Table of Contents

BigObject PointConfiguration<Scalar>

from application polytope

The POINTS of an object of type PointConfiguration encode a not necessarily convex finite point set. The difference to a parent VectorConfiguration is that the points have homogeneous coordinates, i.e. they will be normalized to have first coordinate 1 without warning.

Type Parameters:

Scalar: default: Rational

derived from:
VectorConfiguration
Specializations:

PointConfiguration::ExactCoord: A point configuration with an exact coordinate type, like Rational.

Properties

Input property

These properties are for input only. They allow redundant information.

POINTS

The points of the configuration. Multiples allowed. Alias for property VECTORS.

Type:
Matrix<Scalar,NonSymmetric>

Combinatorics

These properties capture combinatorial information of the object. Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice.

COCIRCUIT_EQUATIONS

Tells the cocircuit equations that hold for the configuration, one for each interior ridge

Type:
SparseMatrix<Rational,NonSymmetric>

GRAPH

Graph of the point configuration. Two points are adjacent if they are neigbours in a edge of the CONVEX_HULL.

Type:
Graph<Undirected>

INTERIOR_RIDGE_SIMPLICES

Tells the number of codimension 1 simplices that are not on the boundary

Type:
Array<Set<Int>>

MAX_BOUNDARY_SIMPLICES

Tells the full-dimensional simplices on the boundary that contain no points except for the vertices.

Type:
Array<Set<Int>>

MAX_INTERIOR_SIMPLICES

Tells the full-dimensional simplices that contain no points except for the vertices.

Type:
Array<Set<Int>>

N_MAX_BOUNDARY_SIMPLICES

Tells the number of MAX_BOUNDARY_SIMPLICES

Type:
Int

N_MAX_INTERIOR_SIMPLICES

Tells the number of MAX_INTERIOR_SIMPLICES

Type:
Int

SIMPLEXITY_LOWER_BOUND

A lower bound for the minimal number of simplices in a triangulation

Type:
Int

SPLITS

The splits of the point configuration, i.e., hyperplanes cutting the configuration in two parts such that we have a regular subdivision.

Type:
Matrix<Scalar,NonSymmetric>

SPLIT_COMPATIBILITY_GRAPH

Two SPLITS are compatible if the defining hyperplanes do not intersect in the interior of the point configuration. This defines a graph.

Type:
Graph<Undirected>

Geometry

These properties capture geometric information of the object. Geometric properties depend on geometric information of the object, like, e.g., vertices or facets.

AFFINE_HULL

Dual basis of the affine hull of the point configuration

Type:
Matrix<Scalar,NonSymmetric>

BARYCENTER

The center of gravity of the point configuration.

Type:
Vector<Scalar>

BOUNDED

True if the point configuration is bounded.

Type:
Bool

CENTERED

True if (1, 0, 0, …) is in the relative interior.

Type:
Bool

CONVEX

True if the POINTS are in convex position.

Type:
Bool

CONVEX_HULL
Type:
Polytope<Scalar>
Properties of CONVEX_HULL:
VERTEX_POINT_MAP

Indices of VERTICES of the CONVEX_HULL as POINTS.

Type:
Array<Int>

FAR_POINTS

Indices of POINTS that are rays.

Type:
Set<Int>

MULTIPLE_POINTS

Tells if multiple points exist. Alias for property MULTIPLE_VECTORS.

Type:
Bool

NON_VERTICES

POINTS that are not VERTICES of the CONVEX_HULL

Type:
Set<Int>

N_POINTS

Number of POINTS. Alias for property N_VECTORS.

Type:
Int

VERTEX_POINT_MAP

Indices of VERTICES of the CONVEX_HULL as POINTS

Type:
Array<Int>

Symmetry

These properties capture information of the object that is concerned with the action of permutation groups.

GROUP
derived from:
GROUP
Type:
Group
Properties of GROUP:
MATRIX_ACTION
Type:
MatrixActionOnVectors<Scalar>
Properties of MATRIX_ACTION:
POINTS_ORBITS

Alias for property VECTORS_ORBITS.

Type:
Array<Set<Int>>
POINTS_ACTION
Type:
PermutationAction<Int,Rational>
Properties of POINTS_ACTION:
SYMMETRIZED_COCIRCUIT_EQUATIONS

The cocircuit equations, projected to a certain direct sum of isotypic components

Type:
SymmetrizedCocircuitEquations
REPRESENTATIVE_BOUNDARY_RIDGE_SIMPLICES

One representative for each orbit of boundary ridge simplices

Type:
Array<Bitset<Int>>
REPRESENTATIVE_INTERIOR_RIDGE_SIMPLICES

One representative for each orbit of interior ridge simplices

Type:
Array<Bitset<Int>>
REPRESENTATIVE_MAX_BOUNDARY_SIMPLICES

One representative for each orbit of maximal-dimensional boundary simplices

Type:
Array<Bitset<Int>>
REPRESENTATIVE_MAX_INTERIOR_SIMPLICES

One representative for each orbit of maximal-dimensional interior simplices

Type:
Array<Bitset<Int>>

Triangulation and volume

These properties collect information about triangulations of the object and properties usually computed from such, as the volume.

POLYTOPAL_SUBDIVISION
Type:
SubdivisionOfPoints<Scalar>
Properties of POLYTOPAL_SUBDIVISION:
REFINED_SPLITS

The splits that are coarsenings of the subdivision. If the subdivision is regular these form the unique split decomposition of the corresponding weight function.

Type:
Set<Int>

TRIANGULATION
Type:
GeometricSimplicialComplex<Scalar>
Properties of TRIANGULATION:
BOUNDARY
derived from:
GeometricSimplicialComplex
Type:
GeometricSimplicialComplex
Properties of BOUNDARY:
FACET_TRIANGULATIONS

DOC_FIXME: Incomprehensible description! For each facet the set of simplex indices of BOUNDARY that triangulate it.

Type:
Array<Set<Int>>
GKZ_VECTOR

GKZ-vector

> See Chapter 7 in Gelfand, Kapranov, and Zelevinsky: 

    > Discriminants, Resultants and Multidimensional Determinants, Birkhäuser 1994
      ? Type:
      :''[[..:common#Vector |Vector]]<Scalar>''
    ?  **''MASSIVE_GKZ_VECTOR''**
    :: Calculate the massive GKZ vectors of the triangulations of a integral PointConfiguration //A//. For a definition see Chapter 11 of Gelfand, Kapranov, and Zelevinsky: Discriminants, Resultants and Multidimensional Determinants, Birkhäuser 1994.
      ? Type:
      :''[[..:common#Vector |Vector]]<Scalar>''
      ? Example:
      :: To calculate the massive GKZ vector of a triangulation of a point configuration. This example is from the book mentioned above (p. 369, top right example).
      :: <code perl> > $A=new PointConfiguration(POINTS=>[[1,0,0],[1,1,0],[1,2,0],[1,3,0],[1,0,1],[1,1,1],[1,0,2]]);
   > $A->add("TRIANGULATION", WEIGHTS=>[0,1,0,1,1,1,0]);
   > print $A->TRIANGULATION->MASSIVE_GKZ_VECTOR;
   1 0 3 1 0 0 4
  </code>
    ?  **''REFINED_SPLITS''**
    :: The splits that are coarsenings of the current ''[[..:polytope:PointConfiguration#TRIANGULATION |TRIANGULATION]]''. If the triangulation is regular these form the unique split decomposition of the corresponding weight function.
      ? Type:
      :''[[..:common#Set |Set]]<[[..:common#Int |Int]]>''
    ?  **''REGULAR''**
    :: Checks regularity of ''[[..:polytope:PointConfiguration#TRIANGULATION |TRIANGULATION]]''.
      ? Type:
      :''[[..:common#Bool |Bool]]''
    ?  **''WEIGHTS''**
    :: Weight vector to construct a regular ''[[..:polytope:PointConfiguration#TRIANGULATION |TRIANGULATION]]''. Must be generic.
      ? Type:
      :''[[..:common#Vector |Vector]]<Scalar>''

Visualization

These properties are for visualization.

PIF_CYCLIC_NORMAL

VIF_CYCLIC_NORMAL of the CONVEX_HULL, but with the indices form POINTS instead of VERTICES

Type:
Array<Array<Int>>

POINT_LABELS

Unique names assigned to the POINTS. Alias for property LABELS.

Type:
Array<String>

Methods

Combinatorics

These methods capture combinatorial information of the object. Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice.

faces_of_dim(PointConfiguration p)

Output the faces of a given dimension

Parameters:

PointConfiguration p: the input point configuration

Returns:
Array<Set<Int>>

Geometry

These methods capture geometric information of the object. Geometric properties depend on geometric information of the object, like, e.g., vertices or facets.

AMBIENT_DIM()

Ambient dimension of the point configuration (without the homogenization coordinate). Similar to AMBIENT_DIM.

Returns:
Int

DIM()

Affine dimension of the point configuration. Similar to DIM.

Returns:
Int

Visualization

These methods are for visualization.

VISUAL()

Visualize a point configuration.

Options:
option list Visual::Polygons::decorations
option list geometric_options
Returns:
Visual::PointConfiguration

VISUAL_POINTS()

Visualize the POINTS of a point configuration.

Options:
option list Visual::Polygons::decorations
option list geometric_options
Returns:
Visual::Object