BigObject Cone<Scalar>

from application polytope

A polyhedral cone, not necessarily pointed. Note that in contrast to the vertices of a polytope, the RAYS are given in affine coordinates.

Type Parameters:

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

Specializations:

Cone::ExactCoord: An affine cone with an exact coordinate type, like Rational.

Cone<Float>: An affine cone with float coordinates realized in R^d.

Cone<Rational>: An affine rational cone realized in R^d.

Permutations:

FacetPerm:

permuting the FACETS

VertexPerm:

permuting the RAYS TODO: rename into RaysPerm and introduce an overriding alias in Polytope

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

---

EQUATIONS

Equations that hold for all INPUT_RAYS of the cone. All vectors in this section must be non-zero. Input section only. Ask for LINEAR_SPAN if you want to see an irredundant description of the linear span.

Type:

Matrix<Scalar,NonSymmetric>

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INEQUALITIES

Inequalities giving rise to the cone; redundancies are allowed. All vectors in this section must be non-zero. Dual to INPUT_RAYS. Input section only. Ask for FACETS if you want to compute an H-representation from a V-representation.

Type:

Matrix<Scalar,NonSymmetric>

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INPUT_LINEALITY

(Non-homogenous) vectors whose linear span defines a subset of the lineality space of the cone; redundancies are allowed. All vectors in the input must be non-zero. Dual to EQUATIONS. Input section only. Ask for LINEALITY_SPACE if you want to compute a V-representation from an H-representation.

Type:

Matrix<Scalar,NonSymmetric>

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INPUT_RAYS

(Non-homogenous) vectors whose positive span form the cone; redundancies are allowed. Dual to INEQUALITIES. All vectors in the input must be non-zero. Input section only. Ask for RAYS if you want to compute a V-representation from an H-representation.

Type:

Matrix<Scalar,NonSymmetric>

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

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ALTSHULER_DET

Let M be the vertex-facet incidence matrix, then the Altshuler determinant is defined as max{det(M ∗ M^T), det(M^T ∗ M)}.

Type:

Integer

Example:

This prints the Altshuler determinant of the built-in pentagonal pyramid (Johnson solid 2):

 > print johnson_solid("pentagonal_pyramid")->ALTSHULER_DET;
 25

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COCIRCUIT_EQUATIONS

A matrix whose rows contain the cocircuit equations of P. The columns correspond to the MAX_INTERIOR_SIMPLICES.

Type:

SparseMatrix<Rational,NonSymmetric>

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COMBINATORIAL_DIM

Combinatorial dimension This is the dimension all combinatorial properties of the cone like e.g. RAYS_IN_FACETS or the HASSE_DIAGRAM refer to. Geometrically, the combinatorial dimension is the dimension of the intersection of the pointed part of the cone with a hyperplane that creates a bounded intersection.

Type:

Int

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DUAL_GRAPH

Facet-ridge graph. Dual to GRAPH.

Type:

Graph<Undirected>

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ESSENTIALLY_GENERIC

All intermediate polytopes (with respect to the given insertion order) in the beneath-and-beyond algorithm are simplicial. We have the implications: RAYS in general position ⇒ ESSENTIALLY_GENERIC ⇒ SIMPLICIAL

Type:

Bool

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EXCESS_FACET_DEGREE

Measures the deviation of the cone from being simple in terms of the DUAL_GRAPH.

Type:

Int

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EXCESS_RAY_DEGREE

Measures the deviation of the cone from being simple in terms of the GRAPH.

Type:

Int

---

F2_VECTOR

The vector counting the number of incidences between pairs of faces. `f<sub>ik</sub>` is the number of incident pairs of `(i+1)`-faces and `(k+1)`-faces. The main diagonal contains the F_VECTOR.

Type:

Matrix<Integer,NonSymmetric>

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FACETS_THRU_RAYS

Transposed to RAYS_IN_FACETS. Notice that this is a temporary property; it will not be stored in any file.

Type:

IncidenceMatrix<NonSymmetric>

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FACET_SIZES

Number of incident rays for each facet.

Type:

Array<Int>

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FLAG_VECTOR

Condensed form of the flag vector, containing all entries indexed by sparse sets in {0, …, COMBINATORIAL_DIM-1} in the following order: (1, f₀, f₁, f₂, f₀₂, f₃, f₀₃, f₁₃, f₄, f₀₄, f₁₄, f₂₄, f₀₂₄, f₅, …). Use Dehn-Sommerville equations, via user function N_FLAGS, to extend.

Type:

Vector<Integer>

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FOLDABLE_COCIRCUIT_EQUATIONS

A matrix whose rows contain the foldable cocircuit equations of P. The columns correspond to 2 * MAX_INTERIOR_SIMPLICES. col 0 = 0, col 1 = first simplex (black copy), col 2 = first simplex (white copy), col 3 = second simplex (black copy), …

Type:

SparseMatrix<Rational,NonSymmetric>

---

F_VECTOR

The vector counting the number of faces (`f<sub>k</sub>` is the number of `(k+1)`-faces).

Type:

Vector<Integer>

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GRAPH

Vertex-edge graph obtained by intersecting the cone with a transversal hyperplane.

Type:

Graph<Undirected>

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HASSE_DIAGRAM

Type:

Lattice<BasicDecoration,Sequential>

Methods of HASSE_DIAGRAM:

dim()

The dimension of the underlying object

Returns:

Int

nodes_of_dim(Int d)

The indices of nodes in the HASSE_DIAGRAM corresponding to faces of dimension d in the underlying object

Parameters:

Int d: dimension

Returns:

Set<Int>

nodes_of_dim_range(Int d1, Int d2)

The indices of nodes in the HASSE_DIAGRAM corresponding to faces with dimension in the range (d1,d2) in the underlying object

Parameters:

Int d1: lower dimension of the range

Int d2: upper dimension of the range

Returns:

Set<Int>

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INTERIOR_RIDGE_SIMPLICES

The (d-1)-dimensional simplices in the interior.

Type:

Array<Set<Int>>

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MAX_BOUNDARY_SIMPLICES

The boundary (d-1)-dimensional simplices of a cone of combinatorial dimension d

Type:

Array<Set<Int>>

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MAX_INTERIOR_SIMPLICES

The interior d-dimensional simplices of a cone of combinatorial dimension d

Type:

Array<Set<Int>>

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N_EDGES

The number of edges of the GRAPH

Type:

Int

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N_FACETS

The number of FACETS.

Type:

Int

---

N_RAYS

The number of RAYS

Type:

Int

---

N_RAY_FACET_INC

Number of pairs of incident vertices and facets.

Type:

Int

---

N_RIDGES

The number of ridges (faces of codimension 2) equals the number of edges of the DUAL_GRAPH

Type:

Int

---

RAYS_IN_FACETS

Ray-facet incidence matrix, with rows corresponding to facets and columns to rays. Rays and facets are numbered from 0 to N_RAYS-1 rsp. N_FACETS-1, according to their order in RAYS rsp. FACETS.

Type:

IncidenceMatrix<NonSymmetric>

---

RAYS_IN_RIDGES

Ray-ridge incidence matrix, with rows corresponding to ridges and columns to rays. Rays and ridges are numbered from 0 to N_RAYS-1 rsp. N_RIDGES-1, according to their order in RAYS rsp. RIDGES.

Type:

IncidenceMatrix<NonSymmetric>

---

RAY_SIZES

Number of incident facets for each ray.

Type:

Array<Int>

---

SELF_DUAL

True if the cone is self-dual.

Type:

Bool

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SIMPLE

True if the facets of the cone are simple. Dual to SIMPLICIAL.

Type:

Bool

---

SIMPLICIAL

True if the facets of the cone are simplicial.

Type:

Bool

---

SIMPLICIAL_CONE

True if the cone is simplicial.

Type:

Bool

---

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

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CONE_AMBIENT_DIM

The dimension of the space in which the cone lives.

Type:

Int

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CONE_DIM

Dimension of the linear span of the cone = dimension of the cone. If the cone is given purely combinatorially, this is the dimension of a minimal embedding space deduced from the combinatorial structure.

Type:

Int

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EPSILON

Threshold for zero test for scalar products (e.g. vertex * facet normal)

Type:

Float

---

FACETS

Facets of the cone, encoded as inequalities. All vectors in this section must be non-zero. Dual to RAYS. This section is empty if and only if the cone is trivial (e.g. if it encodes an empty polytope). Notice that a polytope which is a single point defines a one-dimensional cone, the face at infinity is a facet. The property FACETS appears only in conjunction with the property LINEAR_SPAN, or AFFINE_HULL, respectively. The specification of the property FACETS requires the specification of LINEAR_SPAN, or AFFINE_HULL, respectively, and vice versa.

Type:

Matrix<Scalar,NonSymmetric>

---

FACETS_THRU_INPUT_RAYS

Transposed to INPUT_RAYS_IN_FACETS. Notice that this is a temporary property; it will not be stored in any file.

Type:

IncidenceMatrix<NonSymmetric>

---

FULL_DIM

CONE_AMBIENT_DIM and CONE_DIM coincide. Notice that this makes sense also for the derived Polytope class.

Type:

Bool

---

INEQUALITIES_THRU_RAYS

transposed RAYS_IN_INEQUALITIES Notice that this is a temporary property; it will not be stored in any file.

Type:

IncidenceMatrix<NonSymmetric>

---

INPUT_RAYS_IN_FACETS

Input ray-facet incidence matrix, with rows corresponding to facet and columns to input rays. Input_rays and facets are numbered from 0 to N_INPUT_RAYS-1 rsp. N_FACETS-1, according to their order in INPUT_RAYS rsp. FACETS.

Type:

IncidenceMatrix<NonSymmetric>

---

LINEALITY_DIM

Dimension of the LINEALITY_SPACE (>0 in the non-POINTED case)

Type:

Int

---

LINEALITY_SPACE

Basis of the linear subspace orthogonal to all INEQUALITIES and EQUATIONS All vectors in this section must be non-zero. The property LINEALITY_SPACE appears only in conjunction with the property RAYS, or VERTICES, respectively. The specification of the property RAYS or VERTICES requires the specification of LINEALITY_SPACE, and vice versa.

Type:

Matrix<Scalar,NonSymmetric>

---

LINEAR_SPAN

Dual basis of the linear span of the cone. All vectors in this section must be non-zero. The property LINEAR_SPAN appears only in conjunction with the property FACETS. The specification of the property FACETS requires the specification of LINEAR_SPAN, or AFFINE_HULL, respectively, and vice versa.

Type:

Matrix<Scalar,NonSymmetric>

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N_EQUATIONS

The number of EQUATIONS.

Type:

Int

---

N_INPUT_LINEALITY

The number of INPUT_LINEALITY.

Type:

Int

---

N_INPUT_RAYS

The number of INPUT_RAYS.

Type:

Int

---

ONE_RAY

A ray of a pointed cone.

Type:

Vector<Scalar>

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POINTED

True if the cone does not contain a non-trivial linear subspace.

Type:

Bool

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POSITIVE

True if all RAYS of the cone have non-negative coordinates, that is, if the pointed part of the cone lies entirely in the positive orthant.

Type:

Bool

---

RAYS

Rays of the cone. No redundancies are allowed. All vectors in this section must be non-zero. The property RAYS appears only in conjunction with the property LINEALITY_SPACE. The specification of the property RAYS requires the specification of LINEALITY_SPACE, and vice versa.

Type:

Matrix<Scalar,NonSymmetric>

---

RAYS_IN_INEQUALITIES

Ray-inequality incidence matrix, with rows corresponding to facets and columns to rays. Rays and inequalities are numbered from 0 to N_RAYS-1 rsp. number of INEQUALITIES-1, according to their order in RAYS rsp. INEQUALITIES.

Type:

IncidenceMatrix<NonSymmetric>

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RAY_SEPARATORS

The i-th row is the normal vector of a hyperplane separating the i-th vertex from the others. This property is a by-product of redundant point elimination algorithm.

Type:

Matrix<Scalar,NonSymmetric>

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REL_INT_POINT

A point in the relative interior of the cone.

Type:

Vector<Scalar>

---

TRIVIAL

True if the only valid point in the cone is the unique non-sensical point (0,…,0)

Type:

Bool

---

These properties capture information that depends on the lattice structure of the cone. polymake always works with the integer lattice.

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CONE_TORIC_IDEAL

The ideal whose vanishing set is the affine toric variety given by the cone. In other words the rows of its BINOMIAL_GENERATORS give the relations between the Hilbert basis elements.

Type:

Ideal

depends on extension:

4ti2

---

DEGREE_ONE_GENERATORS

Elements of the HILBERT_BASIS for the cone of degree 1 with respect to the MONOID_GRADING.

Type:

Matrix<Integer,NonSymmetric>

---

GORENSTEIN_CONE

A cone is Gorenstein if it is Q-Gorenstein with index one

Type:

Bool

---

HILBERT_BASIS_GENERATORS

Generators for the HILBERT_BASIS of a posiibly non-pointed cone the first matrix is a Hilbert basis of a pointed part of the cone the second matrix is a lattice basis of the lineality space note: the pointed part used in this property need not be the same as the one described by RAYS or INPUT_RAYS it will be if the cone is pointed (the polytope is bounded)

Type:

Array<Matrix<Integer,NonSymmetric>>

depends on extension:

4ti2 (unbounded) or libnormaliz (bounded)

---

HILBERT_SERIES

Hilbert series of the monoid, given by the intersection of the cone with the lattice Z^d with respect to the MONOID_GRADING

Type:

RationalFunction<Rational,Int>

depends on extension:

libnormaliz

---

HOMOGENEOUS

True if the primitive generators of the rays lie on an affine hyperplane in the span of the rays.

Type:

Bool

---

H_STAR_POLYNOMIAL

The Hilbert polynomial, the h^*-polynomial for lattice polytopes. The degree is at most on less then the h^*-vector.

Type:

UniPolynomial<Rational,Int>

depends on extension:

latte or libnormaliz

---

H_STAR_VECTOR

The coefficients of the Hilbert polynomial, the h^*-polynomial for lattice polytopes, with respect to the MONOID_GRADING, starting at the constant coefficient. For lattice polytopes the length of this vector is CONE_DIM. In general the length is one less than the degree of the denominator of the HILBERT_SERIES.

Type:

Vector<Integer>

depends on extension:

latte or libnormaliz

---

MONOID_GRADING

A grading for the monoid given by the intersection of the cone with the lattice Z^d, should be positive for all generators. If this property is not specified by the user there are two defaults: For rational polytopes the affine hyperplane defined by (1,0,\ldots,0) will be used. For HOMOGENEOUS cones the affine hyperplane containing the primitive generators will be used.

Type:

Vector<Integer>

---

N_HILBERT_BASIS

The number of elements of the HILBERT_BASIS.

Type:

Int

---

Q_GORENSTEIN_CONE

A cone is Q-Gorenstein if all primitive generators of the cone lie in an affine hyperplane spanned by a lattice functional in the dual cone (but not in the lineality space of the dual cone).

Type:

Bool

---

Q_GORENSTEIN_CONE_INDEX

If a cone is Q-Gorenstein, then its index is the common lattice height of the primitive generators with respect to the origin. Otherwise Q_GORENSTEIN_CONE_INDEX is undefined.

Type:

Int

---

SMOOTH_CONE

A cone is smooth if the primitive generators are part of a lattice basis.

Type:

Bool

---

These properties capture information of the object that is concerned with the action of permutation groups.

---

GROUP

Type:

Group

Methods of GROUP:

REPRESENTATIVE_INEQUALITIES()

explicit representatives of equivalence classes of INEQUALITIES under a group action

Returns:

Matrix

REPRESENTATIVE_INPUT_RAYS()

explicit representatives of equivalence classes of INPUT_RAYS under a group action

Returns:

Matrix

Properties of GROUP:

BITSET_ACTION

Type:

ImplicitActionOnSets

MATRIX_ACTION

Type:

MatrixActionOnVectors<Scalar>

Properties of MATRIX_ACTION:

RAYS_ORBITS

Alias for property VECTORS_ORBITS.

Type:

Array<Set<Int>>

REPRESENTATIVE_BOUNDARY_RIDGE_SIMPLICES

One representative for each orbit of boundary ridge simplices

Type:

Array<Bitset<Int>>

REPRESENTATIVE_FACETS

Type:

Matrix<Scalar,NonSymmetric>

REPRESENTATIVE_INTERIOR_RIDGE_SIMPLICES

One representative for each orbit of interior ridge simplices

Type:

Array<Bitset<Int>>

REPRESENTATIVE_MAX_BOUNDARY_SIMPLICES

One representative for each orbit of maximal-dimensional boundary simplices

Type:

Array<Bitset<Int>>

REPRESENTATIVE_MAX_INTERIOR_SIMPLICES

One representative for each orbit of maximal-dimensional interior simplices

Type:

Array<Bitset<Int>>

REPRESENTATIVE_RAYS

Type:

Matrix<Scalar,NonSymmetric>

---

These properties collect information about triangulations of the object and properties usually computed from such, as the volume.

---

TRIANGULATION

Type:

GeometricSimplicialComplex<Scalar>

Properties of TRIANGULATION:

BOUNDARY

derived from:

GeometricSimplicialComplex

Type:

GeometricSimplicialComplex

Properties of BOUNDARY:

FACET_TRIANGULATIONS

For each facet the set of simplex indices of BOUNDARY that triangulate it.

Type:

Array<Set<Int>>

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:

Set<Int>

WEIGHTS

Weight vector to construct a regular TRIANGULATION. Must be generic.

Type:

Vector<Scalar>

---

TRIANGULATION_INT

Conceptually, similar to TRIANGULATION, but using INPUT_RAYS. However, here we use a small object type. The main reason for the existence of this property (in this form) is the beneath_beyond algorithm, which automatically produces this data as a by-product of the conversion from INPUT_RAYS to FACETS. And that data is too valuable to throw away. Use big objects of type VectorConfiguration if you want to work with triangulations using redundant points.

Type:

Array<Set<Int>>

---

These properties are for visualization.

---

COORDINATE_LABELS

Unique names assigned to the coordinate directions, analogous to RAY_LABELS. For Polytopes this should contain “inhomog_var” for the homogenization coordinate and this will be added automatically if necessary and CONE_AMBIENT_DIM can be computed.

Type:

Array<String>

---

FACET_LABELS

Unique names assigned to the FACETS, analogous to RAY_LABELS.

Type:

Array<String>

---

FTR_CYCLIC_NORMAL

Reordered transposed RAYS_IN_FACETS. Dual to RIF_CYCLIC_NORMAL.

Type:

Array<Array<Int>>

---

INEQUALITY_LABELS

Unique names assigned to the INEQUALITIES, analogous to RAY_LABELS.

Type:

Array<String>

---

INPUT_RAY_LABELS

Unique names assigned to the INPUT_RAYS, analogous to RAY_LABELS.

Type:

Array<String>

---

LABELED_FACETS

Print RAYS_IN_FACETS using RAY_LABELS

Type:

Array<String>

---

NEIGHBOR_FACETS_CYCLIC_NORMAL

Reordered DUAL_GRAPH for 3d-cones. The neighbor facets are listed in the order corresponding to RIF_CYCLIC_NORMAL, so that the first two vertices in RIF_CYCLIC_NORMAL make up the ridge to the first neighbor facet and so on.

Type:

Array<Array<Int>>

---

NEIGHBOR_RAYS_CYCLIC_NORMAL

Reordered GRAPH. Dual to NEIGHBOR_FACETS_CYCLIC_NORMAL.

Type:

Array<Array<Int>>

---

RAY_LABELS

Unique names assigned to the RAYS. If specified, they are shown by visualization tools instead of ray indices. For a cone built from scratch, you should create this property by yourself, either manually in a text editor, or with a client program. If you build a cone with a construction client taking some other input cone(s), you can create the labels automatically if you call the client with a relabel option. The exact format of the labels is dependent on the construction, and is described by the corresponding client.

Type:

Array<String>

---

RIF_CYCLIC_NORMAL

Reordered RAYS_IN_FACETS for 2d and 3d-cones. Rays are listed in the order of their appearance when traversing the facet border counterclockwise seen from outside of the origin.

Type:

Array<Array<Int>>

---

These methods are provided for backward compatibility with older versions of polymake only. They should not be used in new code.

---

DIAMETER()

The diameter of the GRAPH of the cone

Returns:

Int

---

DUAL_DIAMETER()

The diameter of the DUAL_GRAPH

Returns:

Int

---

DUAL_TRIANGLE_FREE()

True if the DUAL_GRAPH contains no triangle

Returns:

Bool

---

TRIANGLE_FREE()

True if the GRAPH contains no triangle

Returns:

Bool

---

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

---

CONNECTIVITY()

Connectivity of the GRAPH this is the minimum number of nodes that have to be removed from the GRAPH to make it disconnected

Returns:

Int

---

DUAL_CONNECTIVITY()

Connectivity of the DUAL_GRAPH this is the minimum number of nodes that have to be removed from the DUAL_GRAPH to make it disconnected

Returns:

Int

---

DUAL_EVEN()

True if the DUAL_GRAPH is bipartite

Returns:

Bool

---

EVEN()

True if the GRAPH is bipartite

Returns:

Bool

---

FACET_DEGREES()

Facet degrees of the polytope. The degree of a facet is the number of adjacent facets.

Returns:

Vector<Int>

---

N_FLAGS(Int type …)

Determine the number of flags of a given type. type must belong to {0,…,COMBINATORIAL_DIM-1}. Example: “N_FLAGS(0,3,4)” determines the entry f₀₃₄ of the flag vector.

Parameters:

Int type …: flag type

Returns:

Int

---

VERTEX_DEGREES()

Ray degrees of the cone

Returns:

Vector<Int>

---

faces_of_dim(Int d)

Output the faces of a given dimension

Parameters:

Int d: dimension

Returns:

Array<Set<Int>>

---

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()

returns the dimension of the ambient space of the cone

Returns:

Int

---

DIM()

returns the geometric dimension of the cone (including the lineality space) for the dimension of the pointed part ask for COMBINATORIAL_DIM

Returns:

Int

---

contains(Cone<Scalar> C_in)

checks whether a given cone is containeed in another

Parameters:

Cone<Scalar> C_in

Returns:

Bool

contains(Vector<Scalar> v)

checks whether a given point is contained in a cone

Parameters:

Vector<Scalar> v: point

Returns:

Bool

---

contains_in_interior(Vector<Scalar> v)

checks whether a given point is contained in the strict interior of a cone

Parameters:

Vector<Scalar> v: point

Returns:

Bool

---

These methods capture information that depends on the lattice structure of the cone. polymake always works with the integer lattice.

---

HILBERT_BASIS()

for a cone this method returns a Hilbert basis of the cone for a polytope this method returns a Hilbert basis of the homogenization cone of the polytope note: if the cone is not pointed (the polytope is not bounded) then the returned basis is not unique and usually not minimal

Returns:

Matrix<Integer>

---

The following methods compute topological invariants.

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DUAL_GRAPH_SIGNATURE()

Difference of the black and white nodes if the DUAL_GRAPH is BIPARTITE. Otherwise -1.

Returns:

Int

---

GRAPH_SIGNATURE()

Difference of the black and white nodes if the GRAPH is BIPARTITE. Otherwise -1.

Returns:

Int

---

These methods are for visualization.

---

VISUAL()

Visualizes the cone, intersected with the unit ball.

Options:

option list Visual::Polygons::decorations

option list geometric_options_linear

Returns:

Visual::Cone

---

VISUAL_DUAL_FACE_LATTICE()

Visualize the dual face lattice of a cone as a multi-layer graph.

Options:

Int seed: random seed value for the node placement

option list Visual::Lattice::decorations

Returns:

Visual::Lattice

---

VISUAL_FACE_LATTICE()

Visualize the HASSE_DIAGRAM of a cone as a multi-layer graph.

Options:

Int seed: random seed value for the node placement

option list Visual::Lattice::decorations

Returns:

Visual::Lattice

---