CPINTRO(1)             Mathematica package             CPINTRO(1)



NAME
     cpintro - introduction to CirclePack.m and DrawPacking.m

DESCRIPTION
     (If you have a NeXT workstation have a look at the notebook
     "demo.ma" first.)

     The first few lines of Triangulation.m, CirclePack.m and
     DrawPacking.m will tell you what these files contain over
     and above what is mentioned below.

     Start Mathematica, then type "<<DrawPacking.m" followed by
     "cpexamples.m".  "DrawPacking.m" loads all the packages
     required to calculate and display circle packings. Contained
     in "cpexamples.m" is a triangulation of a genus 2 surface
     which we will look at to begin with.

     Typing "CirclePack[genus2]" will cause a CirclePacking
     object to be created. Please be patient, Mathematica isn't
     too quick at numerical calculations and this will probably
     take about 30 seconds.

     Various functions are available to examine CirclePackings.
     The two most important are "DrawPacking[]" and "DrawN-
     erve[]". Try applying these functions to the CirclePacking.
     The result should be a picture of the packing on which is
     shown the original triangulation and a picture of just the
     triangulation with the edges numbered.

     You can control which edge is at the middle of the picture
     by typing "DrawPacking[-circlepacking-, 23]" for example.
     This says that the 23rd edge listed in "alledges[genus2]"
     should go at the centre.

     Looking at the output of "DrawNerve[-circlepacking-, 23]"
     you will see that some of the edges have duplicate numbers
     on them. This is because these edges have to be identified
     to get a triangulation of a genus 2 surface. All the ver-
     tices round the edge of the picture have been identified. So
     in fact all the disks around the edge of the packing are in
     fact the same. They have been coloured the same in order to
     show this.

     We can also create packings in which circles overlap at
     specified angles rather than simply being tangent. To do
     this we need to include the option "Intersects -> {0.5, 1,
     0, ... }" in the "CirclePack[]" invocation. This says that
     the circles at each end of the edge labelled 1 should cross
     at angle Pi/3. In general the table should contain as many
     entries as the nerve has uniquely numbered edges. The i'th
     entry gives the Cosine of the required intersection angle at
     the i'th edge. (The default is "Table[1., {ne[genus2]}]".)

     Other examples in "cpexamples.m" include an octahedron
     (oct), an icosahedron (icos) and a torus cut into 4 trian-
     gles (torus). With "oct" and "icos" there are two possible
     ways to display the packing. You can either look at it
     stereographically projected to the plane or look at it in
     3D. (Both pictures unfortunately tend to be a bit confusing
     and may take a moment to sort out.)


AUTHOR
     Contact: Oliver Goodman, (oag@geom.umn.edu) if you have any
     comments or suggestions regarding this package.


SEE ALSO
     Triangulation, CirclePack