application: fan

This application deals with polyhedral fans. You can define a fan, e.g. via its RAYS and MAXIMAL_CONES and compute several properties like HASSE_DIAGRAM and F_VECTOR.

imports from: common, graph, polytope
uses: group, ideal, topaz

Objects

User Functions

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      common_refinement (f1, f2) → PolyhedralFan

      Computes the common refinement of two fans.

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      projection (P, indices) → PolyhedralFan

      Orthogonally project a pointed fan to a coordinate subspace.

      The subspace the fan P is projected on is given by indices in the set indices. The option revert inverts the coordinate list.

      Parameters
      PolyhedralFanP
      Array<Int>indices
      Options
      Boolrevert
      inverts the coordinate list
      Returns
      PolyhedralFan
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      projection_full (P) → PolyhedralFan

      Orthogonally project a fan to a coordinate subspace such that redundant columns are omitted, i.e., the projection becomes full-dimensional without changing the combinatorial type.

      Parameters
      PolyhedralFanP
      Options
      Boolrelabel
      copy labels to projection
      Returns
      PolyhedralFan
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      mixed_subdivision (P_0, P_1, VIF, t_0, t_1) → PolytopalComplex

      Create a weighted mixed subdivision of the Minkowski sum of two polytopes, using the Cayley trick. All polytopes must have the same dimension, at least one of them must be pointed. The vertices of the first polytope P_0 are weighted with t_0, and the vertices of the second polytope P_1 with t_1.

      Default values are t_0=t_1=1.

      The option relabel creates an additional section VERTEX_LABELS.

      Parameters
      PolytopeP_0
      the first polytope
      PolytopeP_1
      the second polytope
      Array<Set>VIF
      the indices of the vertices of the mixed cells
      Scalart_0
      the weight for the vertices of P_0; default 1
      Scalart_1
      the weight for the vertices of P_1; default 1
      Options
      Boolrelabel
      Returns
      PolytopalComplex
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      mixed_subdivision (m, C, a) → PolytopalComplex

      Create a weighted mixed subdivision of a Cayley embedding of an array of polytopes. Each vertex v of the i-th polytope is weighted with t_i, the i-th entry of the optional array t.

      The option relabel creates an additional section VERTEX_LABELS.

      Parameters
      Intm
      the number of polytopes giving rise to the Cayley embedding
      PolytopeC
      the Cayley embedding of the input polytopes
      Array<Set>a
      triangulation of C
      Options
      Array<Scalar>t
      scaling for the Cayley embedding; defaults to the all-1 vector
      Boolrelabel
      Returns
      PolytopalComplex
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      mixed_subdivision (A) → PolytopalComplex

      Create a weighted mixed subdivision of an array (P1,...,Pm) of polytopes. All polytopes must have the same dimension, at least one of them must be pointed, and all must be defined over the same number type. Each vertex v of the i-th polytope is weighted with t_i, the i-th entry of the optional array t.

      The option relabel creates an additional section VERTEX_LABELS.

      Parameters
      Array<Polytope>A
      the input polytopes
      Options
      Array<Scalar>t
      scaling for the Cayley embedding; defaults to the all-1 vector
      Boolrelabel
      Returns
      PolytopalComplex
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      tiling_quotient <Coord> (P, Q) → PolyhedralComplex

      Calculates the quotient of P by Q+L, where Q+L is a lattice tiling. The result is a polytopal complex inside Q.

      Type Parameters
      Coord
      Parameters
      PolytopeP
      a polytope
      PolytopeQ
      a polytope that tiles space
      Returns
      PolyhedralComplex
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