Provided by: qhull-bin_2020.2-6build1_amd64 
NAME
qhull - convex hull, Delaunay triangulation, Voronoi diagram, 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
d Qu - furthest-site Delaunay triangulation (upper convex hull)
v - Voronoi diagram as the dual of the Delaunay triangulation
v Qu - furthest-site Voronoi diagram
H1,1 - Halfspace intersection about [1,1,0,...] via polar duality
Qt - triangulated output
QJ - joggled input instead of merged facets
Tv - verify result: structure, convexity, and point inclusion
. - concise list of all options
- - one-line description of each option
-? - this message
-V - version
Output options (subset):
s - summary of results (default)
i - vertices incident to each facet
n - normals with offsets
p - vertex coordinates (if 'Qc', includes coplanar points)
if 'v', Voronoi vertices
FA - report total area and volume
Fp - halfspace intersections
FS - total area and volume
Fx - extreme points (convex hull vertices)
G - Geomview output (2-d, 3-d and 4-d)
m - Mathematica output (2-d and 3-d)
o - OFF format (if 'v', outputs Voronoi regions)
QVn - print facets that include point n, -n if not
TI file - input file, may be enclosed in single quotes
TO file - output file, may be enclosed in single quotes
examples:
rbox D4 | qhull Tv rbox 1000 s | qhull Tv s FA
rbox 10 D2 | qhull d QJ s i TO result rbox 10 D2 | qhull v Qbb Qt p
rbox 10 D2 | qhull d Qu QJ m rbox 10 D2 | qhull v Qu QJ o
rbox c d D2 | qhull Qc s f Fx | more rbox c | qhull FV n | qhull H Fp
rbox d D12 | qhull QR0 FA rbox c D7 | qhull FA TF1000
rbox y 1000 W0 | qhull Qc rbox c | qhull n
- html manual: html/index.htm
- installation: README.txt
- see also: COPYING.txt, REGISTER.txt, Changes.txt
- WWW: <http://www.qhull.org>
- GIT: <git@github.com:qhull/qhull.git>
- news: <http://www.qhull.org/news>
- Geomview: <http://www.geomview.org>
- news group: <news:comp.graphics.algorithms>
- FAQ: <http://www.faqs.org/faqs/graphics/algorithms-faq/>
- email: qhull@qhull.org
- bug reports: qhull_bug@qhull.org
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
(html/index.htm).
INTRODUCTION
Qhull is a general dimension code for computing convex hulls, Delaunay triangulations, Voronoi diagram,
furthest‐site Voronoi diagram, furthest‐site Delaunay triangulations, and halfspace intersections about a
point. It implements the Quickhull algorithm for computing the convex 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, tracing, 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.
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 precision 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 handle precision problems by merging facets. Alternatively, use option 'QJ' to joggle the input.
With 'C-0', Qhull merges non‐convex facets while constructing the hull. The remaining facets are clearly
convex. 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 -?'. 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:
output formats ('F'), Geomview output ('G'), printing ('P'), Qhull 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 triangulation. The 'p' option prints the Voronoi
vertices. 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 indicates 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
feasible point when the first number is the dimension, the second number is "1", and the
coordinates complete 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 guarantees triangular otuput. You can also
use 'Ft' to triangulate via the centrums of non‐simplicial 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 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.
Input/Output options:
f Print 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 corresponding 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
vertices 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 4d and higher, Qhull triangulates non‐simplicial facets. Each apex (the first vertex) is a
created point that corresponds to the facet's centrum. Its index is greater than the indices of
the input points. Each base corresponds to a simplicial ridge between two facets. To print the
vertices without triangulation, use option 'Fv'. To print the centrum coordinates, use option
'Ft'. The centrum indices for option 'i' are one more than the centrum indices for option 'Ft'.
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 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 coordinates of each Voronoi vertex.
s Print a summary to stderr. If no output options are specified, 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 positive, 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 constructed 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 positive, Qhull tests for convexity after
constructing 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 maxcoord*(1-An).
Additional input/output formats
Fa Print area for each facet. For Delaunay triangulations, 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.
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
number 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 oriented toward 'QVn' (if defined), or the first input site of the pair. Use 'Tv' to verify
that the hyperplanes are perpendicular 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.
FM Output the hull in Maple format. Qhull writes a Maple file for 2‐d and 3‐d convex hulls and for
2‐d Delaunay triangulations. Qhull produces a '.mpl' file for displaying with display3d().
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 number of neighbors followed by the corresponding facet indices (see 'Fn').
Fo Print outer planes for each facet in the same format 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
number 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 oriented toward 'QVn' (if defined), or the first input site of the pair. Use 'Tv' to verify
that the hyperplanes are perpendicular 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
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 ("8"), followed by the
dimension, 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, number of simplicial, unmerged facets in 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 consists of the number of integers ("0"). The second
line consists of the number of reals ("2"), followed by the total facet area, and the total
volume. Later versions of Qhull may produce additional integers or reals.
The total volume measures the volume of the intersection 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 'Qz', the
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 number of indices. The first pair lists
adjacent 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 feasible 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 hyperplanes.
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 outer plane. Its edges are determined by the vertices.
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 automatically
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
automatically set.
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 'd', 'PAn', 'PFn', and 'PMn'.
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
Qa Allow input with fewer or more points than coordinates
Qbk:0Bk:0
Drop dimension k from the input points. This allows the user to take convex hulls of sub‐
dimensional 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, scaling 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.
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. Output 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 visible 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. Output 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 triangulations, 'QJn' sets 'Qbb' to scale the last
coordinate (not if 'Qbk:n' or 'QBk:n' is set). ´QJn' is deprecated for Voronoi diagrams. 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
randomized 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 triangulations ('d' and 'v'), rotate about the last axis.
Qs Search all points for the initial simplex.
Qt Triangulated output. Triangulate all non‐simplicial facets. ´Qt' is deprecated for Voronoi
diagrams. See also 'Qt'.
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.
Qw Allow option warnings. Otherwise Qhull returns an error after most option warnings
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 merging flipped facets. Coplanar merges and angle coplanar merges ('A-n') are not
performed. Concavity 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 merges by mergetype/angle instead of mergetype/distance.
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 vertices.
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.
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 outside sets at each iteration.
Q10 With 'Q10', Qhull does not use special processing for narrow distributions.
Q11 With 'Q11', Qhull copies normals and recompute centrums for tricoplanar facets.
Q12 With 'Q12', Qhull allows wide facets and wide dupridge.
Q14 With 'Q14', Qhull merges pinched vertices that create a dupridge.
Q15 With 'Q15', Qhull checks for duplicate ridges with the same vertices.
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
topological events. 'T4' traces the algorithm. 'T5' includes information about memory allocation
and Gaussian elimination.
Ta Annotate output with codes that identify the corresponding qh_fprintf() statement.
TAn Stop Qhull after adding n vertices.
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'.
Tf Flush output after each qh_fprintf. Use 'Tf' for debugging segfaults. See 'Tz' for redirecting
stderr.
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.
TMn Turn on tracing at n'th merge.
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.
TP-1 Turn on tracing after qh_buildhull and qh_postmerge.
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 topological structure, facet convexity, and point
inclusion. If precision problems occurred, facet convexity is tested whether or not 'Tv' is
selected. Option 'Tv' does not check point inclusion if forcing output with 'Po', or if 'Q5' is
set.
For point inclusion testing, Qhull verifies that all points are below all outer planes
(facet->maxoutside). Point inclusion is exhaustive if merging or if the facet‐point product is
small enough; otherwise Qhull verifies each point with a directed search (qh_findbest).
Point inclusion testing occurs after producing output. 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'
TWn Trace merge facets when the width is greater than n.
Tz Redirect stderr to stdout. See 'Tf' for flushing writes.
BUGS
Please report bugs to Brad Barber at qhull_bug@qhull.org.
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 alignment. 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.
E‐MAIL
Please send correspondence to qhull@qhull.org and report bugs to qhull_bug@qhull.org. 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.graphics.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
Mathematical Software, 22(4):469–483, Dec. 1996. http://portal.acm.org/citation.cfm?doid=235815.235821
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.117.405
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@shore.net hannu@qhull.org
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ACKNOWLEDGEMENTS
A special thanks to Albert Marden, Victor Milenkovic, the Geometry Center, Harvard University, and
Endocardial Solutions, Inc. for supporting this work.
Qhull 1.0 and 2.0 were developed under National Science Foundation grants NSF/DMS‐8920161 and NSF‐
CCR‐91‐15793 750‐7504. David Dobkin guided the original work at Princeton University. 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 University of Minnesota, and by Minnesota
Technology, Inc.
Qhull is available from http://www.qhull.org
Geometry Center 2003/12/30 qhull(1)