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**PROGRAM:**

**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

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

Qt - triangulated output

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 (centers for Voronoi)

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.qhull.org>

- GIT: <[email protected]:qhull/qhull.git>

- mirror:

<http://www6.uniovi.es/ftp/pub/mirrors/geom.umn.edu/software/ghindex.html>

- 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: [email protected]

- bug reports: [email protected]

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

- SEE ALSO

- AUTHORS

- ACKNOWLEGEMENTS

This man page briefly describes all Qhull options. Please report any mismatches with

Qhull's html manual (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' 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 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 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 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'.

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 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 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 '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‐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.

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 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 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 does not report a very wide merge due to a duplicated ridge with

nearly coincident 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.

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 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'

TMn Turn on tracing at n'th merge.

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

Tz Redirect stderr to stdout.

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