Available versions of this document: latest release, release 4.13, release 4.12, release 4.11, release 4.10, release 4.9, release 4.8, release 4.7, release 4.6, release 4.5, release 4.4, release 4.3, release 4.2, release 4.1, release 4.0, release 3.6, release 3.5, nightly master
Reference documentation for older polymake versions: release 3.4, release 3.3, release 3.2
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 isRational
.- Example:
Take the 2-dimensional positive orthant and slice it along the ray through (1,1)
> $HA = new HyperplaneArrangement(INPUT_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}
- Permutations:
- ConesPerm:
permuting the
RAYS
Properties
Input property
These properties are for input only. They allow redundant information.
-
INPUT_HYPERPLANES
A matrix containing the input hyperplanes of the arrangement as rows.
- Type:
Matrix<Scalar,NonSymmetric>
-
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(INPUT_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(INPUT_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
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.
-
CELL_SIGNATURES
The i-th entry is the signature of the i-th maximal cone of the
CELL_DECOMPOSITION
.- Type:
- Example:
Take the 2-dimensional positive orthant and slice it along the ray through (1,1)
> $HA = new HyperplaneArrangement(INPUT_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}
Geometry
These properties capture geometric information of the object. Geometric properties depend on geometric information of the object, like, e.g., vertices or facets.
-
CELL_DECOMPOSITION
Slicing the
SUPPORT
along every hyperplane ofHYPERPLANES
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(INPUT_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. This matrix is obtained from
INPUT_HYPERPLANES
by removing duplicates and also removing hyperplanes that are obsolete wrt theSUPPORT
cone.- Type:
Matrix<Scalar,NonSymmetric>
- Example:
The same hyperplane with opposing directions.
> $HA = new HyperplaneArrangement(INPUT_HYPERPLANES=>[[1,-1],[-1,1],[1,1]]); > print $HA->HYPERPLANES; 1 -1 1 1
- Example:
A hyperplane that does not cut through the
SUPPORT
> $HA = new HyperplaneArrangement(INPUT_HYPERPLANES=>[[1,1]], "SUPPORT.INEQUALITIES"=>unit_matrix(2)); > print $HA->HYPERPLANES;
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.
-
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 theHYPERPLANES
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(INPUT_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 ofCELL_DECOMPOSITION
associated to this signature. The result the index of the maximal cone in the maximal cones ofCELL_DECOMPOSITION
.- Example:
Take the 2-dimensional positive orthant and slice it along the ray through (1,1)
> $HA = new HyperplaneArrangement(INPUT_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}