topcomIsRegularTriangulation -- determine if a given triangulation is a regular triangulation

Synopsis

Usage:

topcomIsRegularTriangulation(A, tri)
Inputs:
- A, a matrix, A $d \times n$ matrix over ZZ. Each column represents one of the points which can be used in a triangulation
- tri, a list, A triangulation of the point set C
Optional inputs:
- Homogenize => a Boolean value, default value true, If true, $A$ determines a point configuration, and the matrix is then "homogenized": a row of 1's is appended to $A$, creating a vector configuration of $n$ vectors in $\mathbb{R}^{d+1}$.
Outputs:
- a Boolean value, whether the given triangulation is regular

Description

The following example is one of the simplest examples of a non-regular triangulation. Notice that tri is a triangulation of the polytope which is the convex hull of the columns of $A$, which are the only points allowed in the triangulation.

i1 : A = transpose matrix {{0,3},{0,1},{-1,-1},{1,-1},{-4,-2},{4,-2}}

o1 = | 0 0 -1 1  -4 4  |
     | 3 1 -1 -1 -2 -2 |

              2       6
o1 : Matrix ZZ  <-- ZZ

i2 : tri = {{0,1,2}, {1,3,5}, {2,3,4}, {0,1,5},
         {0,2,4}, {3,4,5}, {1,2,3}}

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

o2 : List

i3 : topcomIsRegularTriangulation(A,tri)

o3 = false

i4 : assert not topcomIsRegularTriangulation(A,tri)

i5 : assert topcomIsTriangulation(A, tri)

Setting debugLevel to either 1,2, or 5 will give more detail about what files are written to Topcom, and what the executable is. Setting debugLevel to 0 means that the function will run silently.

Caveat

Do we check that the triangulation is actually well defined?

Ways to use topcomIsRegularTriangulation :

topcomIsRegularTriangulation(Matrix,List)

For the programmer