BigObject HyperplaneArrangement<Scalar>

from application fan

A hyperplane arrangement. The hyperplane arrangement is given by a matrix HYPERPLANES whose rows are the linear equations of the hyperplanes and an optional support cone. The support cone defaults to being the whole space. Duplicate hyperplanes are ignored, as well as hyperplanes that intersect the support cone trivially. The support cone is subdivided by the hyperplanes resulting in a fan CELL_DECOMPOSITION.

Type Parameters:

Scalar: numeric data type used for the coordinates, must be an ordered field. Default is Rational.

Example:

Take the 2-dimensional positive orthant and slice it along the ray through (1,1)

 > $HA = new HyperplaneArrangement(HYPERPLANES=>[[-1,1]], "SUPPORT.INPUT_RAYS"=>[[1,0],[0,1]]);
 > $CD = $HA->CELL_DECOMPOSITION;
 > print $CD->RAYS;
 0 1
 1 0
 1 1

 > print $CD->MAXIMAL_CONES;
 {0 2}
 {1 2}

Example:

Coxeter hyperplane arrangement of type E8.

 > $E8 = new HyperplaneArrangement(HYPERPLANES=>root_system("E8")->VECTORS->minor(All,~[0]));

Note that the roots lie “at infinity”, which is why the leading zero column of the root vectors is eliminated.

Permutations:

ConesPerm:

permuting the RAYS

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

SUPPORT

A cone being subdivided by the HYPERPLANES defaults to the whole space.

Type:

Cone<Scalar>

Example:

Take the 2-dimensional positive orthant and slice it along the ray through (1,1)

 > $HA = new HyperplaneArrangement(HYPERPLANES=>[[-1,1]], "SUPPORT.INPUT_RAYS"=>[[1,0],[0,1]]);
 > $CD = $HA->CELL_DECOMPOSITION;
 > print $CD->RAYS;
 0 1
 1 0
 1 1

 > print $CD->MAXIMAL_CONES;
 {0 2}
 {1 2}

Example:

Subdivide the two-dimensional space along the axes

 > $HA = new HyperplaneArrangement(HYPERPLANES=>[[1,0],[0,1]]);
 > $CD = $HA->CELL_DECOMPOSITION;
 > print $CD->RAYS;
 -1 0
 0 -1
 0 1
 1 0

 > print $CD->MAXIMAL_CONES;
 {0 1}
 {0 2}
 {1 3}
 {2 3}

 > print $CD->COMPLETE;
 true

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

CELL_SIGNATURES

The i-th entry is the signature of the i-th maximal cone of the CELL_DECOMPOSITION as Set<Int> # of indices of the HYPERPLANES that evaluate negatively on this cone.

Type:

Array<Set<Int>>

Example:

Take the 2-dimensional positive orthant and slice it along the ray through (1,1)

 > $HA = new HyperplaneArrangement(HYPERPLANES=>[[-1,1]], "SUPPORT.INPUT_RAYS"=>[[1,0],[0,1]]);
 > $CD = $HA->CELL_DECOMPOSITION;
 > print $CD->MAXIMAL_CONES;
 {0 2}
 {1 2}

 > print $HA->CELL_SIGNATURES;
 {}
 {0}

 > print $HA->cell_to_signature($CD->MAXIMAL_CONES->[0]);
 {}

 > print $HA->cell_to_signature($CD->MAXIMAL_CONES->[1]);
 {0}

 > print $HA->signature_to_cell($HA->CELL_SIGNATURES->[0]);
 0

 > print $CD->MAXIMAL_CONES->[$HA->signature_to_cell($HA->CELL_SIGNATURES->[0])];
 {0 2}

N_HYPERPLANES

Number of HYPERPLANES.

Type:

Int

Example:

Coordinate hyperplane arrangement in the plane.

 > $HA = new HyperplaneArrangement(HYPERPLANES=>[[1,0],[0,1]]);
 > print $HA->N_HYPERPLANES;
 2

RAYS_IN_HYPERPLANES

Incidences between RAYS and HYPERPLANES.

Type:

IncidenceMatrix<NonSymmetric>

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

AMBIENT_DIM

Dimension of the space which contains the hyperplane arrangement.

Type:

Int

CELL_DECOMPOSITION

Slicing the SUPPORT along every hyperplane of HYPERPLANES one gets a polyhedral fan.

Type:

PolyhedralFan<Scalar>

Example:

Take the 2-dimensional positive orthant and slice it along the ray through (1,1)

 > $HA = new HyperplaneArrangement(HYPERPLANES=>[[-1,1]], "SUPPORT.INPUT_RAYS"=>[[1,0],[0,1]]);
 > $CD = $HA->CELL_DECOMPOSITION;
 > print $CD->RAYS;
 0 1
 1 0
 1 1

 > print $CD->MAXIMAL_CONES;
 {0 2}
 {1 2}

HYPERPLANES

A matrix containing the hyperplanes of the arrangement as rows.

Type:

Matrix<Scalar,NonSymmetric>

Example:

The same hyperplane with opposing directions.

 > $HA = new HyperplaneArrangement(HYPERPLANES=>[[1,-1],[-1,1],[1,1]]);
 > print $HA->HYPERPLANES;
 1 -1
 -1 1
 1 1

Example:

A hyperplane that does not cut through the SUPPORT

 > $HA = new HyperplaneArrangement(HYPERPLANES=>[[1,1]], "SUPPORT.INEQUALITIES"=>unit_matrix(2));
 > print $HA->HYPERPLANES;
 1 1

LINEALITY_SPACE

A basis of the lineality space of the hyperplane arrangement.

Type:

Matrix<Scalar,NonSymmetric>

HYPERPLANE_LABELS

For a polyhedral fan built from scratch, you should create this property by yourself, either manually in a text editor, or with a client program.

Type:

Array<String>

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

cell_to_signature

Given a maximal cone of CELL_DECOMPOSITION as Set<Int> containing the indices of the rays spanning it, return the signature of the cone as Set<Int> of indices of the HYPERPLANES that evaluate negatively on this cone.

Example:

Take the 2-dimensional positive orthant and slice it along the ray through (1,1)

 > $HA = new HyperplaneArrangement(HYPERPLANES=>[[-1,1]], "SUPPORT.INPUT_RAYS"=>[[1,0],[0,1]]);
 > $CD = $HA->CELL_DECOMPOSITION;
 > print $CD->MAXIMAL_CONES;
 {0 2}
 {1 2}

 > print $HA->CELL_SIGNATURES;
 {}
 {0}

 > print $HA->cell_to_signature($CD->MAXIMAL_CONES->[0]);
 {}

 > print $HA->cell_to_signature($CD->MAXIMAL_CONES->[1]);
 {0}

 > print $HA->signature_to_cell($HA->CELL_SIGNATURES->[0]);
 0

 > print $CD->MAXIMAL_CONES->[$HA->signature_to_cell($HA->CELL_SIGNATURES->[0])];
 {0 2}

signature_to_cell

Given a signature as a Set<Int> of indices that indicate which HYPERPLANES should evaluate negatively (the remaining evaluate positively), return the maximal cone of CELL_DECOMPOSITION associated to this signature. The result the index of the maximal cone in the maximal cones of CELL_DECOMPOSITION.

Example:

Take the 2-dimensional positive orthant and slice it along the ray through (1,1)

 > $HA = new HyperplaneArrangement(HYPERPLANES=>[[-1,1]], "SUPPORT.INPUT_RAYS"=>[[1,0],[0,1]]);
 > $CD = $HA->CELL_DECOMPOSITION;
 > print $CD->MAXIMAL_CONES;
 {0 2}
 {1 2}

 > print $HA->CELL_SIGNATURES;
 {}
 {0}

 > print $HA->cell_to_signature($CD->MAXIMAL_CONES->[0]);
 {}

 > print $HA->cell_to_signature($CD->MAXIMAL_CONES->[1]);
 {0}

 > print $HA->signature_to_cell($HA->CELL_SIGNATURES->[0]);
 0

 > print $CD->MAXIMAL_CONES->[$HA->signature_to_cell($HA->CELL_SIGNATURES->[0])];
 {0 2}

These methods are for visualization.

VISUAL()

Visualizes the fan, intersected with the unit ball.

Options:

option list geometric_options_linear

Returns:

Visual::PolyhedralFan