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.
.TP
En
Maximum roundoff error for distance computations.
.TP
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'.
.TP
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'.
.TP
Un
Maximum distance below a facet for a point to be coplanar to the facet.  The
default value is 'Vn'.
.TP
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).
.PP
.TP
Additional input/output formats
.TP
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.
.TP
FA
Compute the total area and volume for option 's'.  It is an approximation
for non-simplicial facets (see 'Fa').
.TP
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). 
.TP
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.
.TP
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.
.TP
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.
.TP
FF
Print facets (as in 'f') without printing the ridges.
.TP
Fi
Print inner planes for each facet.  The inner plane is below all vertices.
.TP
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.
.TP
FI
Print facet identifiers.
.TP
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.
.TP
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.
.TP
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').
.TP
Fo
Print outer planes for each facet in the same format as 'n'.  
The outer plane is above all points.
.TP
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.
.TP
FO
List all options to stderr, including the default values.  Additional 'FO's
are printed to stdout.
.TP
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.
.TP
FP
For each coplanar point ('Qc') print the point id of the nearest vertex,
the point id, the facet id, and the distance.
.TP
FQ
Print command used for qhull and input.
.TP
Fs
Print a summary.  The first line consists of the number of integers ("7"), 
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.

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.
.TP
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'.
.TP
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.
.TP
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).
.TP
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.
.TP
FV
Print average vertex.  The average vertex is a feasible point 
for halfspace intersection. 
.TP
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.
.PP
.TP
Geomview options
.TP
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.
.TP
Ga
Display all input points as dots.
.TP
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'.
.TP
GDn
Drop dimension n in 3-d or 4-d.  The result is a 2-d or 3-d object.  
.TP
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.
.TP
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.
.TP
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.
.TP
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.
.TP
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.
.TP
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.
.TP
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'.
.TP
Gv
Display vertices as spheres.  The radius of the sphere corresponds to
the imprecision of the data.  See 'Gp' for determining the radius.
.PP
.TP
Print options
.TP
PAn
Only the n largest facets are marked good for printing.  
Unless 'PG' is set, 'Pg' is automatically set. 
.TP
Pdk:n
Drop facet from output if normal[k] <= n.  The option 'Pdk' uses the
default value of 0 for n.
.TP
PDk:n
Drop facet from output if normal[k] >= n.  The option 'PDk' uses the
default value of 0 for n.
.TP
PFn
Only facets with area at least 'n' are marked good for printing.  
Unless 'PG' is set, 'Pg' is automatically set. 
.TP
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'.
.TP
PG
Print neighbors of good facets.