Available versions of this document: latest release, release 3.5, nightly master
Reference documentation for older polymake versions: release 3.4, release 3.3, release 3.2
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:
 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:

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

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

MAX_BOUNDARY_SIMPLICES
Tells the fulldimensional simplices on the boundary that contain no points except for the vertices.
 Type:

MAX_INTERIOR_SIMPLICES
Tells the fulldimensional simplices that contain no points except for the vertices.
 Type:

N_MAX_BOUNDARY_SIMPLICES
Tells the number of MAX_BOUNDARY_SIMPLICES
 Type:

N_MAX_INTERIOR_SIMPLICES
Tells the number of MAX_INTERIOR_SIMPLICES
 Type:

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

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:
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:

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

CONVEX
True if the
POINTS
are in convex position. Type:

CONVEX_HULL
 Type:
Polytope<Scalar>
 Properties of CONVEX_HULL:

VERTEX_POINT_MAP
Indices of
VERTICES
of theCONVEX_HULL
asPOINTS
. Type:


FAR_POINTS
Indices of
POINTS
that are rays. Type:

MULTIPLE_POINTS
Tells if multiple points exist. Alias for property
MULTIPLE_VECTORS
. Type:

NON_VERTICES
POINTS
that are notVERTICES
of theCONVEX_HULL
 Type:

N_POINTS
 Type:

VERTEX_POINT_MAP
Indices of
VERTICES
of theCONVEX_HULL
asPOINTS
 Type:
Symmetry
These properties capture information of the object that is concerned with the action of permutation groups.

GROUP
 derived from:
 Type:
 Properties of GROUP:

MATRIX_ACTION
 Type:
MatrixActionOnVectors<Scalar>
 Properties of MATRIX_ACTION:

POINTS_ORBITS
Alias for property
VECTORS_ORBITS
. Type:


POINTS_ACTION
 Type:
 Properties of POINTS_ACTION:

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


REPRESENTATIVE_BOUNDARY_RIDGE_SIMPLICES
One representative for each orbit of boundary ridge simplices
 Type:

REPRESENTATIVE_INTERIOR_RIDGE_SIMPLICES
One representative for each orbit of interior ridge simplices
 Type:

REPRESENTATIVE_MAX_BOUNDARY_SIMPLICES
One representative for each orbit of maximaldimensional boundary simplices
 Type:

REPRESENTATIVE_MAX_INTERIOR_SIMPLICES
One representative for each orbit of maximaldimensional interior simplices
 Type:

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:


TRIANGULATION
 Type:
GeometricSimplicialComplex<Scalar>
 Properties of TRIANGULATION:

BOUNDARY
 derived from:
 Type:
 Properties of BOUNDARY:

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


GKZ_VECTOR
GKZvector
See Chapter 7 in Gelfand, Kapranov, and Zelevinsky:
Discriminants, Resultants and Multidimensional Determinants, Birkhäuser 1994
 Type:
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:
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).
> $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

REFINED_SPLITS
The splits that are coarsenings of the current
TRIANGULATION
. If the triangulation is regular these form the unique split decomposition of the corresponding weight function. Type:

WEIGHTS
Weight vector to construct a regular
TRIANGULATION
. Must be generic. Type:
Vector<Scalar>

Visualization
These properties are for visualization.

PIF_CYCLIC_NORMAL
VIF_CYCLIC_NORMAL
of theCONVEX_HULL
, but with the indices formPOINTS
instead ofVERTICES
 Type:

POINT_LABELS
 Type:
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:
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:

DIM()
Affine dimension of the point configuration. Similar to
DIM
. Returns:
Visualization
These methods are for visualization.

VISUAL()
Visualize a point configuration.
 Options:
 option list
Visual::Polygons::decorations
 option list
geometric_options
 Returns:

VISUAL_POINTS()
Visualize the
POINTS
of a point configuration. Options:
 option list
Visual::Polygons::decorations
 option list
geometric_options
 Returns: