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}

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>

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>

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

These methods are for visualization.

VISUAL()

Visualizes the SubdivisionOfPoints.

Options:

option list geometric_options

Returns:

Visual::PolyhedralFan