates.
	      For  Delaunay and	Voronoi	diagrams, scaling happens
	      after projection to  the	paraboloid.   It  reduces
	      roundoff error for inputs	with integer coordinates.
	      Under precise arithmetic,	scaling	does  not  change
	      the topology of the convex hull.

       Qbk:n  Scale  the  k'th	coordinate  of	the input points.
	      After scaling, the lower bound of	the input  points
	      will be n.  'Qbk'	scales to -0.5.



Geometry Center		     Aug 20, 2002		       13





qhull(1)						 qhull(1)


       QBk:n  Scale  the  k'th	coordinate  of	the input points.
	      After scaling, the upper bound will  be  n.   'QBk'
	      scales to	+0.5.

       Qc     Keep  coplanar points with the nearest facet.  Out-
	      put formats 'p', 'f', 'Gp', 'Fc',	 'FN',	and  'FP'
	      will print the points.

       Qf     Partition	points to the furthest outside facet.

       Qg     Only  build  good	 facets.   With	 the 'Qg' option,
	      Qhull will only build those facets that it needs to
	      determine	 the  good  facets  in	the  output.  See
	      'QGn', 'QVn', and	'PdD' for defining  good  facets,
	      and  'Pg'	 and  'PG'  for	 printing good facets and
	      their neighbors.

       QGn    A	facet is good (see 'Qg'	and 'Pg') if it	is  visi-
	      ble  from	point n.  If n < 0, a facet is good if it
	      is not visible from point	n.  Point n is not  added
	      to  the  hull  (unless 'TCn' or 'TPn').  With rbox,
	      use the 'Pn,m,r' option to define	 your  point;  it
	      will be point 0 (QG0).

       Qi     Keep  interior points with the nearest facet.  Out-
	      put formats 'p', 'f', 'Gp', 'FN',	 'FP',	and  'Fc'
	      will print the points.

       QJn    Joggle  each  input  coordinate  by adding a random
	      number in	[-n,n].	 If  a	precision  error  occurs,
	      then  qhull  increases  n	and tries again.  It does
	      not increase n beyond a certain value, and it stops
	      after  a	certain	 number	of attempts [see user.h].
	      Option 'QJ' selects a default  value  for	 n.   The
	      output will be simplicial.  For Delaunay triangula-
	      tions, 'QJn' sets	'Qbb' to scale the  last  coordi-
	      nate (not	if 'Qbk:n' or 'QBk:n' is set).  See also
	      'Qt'.

       Qm     Only  process  points that would otherwise increase
	      max_outside.  Other points are treated as	 coplanar
	      or interior points.

       Qr     Process  random  outside points instead of furthest
	      ones.  This makes	Qhull equivalent to  the  random-
	      ized  incremental	 algorithms.   CPU  time  is  not
	      reported since the randomization is inefficient.

       QRn    Randomly rotate the input	points.	 If n=0, use time
	      as  the  random  number seed.  If	n>0, use n as the
	      random number seed.  If n=-1, don't rotate but  use
	      time  as the random number seed.	For Delaunay tri-
	      angulations ('d' and 'v'), rotate	 about	the  last
	      axis.




Geometry Center		     Aug 20, 2002		       14





qhull(1)						 qhull(1)


       Qs     Search all points	for the	initial	simplex.

       Qt     Triangulated output.  Triangulate non-simplicial 
              facets.

       Qv     Test  vertex  neighbors  for  convexity after post-
	      merging.	To use the 'Qv'	option,	you also need  to
	      set a merge option (e.g.,	'Qx' or	'C-0').

       QVn    A	 good facet (see 'Qg' and 'Pg')	includes point n.
	      If n<0, then a good facet	does not include point n.
	      The point	is either in the initial simplex or it is
	      the first	point added to the  hull.   Option  'QVn'
	      may not be used with merging.

       Qx     Perform  exact merges while building the hull.  The
	      "exact" merges are merging a point into a	 coplanar
	      facet  (defined  by 'Vn',	'Un', and 'C-n'), merging
	      concave facets, merging duplicate	ridges,	and merg-
	      ing  flipped  facets.   Coplanar	merges	and angle
	      coplanar merges ('A-n') are not performed.  Concav-
	      ity testing is delayed until a merge occurs.

	      After  the  hull	is built, all coplanar merges are
	      performed	(defined by 'C-n' and 'A-n'), then  post-
	      merges are performed (defined by 'Cn' and	'An').

       Qz     Add  a  point  "at  infinity"  that  is  above  the
	      paraboloid for Delaunay triangulations and  Voronoi
	      diagrams.	  This	reduces	 precision  problems  and
	      allows the triangulation of cospherical points.


       Qhull experiments and speedups

       Q0     Turn off pre-merging as  a  default  option.   With
	      'Q0'/'Qx'	 and  without explicit pre-merge options,
	      Qhull ignores precision issues  while  constructing
	      the  convex  hull.   This	 may  lead  to	precision
	      errors.  If so, a	descriptive warning is generated.

       Q1     With  'Q1',  Qhull  sorts	merges by type (coplanar,
	      angle coplanar, concave) instead of by angle.

       Q2     With 'Q2', Qhull merges all facets at once  instead
	      of  using	 independent  sets  of	merges	and  then
	      retesting.

       Q3     With 'Q3', Qhull does  not  remove  redundant  ver-
	      tices.

       Q4     With 'Q4', Qhull avoids merges of	an old facet into
	      a	new facet.

       Q5     With 'Q5', Qhull does not	correct	outer  planes  at
	      the  end.	 The maximum outer plane is used instead.




Geometry Center		     Aug 20, 2002		       15





qhull(1)						 qhull(1)


       Q6     With 'Q6', Qhull	does  not  pre-merge  concave  or
	      coplanar facets.

       Q7     With  'Q7',  Qhull  processes facets in depth-first
	      order instead of breadth-first order.

       Q8     With 'Q8'	and merging, Qhull does	not retain  near-
	      interior	points	for adjusting outer planes.  'Qc'
	      will probably retain all points that  adjust  outer
	      planes.

       Q9     With 'Q9', Qhull processes the furthest of all out-
	      side sets	at each	iteration.

       Q10    With 'Q10', Qhull does not use special processing 
	      for narrow distributions.

       Q11    With 'Q11', Qhull copies normals and recomputes 
	      centrums for tricoplanar facets.

       Trace options

       Tn     Trace at level n.	 Qhull	includes  full	execution
	      tracing.	 'T-1'	traces	events.	  'T1' traces the
	      overall execution	of the program.	  'T2'	and  'T3'
	      trace overall execution and geometric and	topologi-
	      cal  events.   'T4'  traces  the	algorithm.   'T5'
	      includes	information  about  memory allocation and
	      Gaussian elimination.

       Tc     Check frequently during execution.  This will catch
	      most inconsistency errors.

       TCn    Stop  Qhull  after  building the cone of new facets
	      for point	n.  The	output for 'f' includes	the  cone
	      and the old hull.	 See also 'TVn'.

       TFn    Report  progress	whenever  more	than n facets are
	      created During post-merging, 'TFn' reports progress
	      after more than n/2 merges.

       TI file
	      Input data from 'file'.  The filename may not include 
		  spaces or quotes.

       TO file
	      Output results to	'file'.	 The name may be enclosed
	      in single	quotes.

       TPn    Turn on tracing when point n is added to the  hull.
          Trace partitions of point n. If used with TWn, turn off
          tracing after adding point n to the hull.

       TRn    Rerun  qhull  n  times.  Usually used with 'QJn' to
	      determine	the probability	that a given joggle  will
	      fail.

       Ts     Collect  statistics  and print to	stderr at the end
	      of execution.

       Tv     Verify the convex	hull.  This checks the	topologi-
	      cal  structure,  facet  convexity, and point inclu-
	      sion.  If	precision problems occurred,  facet  con-
	      vexity  is  tested whether or not	'Tv' is	selected.
	      Option 'Tv'  does	 not  check  point  inclusion  if



Geometry Center		     Aug 20, 2002		       16





qhull(1)						 qhull(1)


	      forcing output with 'Po',	or if 'Q5' is set.

	      For  point  inclusion  testing, Qhull verifies that
	      all points are below all outer planes  (facet->max-
	      outside).	 Point inclusion is exhaustive if merging
	      or if the	facet-point product is small enough; oth-
	      erwise  Qhull  verifies  each point with a directed
	      search (qh_findbest).

	      Point inclusion testing occurs after producing out-
	      put.   It	 prints	a message to stderr unless option
	      'Pp' is used.  This allows the  user  to	interrupt
	      Qhull without changing the output.

       TVn    Stop  Qhull  after  adding point n.  If n	< 0, stop
	      Qhull before adding point	n.  Output shows the hull
	      at this time.  See also 'TCn'

       TMn    Turn on tracing at n'th merge.

       TWn    Trace  merge  facets when	the width is greater than
	      n.

       Tz     Redirect stderr to stdout.


BUGS
       Please	 report	   bugs	    to	   Brad	    Barber     at
       qhull_bug@geom.umn.edu.

       If Qhull	does not compile, it is	due to an incompatibility
       between your system and ours.  The first	thing to check is
       that  your compiler is ANSI standard.  If it is,	check the
       man page	for the	best options, or  find	someone	 to  help
       you.  If	you locate the cause of	your problem, please send
       email since it might help others.

       If Qhull	compiles but crashes on	the test case (rbox  D4),
       there's	still  incompatibility	between	 your  system and
       ours.  Typically	it's been due to mem.c and memory  align-
       ment.   You  can	 use qh_NOmem in mem.h to turn off memory
       management.  Please let us know if you figure out  how  to
       fix these problems.

       If  you	do  find  a  problem,  try  to simplify	it before
       reporting the error.  Try different size	inputs to  locate
       the  smallest one that causes an	error.	You're welcome to
       hunt through the	code  using  the  execution  trace  as	a
       guide.	This  is  especially true if you're incorporating
       Qhull into your own program.

       When you	do report an error, please attach a data  set  to
       the  end	of your	message.  This allows us to see	the error
       for ourselves.  Qhull is	maintained part-time.



Geometry Center		     Aug 20, 2002		       17





qhull(1)						 qhull(1)


E-MAIL
       Please  send  correspondence  to	 qhull@geom.umn.edu   and
       report  bugs  to	 qhull_bug@geom.umn.edu.  Let us know how
       you use Qhull.  If you mention it in a paper, please  send
       the reference and an abstract.

       If  you would like to get Qhull announcements (e.g., a new
       version)	and news (any bugs that	get fixed, etc.), let  us
       know  and  we  will  add	 you to	our mailing list.  If you
       would like to communicate with other Qhull users, we  will
       add you to the qhull_users alias.  For Internet news about
       geometric algorithms and	convex hulls, look at comp.graph-
       ics.algorithms and sci.math.num-analysis


SEE ALSO
       rbox(1)

       Barber,	C.  B.,	 D.P.  Dobkin,	and  H.T. Huhdanpaa, "The
       Quickhull Algorithm for Convex Hulls," ACM Trans. on Math-
       ematical		  Software,	      Dec.	    1996.
       http://www.acm.org/pubs/citations/jour-
       nals/toms/1996-22-4/p469-barber/
       http://www.geom.umn.edu/software/qhull/qhull-96.ps

       Clarkson, K.L., K. Mehlhorn, and	R. Seidel, "Four  results
       on  randomized  incremental  construction,"  Computational
       Geometry: Theory	and Applications,  vol.	 3,  p.	 185-211,
       1993.

       Preparata,  F.  and  M.	Shamos,	 Computational	Geometry,
       Springer-Verlag,	New York, 1985.



AUTHORS
	 C. Bradford Barber		       Hannu Huhdanpaa
	 bradb@geom.umn.edu		       hannu@geom.umn.edu

			   c/o The Geometry Center
			   University of Minnesota
			   400 Lind Hall
			   207 Church Street S.E.
			   Minneapolis,	MN 55455



ACKNOWLEDGEMENTS
       A special thanks	to Albert Marden, Victor Milenkovic,  the
       Geometry	Center,	Harvard	University, and	Endocardial Solu-
       tions, Inc. for supporting this work.

       The software was	developed under	National Science  Founda-
       tion grants NSF/DMS-8920161 and NSF-CCR-91-15793	750-7504.



Geometry Center		     Aug 20, 2002		       18





qhull(1)						 qhull(1)


       David Dobkin guided the original	work at	Princeton Univer-
       sity.  If you find it useful, please let	us know.

       The Geometry Center is supported	by grant DMS-8920161 from
       the  National  Science  Foundation,   by	  grant	  DOE/DE-
       FG02-92ER25137  from the	Department of Energy, by the Uni-
       versity of Minnesota, and by Minnesota Technology, Inc.

       Qhull is available at http://www.geom.umn.edu/software/qhull













































Geometry Center		     Aug 20, 2002		       19


