topcomRegularFineTriangulation(Matrix) -- compute a regular triangulation using all of the given points

Synopsis

Function: topcomRegularFineTriangulation
Usage:

topcomRegularFineTriangulation A
Inputs:
- A, a matrix,
Optional inputs:
- Homogenize => a Boolean value, default value true
Outputs:
- a list,

Description

This function returns a regular fine triangulation of the point set $A$ (that is, a list of lists of indices in the range $0, \ldots, n-1$, where $A$ is $d \times n$, and has full rank.

Recall: a fine triangulation is one which uses all of the points, and a regular triangulation is one which is induced as the lower faces of a lift of the points to one dimension higher.

Here we find a regular fine triangulation of the cyclic polytope with 7 vertices in 3-space.

i1 : A = matrix {{0, 1, 2, 3, 4, 5, 6}, {0, 1, 4, 9, 16, 25, 36}, {0, 1, 8, 27, 64, 125, 216}}

o1 = | 0 1 2 3  4  5   6   |
     | 0 1 4 9  16 25  36  |
     | 0 1 8 27 64 125 216 |

              3       7
o1 : Matrix ZZ  <-- ZZ

i2 : tri = topcomRegularFineTriangulation A

o2 = {{0, 1, 2, 3}, {1, 2, 3, 4}, {0, 1, 3, 4}, {2, 3, 4, 5}, {1, 2, 4, 5},
     ------------------------------------------------------------------------
     {0, 1, 4, 5}, {3, 4, 5, 6}, {2, 3, 5, 6}, {1, 2, 5, 6}, {0, 1, 5, 6}}

o2 : List

i3 : assert topcomIsTriangulation(A, tri)

i4 : assert topcomIsRegularTriangulation(A, tri)

i5 : topcomRegularTriangulationWeights(A, tri)

      5   5   1
o5 = {-, --, --, 0, 0, 0, 0}
      4  12  12

o5 : List

Ways to use this method: