.TH CIRCLEPACK 1 "8th April 1992" CirclePack "Mathematica package"
.SH NAME
CirclePack, nerve, radii, lengths, angles, DrawPacking, DrawPacking3D,
DrawNerve
.SH SYNOPSIS
.nf
.ta 1i 2.5i
.B < {0.5, ...}]\fP.
The \f3Intersects\fP parameter is a list of cosines of intersection angles.
The i'th element of the list corresponds to the i'th
edge listed by the invocation \f3edges[nerve[cp]]\fP. (So in particular
the list should contain \f3ne[nerve[cp]]\fP numbers between 0 and 1.)
\f3DrawNerve[cp]\fP provides a more immediate way to see how the
edges are numbered.
.PP
You can control the maximum number of steps in the numerical algorithm
which calculates \f3CirclePackings\fP with the option \f3Steps -> 30\fP
for example. The default is 20.
.TP 4
\f3nerve[cp]\fP
Returns the \f3Triangulation\fP object originally supplied to the
\f3CirclePack\fP call by which \f3cp\fP was created. In other words
the nerve of the packing. See the manual page for \f3Triangulation.m\fP
for more information on \f3Triangulation\fP objects.
.TP 4
\f3radii[cp]\fP
Lists the radii of the circles in \f3cp\fP.
The i'th member of this list is the radius of the circle centered on
the i'th vertex of the nerve as listed by \f3vertices[nerve[cp]]\fP.
.TP 4
\f3lengths[cp]\fP
Lists the lengths of the edges of the nerve of \f3cp\fP.
The i'th member of this list is the length of the i'th edge in the list
\f3edges[nerve[cp]]\fP.
.TP 4
\f3angles[cp]\fP
Lists the angles at each corner of each triangle of the nerve.
The i'th member of this list is the angle at the i'th corner in the
list \f3corners[nerve[cp]]\fP.
.PP
The following functions are defined in \f3DrawPacking.m\fP:
.TP 4
\f3DrawPacking[cp]\fP
Draws the packing in whichever geometry is appropriate. Circle packings
on the sphere are drawn using stereographic projection. Circle packings
on surfaces of genus 2 or higher are drawn in the Poincare disk model
of hyperbolic space. In the Euclidean and hyperbolic cases circles are
shown as coloured disks. Disks which represent the same disk on the
underlying surface are given the same colour.
.TP 4
\f3DrawPacking3D[cp]\fP
Applies only to spherical circle packings. Draws a 3D representation
of the circle packing.
.TP 4
\f3DrawNerve[cp]\fP
Draws the nerve of the packing. The geometry is selected as for
\f3DrawPacking[cp]\fP. Each edge is labelled with its position in the
list \f3edges[nerve[cp]]\fP. This is useful when constructing
circle packings with overlapping circles.
.SH SEE ALSO
Triangulation - A Mathematica package for creating and working with
2D triangulations.
.SH BUGS
None of these functions check that the input triangulation is
actually the triangulation of some closed orientable surface.
.PP
The algorithm used may not succeed in finding every circle packing
on the sphere. (That is, I have not seen it fail but neither have I
been able to prove that it should succeed.)