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someTriangulation -- a triangulation of a point or vector configuration, not necessarily fine or regular

Description

Returns some triangulation of the vector configuration $A$, useful as a seed for flipGraph or generateTriangulations. Unlike regularFineTriangulation with Homogenize => false, this function is reliable on non-acyclic configurations (where the origin lies in the interior of $\mathrm{conv}(A)$). The returned triangulation need not be fine; columns in the strict interior of $\mathrm{conv}(A)$ are omitted. When coneFromVData$(A)$ is pointed, this defers to regularFineTriangulation.

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 = someTriangulation 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

Strategy: triangulate homogenize$(A)$ as a point configuration (which topcomRegularFineTriangulation handles correctly for any $A$); the resulting $d$-dimensional polytope simplices each have $d{+}1$ vertices. The $(d{-}1)$-subsets appearing in exactly one polytope simplex are the boundary $(d{-}1)$-faces of $\mathrm{conv}(A)$. For a non-acyclic $A$ the cones over these boundary faces form a complete simplicial fan.

See also

Ways to use someTriangulation:

  • someTriangulation(Matrix)

For the programmer

The object someTriangulation is a method function.

The source of this document is in Triangulations.m2:1489:0.