regularFineFanTriangulation -- a regular fine star fan triangulation of a vector configuration

Usage:

T = regularFineFanTriangulation A
Inputs:
- A, a matrix, whose columns are nonzero vectors in $\mathbb{Z}^d$ (or $\mathbb{Q}^d$); every column should lie on the boundary of $\mathrm{conv}(A)$ and the origin should be in the interior of $\mathrm{conv}(A)$
Outputs:
- T, an instance of the type Triangulation, a vector-configuration triangulation of $A$, returned with the origin as the implicit (unstored) cone vertex

Description

Implements the classical "star fan" construction:

$\bullet$ compute a regular fine triangulation of $\mathrm{conv}(A)$ as a point configuration (regularFineTriangulation);

$\bullet$ for each polytope simplex $f$ and each boundary facet $g$ of $\mathrm{conv}(A)$, take $f \cap g$; the size-$d$ intersections are the $(d{-}1)$-faces of $f$ that lie on $\partial \mathrm{conv}(A)$;

$\bullet$ return those size-$d$ index sets as a triangulation of $A$ as a vector configuration.

Under the stated assumptions the cones over these boundary faces form a complete simplicial fan, and the resulting triangulation is regular, fine, and star at the origin.

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

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

              4       6
o1 : Matrix ZZ  <-- ZZ

i2 : isPointed posHull A

o2 = false

i3 : T = regularFineFanTriangulation A

o3 = triangulation {{0, 1, 2, 4}, {0, 1, 2, 5}, {0, 1, 4, 5}, {0, 2, 4, 5}, {1, 2, 3, 4}, {1, 2, 3, 5}, {1, 3, 4, 5}, {2, 3, 4, 5}}

o3 : Triangulation

i4 : isWellDefined T

o4 = true

i5 : isFine T

o5 = true

i6 : isRegularTriangulation T

o6 = true

Caveat

If some column of $A$ lies in the interior of $\mathrm{conv}(A)$, the construction still returns a valid triangulation (and is still regular and star-at-origin), but interior columns are silently omitted, so the result will not be fine. Use insertion flips (neighbors with Fine => false) to add them.

Ways to use regularFineFanTriangulation:

regularFineFanTriangulation(Matrix)

For the programmer

The object regularFineFanTriangulation is a method function.

The source of this document is in Triangulations.m2:1441:0.

regularFineFanTriangulation -- a regular fine star fan triangulation of a vector configuration

Description

Caveat

See also

Ways to use regularFineFanTriangulation:

For the programmer