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

Description

A triangulation is called regular if it can be constructed in the following way: place the point set in one higher dimension at various heights in the new variable. Compute the convex hull. Collect the list of facets with downward pointing normal (last coordinate of normal vector is negative). If each of these is a simplex, then these form a triangulation of the original point set. A triangulation which arises this way is called regular. See the book [deLoera et al] for more details and many beautiful properties of such triangulations.

Tested via the engine LP rawConeInteriorPoint: the triangulation is regular iff its secondaryCone has nonempty interior. The topcom-based test, kept for comparison, is available as topcomIsRegularTriangulation.

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 : T = triangulation(A, tri)

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

o3 : Triangulation

We check that T is indeed a triangulation, and whether it is a regular triangulation.

i4 : isWellDefined T

o4 = true
 
i5 : isRegularTriangulation T

o5 = false

See also

Ways to use isRegularTriangulation:

  • isRegularTriangulation(Matrix,List)
  • isRegularTriangulation(Triangulation)

For the programmer

The object isRegularTriangulation is a method function with options.

The source of this document is in /build/reproducible-path/macaulay2-1.26.06+ds/M2/Macaulay2/packages/Triangulations.m2:1150:0.