CIRCLEPACK(1)          Mathematica package          CIRCLEPACK(1)



NAME
     CirclePack, nerve, radii, lengths, angles, DrawPacking,
     DrawPacking3D, DrawNerve

SYNOPSIS
     <<DrawPacking`

     cp = CirclePack[tri, opts]

     nerve[cp]
     radii[cp]
     lengths[cp]
     angles[cp]

     DrawPacking[cp]
     DrawPacking3D[cp]
     DrawNerve[cp]


DESCRIPTION
     The following Mathematica functions are defined by loading
     the package CirclePack.m. These along with additional func-
     tions for displaying circle packings are defined by loading
     DrawPacking.m.

     CirclePack[tri]
         where tri is a Triangulation object (as defined in the
         package Triangulation.m) creates and returns a Cir-
         clePacking object. This may then be used as input to any
         of the functions that follow.

     The default CirclePacking is one in which the circles are
     tangent to one another. To create a CirclePacking in which
     circles overlap with specified intersection angles use
     CirclePack[tri, Intersects -> {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.)