Triangulations -- generating and manipulating triangulations of point or vector configurations

Description

Warning! This package is experimental, documentation is missing, and the interface will be cleaned up and changed. Use only if these issues don't bother you!

Data of a triangulation

max(Triangulation) (missing documentation)
vectors(Triangulation) (missing documentation)

Creating triangulations

triangulation(Matrix,List) (missing documentation)
regularFineTriangulation(Matrix) (missing documentation)
generateTriangulations(Triangulation) (missing documentation)
allTriangulations(Matrix) -- use topcom to generate all triangulations of a point or vector configuration

Properties of triangulations

isWellDefined(Triangulation) (missing documentation)
isRegularTriangulation(Triangulation) -- determine if a given triangulation is a regular triangulation
regularTriangulationWeights(Triangulation) (missing documentation)
isStar(Triangulation) (missing documentation)
isFine(Triangulation) (missing documentation)
naiveIsTriangulation(Triangulation) (missing documentation)

Exploring the set of triangulations

bistellarFlip(Triangulation,List) (missing documentation)
flips(Triangulation) (missing documentation)
neighbors(Triangulation) (missing documentation)
affineCircuits(Triangulation) (missing documentation)
volumeVector(Triangulation) (missing documentation)
volumeVector(Triangulation) (missing documentation)
delaunayWeights(Matrix) (missing documentation)
delaunaySubdivision(Matrix) (missing documentation)

This package is designed to help compute and explore the set of all (or many) triangulations of a point set or polytope.

We give a sample use of this package.

i1 : LP = {{-1, 0, -1, 1}, {-1, 0, 1, 0}, {-1, 0, 2, -1}, {-1, 1, -1, 0}, {1, 0, -1, 0}, {1, 0, 1, 0}, {2, -1, -1, 0}, {0, 0, 1, 0}, {1, 0, 0, 0}, {0,0,0,0}}

o1 = {{-1, 0, -1, 1}, {-1, 0, 1, 0}, {-1, 0, 2, -1}, {-1, 1, -1, 0}, {1, 0,
     ------------------------------------------------------------------------
     -1, 0}, {1, 0, 1, 0}, {2, -1, -1, 0}, {0, 0, 1, 0}, {1, 0, 0, 0}, {0, 0,
     ------------------------------------------------------------------------
     0, 0}}

o1 : List

i2 : A = transpose matrix LP

o2 = | -1 -1 -1 -1 1  1 2  0 1 0 |
     | 0  0  0  1  0  0 -1 0 0 0 |
     | -1 1  2  -1 -1 1 -1 1 0 0 |
     | 1  0  -1 0  0  0 0  0 0 0 |

              4       10
o2 : Matrix ZZ  <-- ZZ

i3 : elapsedTime Ts = allTriangulations(A, Fine => true);
 -- .251174s elapsed

i4 : select(Ts, T -> isStar T)

o4 = {triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0,
     ------------------------------------------------------------------------
     1, 6, 7, 9}, {0, 2, 3, 6, 9}, {0, 3, 4, 6, 9}, {0, 3, 4, 8, 9}, {0, 3,
     ------------------------------------------------------------------------
     5, 7, 9}, {0, 3, 5, 8, 9}, {0, 4, 6, 8, 9}, {0, 5, 6, 7, 9}, {0, 5, 6,
     ------------------------------------------------------------------------
     8, 9}, {1, 2, 3, 7, 9}, {1, 2, 6, 7, 9}, {2, 3, 4, 6, 9}, {2, 3, 4, 8,
     ------------------------------------------------------------------------
     9}, {2, 3, 5, 7, 9}, {2, 3, 5, 8, 9}, {2, 4, 6, 8, 9}, {2, 5, 6, 7, 9},
     ------------------------------------------------------------------------
     {2, 5, 6, 8, 9}}}

o4 : List

i5 : #oo == 1

o5 = true

i6 : #Ts == 51

o6 = true

i7 : T = regularFineTriangulation A

o7 = triangulation {{0, 1, 2, 3, 9}, {0, 1, 2, 6, 9}, {0, 1, 3, 7, 9}, {0, 1, 6, 7, 9}, {0, 2, 3, 4, 6}, {0, 2, 3, 4, 9}, {0, 2, 4, 6, 9}, {0, 3, 4, 7, 8}, {0, 3, 4, 7, 9}, {0, 3, 5, 7, 8}, {0, 4, 6, 7, 8}, {0, 4, 6, 7, 9}, {0, 5, 6, 7, 8}, {1, 2, 3, 7, 9}, {1, 2, 6, 7, 9}, {2, 3, 4, 7, 8}, {2, 3, 4, 7, 9}, {2, 3, 5, 7, 8}, {2, 4, 6, 7, 8}, {2, 4, 6, 7, 9}, {2, 5, 6, 7, 8}}

o7 : Triangulation

i8 : elapsedTime Ts2 = generateTriangulations T;
 -- 2.7263s elapsed

i9 : #Ts2 == #Ts

o9 = true

Author

Mike Stillman <[email protected]>

Version

This documentation describes version 0.2 of Triangulations.

Citation

If you have used this package in your research, please cite it as follows:

@misc{TriangulationsSource,
  title = {{Triangulations: triangulations of polyhedra and point sets. Version~0.2}},
  author = {Mike Stillman},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

Types
- Chirotope (missing documentation)
- Triangulation (missing documentation)
Functions and commands
- affineCircuits (missing documentation)
- allTriangulations -- see allTriangulations(Matrix) -- use topcom to generate all triangulations of a point or vector configuration
- bistellarFlip (missing documentation)
- chirotope (missing documentation)
- delaunaySubdivision (missing documentation)
- delaunayWeights (missing documentation)
- fineStarTriangulation (missing documentation)
- flips (missing documentation)
- generateTriangulations -- generate all triangulations with certain properties
- gkzVector (missing documentation)
- isFine (missing documentation)
- isRegularTriangulation -- determine if a given triangulation is a regular triangulation
- isStar (missing documentation)
- naiveChirotope (missing documentation)
- naiveIsTriangulation (missing documentation)
- neighbors (missing documentation)
- regularFineStarTriangulation (missing documentation)
- regularFineTriangulation (missing documentation)
- regularTriangulationWeights (missing documentation)
- triangulation -- make a Triangulation object
- vectors (missing documentation)
- volumeVector (missing documentation)
Methods
- affineCircuits(Matrix,List) (missing documentation)
- affineCircuits(Triangulation) (missing documentation)
- allTriangulations(Matrix) -- use topcom to generate all triangulations of a point or vector configuration
- bistellarFlip(List,List) (missing documentation)
- bistellarFlip(Triangulation,List) (missing documentation)
- Chirotope == Chirotope (missing documentation)
- chirotope(Matrix) (missing documentation)
- chirotope(String) (missing documentation)
- delaunaySubdivision(Matrix) (missing documentation)
- delaunayWeights(Matrix) (missing documentation)
- fineStarTriangulation(Matrix,List) (missing documentation)
- flips(Triangulation) (missing documentation)
- generateTriangulations(Matrix) -- see generateTriangulations -- generate all triangulations with certain properties
- generateTriangulations(Matrix,List) (missing documentation)
- generateTriangulations(Triangulation) (missing documentation)
- isFine(Matrix,List) (missing documentation)
- isFine(Triangulation) (missing documentation)
- isRegularTriangulation(Triangulation) -- see isRegularTriangulation -- determine if a given triangulation is a regular triangulation
- isRegularTriangulation(Matrix,List) (missing documentation)
- isStar(Matrix,List) (missing documentation)
- isStar(Triangulation) (missing documentation)
- isWellDefined(Triangulation) (missing documentation)
- matrix(Triangulation) (missing documentation)
- max(Triangulation) (missing documentation)
- naiveChirotope(Matrix) (missing documentation)
- naiveIsTriangulation(Matrix,List) (missing documentation)
- naiveIsTriangulation(Matrix,List,List) (missing documentation)
- naiveIsTriangulation(Triangulation) (missing documentation)
- neighbors(Triangulation) (missing documentation)
- net(Triangulation) (missing documentation)
- regularFineStarTriangulation(Matrix) (missing documentation)
- regularFineTriangulation(Matrix) (missing documentation)
- regularTriangulationWeights(Matrix,List) (missing documentation)
- regularTriangulationWeights(Triangulation) (missing documentation)
- toExternalString(Triangulation) (missing documentation)
- toString(Chirotope) (missing documentation)
- toString(Triangulation) (missing documentation)
- Triangulation == Triangulation (missing documentation)
- triangulation(Matrix,List) (missing documentation)
- vectors(Triangulation) (missing documentation)
- volumeVector(Matrix,List) (missing documentation)
- volumeVector(Triangulation) (missing documentation)
Symbols
- ConeIndex (missing documentation)

For the programmer

The object Triangulations is a package, defined in Triangulations.m2.

The source of this document is in Triangulations.m2:560:0.