CIRCLEPACK(1) Mathematica package CIRCLEPACK(1)
NAME
CirclePack, nerve, radii, lengths, angles, DrawPacking,
DrawPacking3D, DrawNerve
SYNOPSIS
< {0.5, ...}]. The Intersects
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 edges[nerve[cp]]. (So in particular the
list should contain ne[nerve[cp]] numbers between 0 and 1.)
DrawNerve[cp] provides a more immediate way to see how the
edges are numbered.
You can control the maximum number of steps in the numerical
algorithm which calculates CirclePackings with the option
Steps -> 30 for example. The default is 20.
nerve[cp]
Returns the Triangulation object originally supplied to
the CirclePack call by which cp was created. In other
words the nerve of the packing. See the manual page for
Triangulation.m for more information on Triangulation
objects.
radii[cp]
Lists the radii of the circles in cp. 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
vertices[nerve[cp]].
lengths[cp]
Lists the lengths of the edges of the nerve of cp. The
i'th member of this list is the length of the i'th edge
in the list edges[nerve[cp]].
angles[cp]
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 corners[nerve[cp]].
The following functions are defined in DrawPacking.m:
DrawPacking[cp]
Draws the packing in whichever geometry is appropriate.
Circle packings on the sphere are drawn using stereo-
graphic 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.
DrawPacking3D[cp]
Applies only to spherical circle packings. Draws a 3D
representation of the circle packing.
DrawNerve[cp]
Draws the nerve of the packing. The geometry is selected
as for DrawPacking[cp]. Each edge is labelled with its
position in the list edges[nerve[cp]]. This is useful
when constructing circle packings with overlapping cir-
cles.
SEE ALSO
Triangulation - A Mathematica package for creating and work-
ing with 2D triangulations.
BUGS
None of these functions check that the input triangulation
is actually the triangulation of some closed orientable sur-
face.
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.)