TRIANGULATION(1)

NAME
Triangulation, faces, edges, alledges, vertices, allvertices, corners, glue, ne, nv, nc, euler, vN, eN

SYNOPSIS
<<Triangulation`

Triangulation[faces, glue]
faces[tri]
edges[tri]
alledges[tri]
vertices[tri]
allvertices[tri]
corners[tri]
glue[tri]
nv[tri]
ne[tri]
nc[tri]
euler[tri]
vN[tri, v]
eN[tri, {v1, v2}]

DESCRIPTION
The following Mathematica functions are defined by the Triangulation package.

tri = Triangulation[faces, glue]
    faces is a list of faces in the form {{v1, v2, v3}, ...} and glue is a list of pairs of edges to be identified, given in the form {{{v1, v2}, {v3, v4}}, ...}. Each vi represents a vertex and must be an integer.

This creates a triangulation object and assigns it to tri. The v's must be integers representing vertices of the triangulation. Faces of the triangulation are triples of vertices. Edges of the triangulation will be identified if they join the same two vertices or if they are specified in the glue parameter.

The following functions apply to Triangulations.

faces[tri]
    gives the original list of faces.

edges[tri]
    gives a list of edges after identified edges have been deleted. Each edge is given as a pair of vertices {v1, v2}.

alledges[tri]
    gives the full list of edges without deleting identified edges. This will be the same as edges[tri] if and only if glue is empty.

vertices[tri]
    gives a list of vertices after identified vertices have been removed. This is simply a list of integers {v1, ...}.

allvertices[tri]
    gives the full list of vertices with which the triangulation was originally specified. Ie. The union of all integers appearing in the faces argument of Triangulation.

corners[tri]
    gives a list of the corners of faces in tri. The corner at v2 of {v1, v2, v3} is given as {v2, v3, v1}.

glue[tri]
    gives the list of identifications as supplied to Triangulation.

nv[tri], ne[tri], nc[tri]
    give the number of vertices, edges and corners respectively in tri after performing identifications.

euler[tri]
    gives the Euler characteristic of the triangulation.

vN[tri, v]
    returns a number in the range 1-nv[tri]. If v appears in the list vertices[tri] then vN[tri, v] returns the position of v in that list. vN[tri, v] returns the same value for all vertices identified with v.

eN[tri, {v1, v2}]
    returns a number in the range 1-ne[tri] assuming that {v1, v2} is an edge of tri. If Sort[{v1, v2}] appears in edges[tri] then eN[tri, {v1, v2}] returns its position in that list. eN[tri, {v1, v2}] returns the same value for all edges identified with {v1, v2}.

SEE ALSO
CIRCLEPACK - a Mathematica package for finding circle packings on a given triangulated surface.

BUGS
Triangulation, vN and eN do not check their arguments.