Table of Contents

BigObject SubdivisionOfPoints<Scalar>

from application fan

The inhomogeneous variant of SubdivisionOfVectors, similar to the derivation of PointConfiguration from VectorConfiguration.

Type Parameters:

Scalar: default: Rational

derived from:
SubdivisionOfVectors
Example:

To produce a regular subdivision of the vertices of a square: 

 > $c=new SubdivisionOfPoints(POINTS=>polytope::cube(2)->VERTICES,WEIGHTS=>[0,0,0,1]);
 > print $c->MAXIMAL_CELLS;
 {0 1 2}
 {1 2 3}

Properties

Geometry

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

N_POINTS

The number of POINTS in the configuration. Alias for property N_VECTORS.

Type:
Int
POINTS

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

Type:
Matrix<Scalar,NonSymmetric>
POLYHEDRAL_COMPLEX

The polyhedral complex induced by the cells of the subdivision.

Type:
PolyhedralComplex<Scalar>
REGULAR

Whether the subdivision is regular, i.e. induced by a weight vector.

Type:
Bool
TIGHT_SPAN

The tight span of the subdivision.

Type:
PolyhedralComplex<Scalar>
UNIMODULAR

A subdivision is unimodular if it is a triangulation such that each maximal simplex has unit normalized volume.

Type:
Bool
Example:

Unit square, triangulated. 

 > $S = new SubdivisionOfPoints(POINTS=>cube(2,0)->VERTICES, WEIGHTS=>[0,0,0,1]);
 > print $S->UNIMODULAR
 true
Example:

Unit 3-cube, triangulation induced by four compatible vertex splits. 

 > $S = new SubdivisionOfPoints(POINTS=>cube(3,0)->VERTICES, WEIGHTS=>[0,1,1,0,1,0,1,0]);
 > print $S->UNIMODULAR
 false

WEIGHTS

Vector assigning a weight to each point to get a regular subdivision.

Type:
Vector<Scalar>

Visualization

These properties are for visualization.

POINT_LABELS

Unique names assigned to the POINTS. If specified, they are shown by visualization tools instead of point indices. Alias for property LABELS.

Type:
Array<String>

Methods

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.

DIM

Affine dimension of the point configuration. Similar to DIM.

cell(Int i)

Returns the i-th cell of the complex as a PointConfiguration

Parameters:

Int i

Returns:
PointConfiguration

Visualization

These methods are for visualization.

VISUAL()

Visualizes the SubdivisionOfPoints.

Options:
option list geometric_options
Returns:
Visual::PolyhedralFan