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Triangulation -- a triangulation of a point or vector configuration

Description

A Triangulation packages a point (or vector) configuration $A$ together with a triangulation of the columns of $A$, given as a list of maximal simplices. Each simplex is a sorted list of column indices into $A$, and the outer list is sorted as well.

Construct a Triangulation with triangulation from an explicit matrix and list, or with regularFineTriangulation from just a matrix. Inspect one with the accessors max(Triangulation), vectors(Triangulation), and matrix(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 : T = regularFineTriangulation A

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

o2 : Triangulation
 
i3 : max T

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

o3 : List
 
i4 : matrix T

o4 = | 0 0 -1 1  -4 4  |
     | 3 1 -1 -1 -2 -2 |
     | 1 1 1  1  1  1  |

              3       6
o4 : Matrix ZZ  <-- ZZ
 
i5 : vectors T

o5 = {{0, 3, 1}, {0, 1, 1}, {-1, -1, 1}, {1, -1, 1}, {-4, -2, 1}, {4, -2, 1}}

o5 : List

By default, the matrix stored in T is the homogenized configuration, with a final row of $1$'s appended; most algorithms in this package operate on homogenized vectors. Pass Homogenize => false to triangulation if the input is already a vector configuration.

See also

Functions and methods returning an object of class Triangulation:

  • makeFine(Triangulation) -- see makeFine -- walk to a fine triangulation by support-increasing bistellar flips
  • regularFineFanTriangulation(Matrix) -- see regularFineFanTriangulation -- a regular fine star fan triangulation of a vector configuration
  • regularFineTriangulation(Matrix) -- see regularFineTriangulation -- a regular fine triangulation of a point or vector configuration
  • regularFineTriangulation(Triangulation) -- see regularFineTriangulation -- a regular fine triangulation of a point or vector configuration
  • someTriangulation(Matrix) -- see someTriangulation -- a triangulation of a point or vector configuration, not necessarily fine or regular
  • triangulation(List,List) -- see triangulation -- make a Triangulation object
  • triangulation(Matrix,List) -- see triangulation -- make a Triangulation object

Methods that use an object of class Triangulation:

  • bistellarFlip(Triangulation,List) -- see bistellarFlip -- perform a bistellar flip on a triangulation
  • degreeMatrix(Triangulation) -- see degreeMatrix -- charge matrix Q whose rows generate ker A
  • flipCandidates(Triangulation) -- see flipCandidates -- candidate affine circuits for bistellar flips of a triangulation
  • flipGraph(Triangulation) -- see flipGraph -- bistellar-flip graph of a point or vector configuration
  • flips(Triangulation) -- see flips -- legal bistellar flips of a triangulation, computed via topcom
  • generateTriangulations(Triangulation) (missing documentation)
  • isFine(Triangulation) -- see isFine -- test whether a triangulation uses every point in the configuration
  • isRegularTriangulation(Triangulation) -- see isRegularTriangulation -- determine if a given triangulation is a regular triangulation
  • isStar(Triangulation) -- see isStar -- test whether a triangulation is a star with respect to its last point
  • isWellDefined(Triangulation) -- test whether the simplices of a triangulation form a valid triangulation
  • matrix(Triangulation) -- the matrix whose columns are the points of a triangulation
  • max(Triangulation) -- the maximal simplices of a triangulation
  • neighbors(Triangulation) -- see neighbors -- triangulations adjacent to a given one in the bistellar-flip graph
  • net(Triangulation) (missing documentation)
  • regularTriangulationWeights(Triangulation) -- see regularTriangulationWeights -- height vector inducing a regular triangulation, if one exists
  • secondaryCone(Triangulation) -- see secondaryCone -- secondary cone of a triangulation, in charge-lattice coordinates
  • toExternalString(Triangulation) (missing documentation)
  • topcomIsRegularTriangulation(Triangulation) (missing documentation)
  • topcomRegularTriangulationWeights(Triangulation) (missing documentation)
  • toString(Triangulation) (missing documentation)
  • Triangulation == Triangulation (missing documentation)
  • vectors(Triangulation) -- see vectors -- the column vectors of the configuration underlying a triangulation
  • volumeVector(Triangulation) -- see volumeVector -- the GKZ vector (per-vertex/per-ray volume sum) of a triangulation
  • wallCircuits(Triangulation) -- see wallCircuits -- distinct codim-2 wall circuits of a triangulation, with sign convention

For the programmer

The object Triangulation is a type, with ancestor classes HashTable < Thing.

The source of this document is in /build/reproducible-path/macaulay2-1.26.06+ds/M2/Macaulay2/packages/Triangulations.m2:1187:0.