====== BigObject Cone<Scalar> ======
//from application [[..:polytope|polytope]]//\\
\\
 A polyhedral cone, not necessarily pointed. Note that in contrast to the vertices of a polytope, the ''[[..:polytope:Cone#RAYS |RAYS]]'' are given in affine coordinates.

  ? Type Parameters:
  :: ''Scalar'': numeric data type used for the coordinates, must be an ordered field. Default is ''[[..:common#Rational |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<sup>d</sup>.
  :: ''Cone<Rational>'':  An affine rational cone realized in R<sup>d</sup>.
  ? Permutations:
  :
    ? FacetPerm:
    :: permuting the ''[[..:polytope:Cone#FACETS |FACETS]]''
    ? VertexPerm:
    :: permuting the ''[[..:polytope:Cone#RAYS |RAYS]]'' TODO: rename into RaysPerm and introduce an overriding alias in Polytope
===== Properties =====

==== Input property ====
 These properties are for input only. They allow redundant information.
----
{{anchor:equations:}}
  ?  **''EQUATIONS''**
  :: Equations that hold for all ''[[..:polytope:Cone#INPUT_RAYS |INPUT_RAYS]]'' of the cone. All vectors in this section must be non-zero. Input section only.  Ask for ''[[..:polytope:Cone#LINEAR_SPAN |LINEAR_SPAN]]'' if you want to see an irredundant description of the linear span.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:inequalities:}}
  ?  **''INEQUALITIES''**
  :: Inequalities giving rise to the cone; redundancies are allowed. All vectors in this section must be non-zero. Dual to ''[[..:polytope:Cone#INPUT_RAYS |INPUT_RAYS]]''. Input section only.  Ask for ''[[..:polytope:Cone#FACETS |FACETS]]'' if you want to compute an H-representation from a V-representation.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:input_lineality:}}
  ?  **''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 ''[[..:polytope:Cone#EQUATIONS |EQUATIONS]]''. Input section only.  Ask for ''[[..:polytope:Cone#LINEALITY_SPACE |LINEALITY_SPACE]]'' if you want to compute a V-representation from an H-representation.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:input_rays:}}
  ?  **''INPUT_RAYS''**
  :: (Non-homogenous) vectors whose positive span form the cone; redundancies are allowed. Dual to ''[[..:polytope:Cone#INEQUALITIES |INEQUALITIES]]''. All vectors in the input must be non-zero. Input section only.  Ask for ''[[..:polytope:Cone#RAYS |RAYS]]'' if you want to compute a V-representation from an H-representation.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''


----

==== 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.
----
{{anchor:altshuler_det:}}
  ?  **''ALTSHULER_DET''**
  :: Let M be the vertex-facet incidence matrix, then the Altshuler determinant is defined as max{det(M ∗ M<sup>T</sup>), det(M<sup>T</sup> ∗ M)}.
    ? Type:
    :''[[..:common#Integer |Integer]]''
    ? Example:
    :: This prints the Altshuler determinant of the built-in pentagonal pyramid (Johnson solid 2):
    :: <code perl> > print johnson_solid("pentagonal_pyramid")->ALTSHULER_DET;
 25
</code>


----
{{anchor:cocircuit_equations:}}
  ?  **''COCIRCUIT_EQUATIONS''**
  :: A matrix whose rows contain the cocircuit equations of P. The columns correspond to the ''[[..:polytope:Cone#MAX_INTERIOR_SIMPLICES |MAX_INTERIOR_SIMPLICES]]''.
    ? Type:
    :''[[..:common#SparseMatrix |SparseMatrix]]<[[..:common#Rational |Rational]],[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:combinatorial_dim:}}
  ?  **''COMBINATORIAL_DIM''**
  :: Combinatorial dimension This is the dimension all combinatorial properties of the cone like e.g. ''[[..:polytope:Cone#RAYS_IN_FACETS |RAYS_IN_FACETS]]'' or the ''[[..:polytope:Cone#HASSE_DIAGRAM |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:
    :''[[..:common#Int |Int]]''


----
{{anchor:dual_graph:}}
  ?  **''DUAL_GRAPH''**
  :: Facet-ridge graph. Dual to ''[[..:polytope:Cone#GRAPH |GRAPH]]''.
    ? Type:
    :''[[..:graph:Graph |Graph]]<[[..:common#Undirected |Undirected]]>''


----
{{anchor:essentially_generic:}}
  ?  **''ESSENTIALLY_GENERIC''**
  :: All intermediate polytopes (with respect to the given insertion order) in the beneath-and-beyond algorithm are simplicial. We have the implications: ''[[..:polytope:Cone#RAYS |RAYS]]'' in general position => ESSENTIALLY_GENERIC => ''[[..:polytope:Cone#SIMPLICIAL |SIMPLICIAL]]''
    ? Type:
    :''[[..:common#Bool |Bool]]''


----
{{anchor:excess_facet_degree:}}
  ?  **''EXCESS_FACET_DEGREE''**
  :: Measures the deviation of the cone from being simple in terms of the ''[[..:polytope:Cone#DUAL_GRAPH |DUAL_GRAPH]]''.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:excess_ray_degree:}}
  ?  **''EXCESS_RAY_DEGREE''**
  :: Measures the deviation of the cone from being simple in terms of the ''[[..:polytope:Cone#GRAPH |GRAPH]]''.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:f2_vector:}}
  ?  **''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 ''[[..:polytope:Cone#F_VECTOR |F_VECTOR]]''.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<[[..:common#Integer |Integer]],[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:facets_thru_rays:}}
  ?  **''FACETS_THRU_RAYS''**
  :: Transposed to ''[[..:polytope:Cone#RAYS_IN_FACETS |RAYS_IN_FACETS]]''. Notice that this is a temporary property; it will not be stored in any file.
    ? Type:
    :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:facet_sizes:}}
  ?  **''FACET_SIZES''**
  :: Number of incident rays for each facet.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Int |Int]]>''


----
{{anchor:flag_vector:}}
  ?  **''FLAG_VECTOR''**
  :: Condensed form of the flag vector, containing all entries indexed by sparse sets in {0, ..., ''[[..:polytope:Cone#COMBINATORIAL_DIM |COMBINATORIAL_DIM]]''-1} in the following order:        (1, f<sub>0</sub>, f<sub>1</sub>, f<sub>2</sub>, f<sub>02</sub>, f<sub>3</sub>, f<sub>03</sub>, f<sub>13</sub>, f<sub>4</sub>, f<sub>04</sub>, f<sub>14</sub>, f<sub>24</sub>, f<sub>024</sub>, f<sub>5</sub>, ...). Use Dehn-Sommerville equations, via user function ''[[..:polytope:Cone#N_FLAGS |N_FLAGS]]'', to extend.
    ? Type:
    :''[[..:common#Vector |Vector]]<[[..:common#Integer |Integer]]>''


----
{{anchor:foldable_cocircuit_equations:}}
  ?  **''FOLDABLE_COCIRCUIT_EQUATIONS''**
  :: A matrix whose rows contain the foldable cocircuit equations of P.  The columns correspond to 2 * ''[[..:polytope:Cone#MAX_INTERIOR_SIMPLICES |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:
    :''[[..:common#SparseMatrix |SparseMatrix]]<[[..:common#Rational |Rational]],[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:f_vector:}}
  ?  **''F_VECTOR''**
  :: The vector counting the number of faces (`f<sub>k</sub>` is the number of `(k+1)`-faces).
    ? Type:
    :''[[..:common#Vector |Vector]]<[[..:common#Integer |Integer]]>''


----
{{anchor:graph:}}
  ?  **''GRAPH''**
  :: Vertex-edge graph obtained by intersecting the cone with a transversal hyperplane.
    ? Type:
    :''[[..:graph:Graph |Graph]]<[[..:common#Undirected |Undirected]]>''


----
{{anchor:hasse_diagram:}}
  ?  **''HASSE_DIAGRAM''**
  ::
    ? Type:
    :''[[..:graph:Lattice |Lattice]]<[[..:graph#BasicDecoration |BasicDecoration]],[[..:graph#Sequential |Sequential]]>''
    ? Methods of HASSE_DIAGRAM:
    :
      ?  **''dim()''**
      :: The dimension of the underlying object
        ? Returns:
        :''[[..:common#Int |Int]]''
      ?  **''nodes_of_dim([[..:common#Int |Int]] d)''**
      :: The indices of nodes in the ''[[..:polytope:Cone#HASSE_DIAGRAM |HASSE_DIAGRAM]]'' corresponding to faces of dimension d in the underlying object
        ? Parameters:
        :: ''[[..:common#Int |Int]]'' ''d'': dimension
        ? Returns:
        :''[[..:common#Set |Set]]<[[..:common#Int |Int]]>''
      ?  **''nodes_of_dim_range([[..:common#Int |Int]] d1, [[..:common#Int |Int]] d2)''**
      :: The indices of nodes in the ''[[..:polytope:Cone#HASSE_DIAGRAM |HASSE_DIAGRAM]]'' corresponding to faces with dimension in the range (d1,d2) in the underlying object
        ? Parameters:
        :: ''[[..:common#Int |Int]]'' ''d1'': lower dimension of the range
        :: ''[[..:common#Int |Int]]'' ''d2'': upper dimension of the range
        ? Returns:
        :''[[..:common#Set |Set]]<[[..:common#Int |Int]]>''


----
{{anchor:interior_ridge_simplices:}}
  ?  **''INTERIOR_RIDGE_SIMPLICES''**
  :: The (//d//-1)-dimensional simplices in the interior.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''


----
{{anchor:max_boundary_simplices:}}
  ?  **''MAX_BOUNDARY_SIMPLICES''**
  :: The boundary (//d//-1)-dimensional simplices of a cone of combinatorial dimension //d//
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''


----
{{anchor:max_interior_simplices:}}
  ?  **''MAX_INTERIOR_SIMPLICES''**
  :: The interior //d//-dimensional simplices of a cone of combinatorial dimension //d//
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''


----
{{anchor:n_edges:}}
  ?  **''N_EDGES''**
  :: The number of edges of the ''[[..:polytope:Cone#GRAPH |GRAPH]]''
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:n_facets:}}
  ?  **''N_FACETS''**
  :: The number of ''[[..:polytope:Cone#FACETS |FACETS]]''.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:n_rays:}}
  ?  **''N_RAYS''**
  :: The number of ''[[..:polytope:Cone#RAYS |RAYS]]''
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:n_ray_facet_inc:}}
  ?  **''N_RAY_FACET_INC''**
  :: Number of pairs of incident vertices and facets.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:n_ridges:}}
  ?  **''N_RIDGES''**
  :: The number of ridges (faces of codimension 2) equals the number of edges of the ''[[..:polytope:Cone#DUAL_GRAPH |DUAL_GRAPH]]''
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:rays_in_facets:}}
  ?  **''RAYS_IN_FACETS''**
  :: Ray-facet incidence matrix, with rows corresponding to facets and columns to rays. Rays and facets are numbered from 0 to ''[[..:polytope:Cone#N_RAYS |N_RAYS]]''-1 rsp. ''[[..:polytope:Cone#N_FACETS |N_FACETS]]''-1, according to their order in ''[[..:polytope:Cone#RAYS |RAYS]]'' rsp. ''[[..:polytope:Cone#FACETS |FACETS]]''.
    ? Type:
    :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:rays_in_ridges:}}
  ?  **''RAYS_IN_RIDGES''**
  :: Ray-ridge incidence matrix, with rows corresponding to ridges and columns to rays. Rays and ridges are numbered from 0 to ''[[..:polytope:Cone#N_RAYS |N_RAYS]]''-1 rsp. ''[[..:polytope:Cone#N_RIDGES |N_RIDGES]]''-1, according to their order in ''[[..:polytope:Cone#RAYS |RAYS]]'' rsp. ''[[..:polytope:SymmetrizedCocircuitEquations#RIDGES |RIDGES]]''.
    ? Type:
    :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:ray_sizes:}}
  ?  **''RAY_SIZES''**
  :: Number of incident facets for each ray.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Int |Int]]>''


----
{{anchor:self_dual:}}
  ?  **''SELF_DUAL''**
  :: True if the cone is self-dual.
    ? Type:
    :''[[..:common#Bool |Bool]]''


----
{{anchor:simple:}}
  ?  **''SIMPLE''**
  :: True if the facets of the cone are simple. Dual to ''[[..:polytope:Cone#SIMPLICIAL |SIMPLICIAL]]''.
    ? Type:
    :''[[..:common#Bool |Bool]]''


----
{{anchor:simplicial:}}
  ?  **''SIMPLICIAL''**
  :: True if the facets of the cone are simplicial.
    ? Type:
    :''[[..:common#Bool |Bool]]''


----
{{anchor:simplicial_cone:}}
  ?  **''SIMPLICIAL_CONE''**
  :: True if the cone is simplicial.
    ? Type:
    :''[[..:common#Bool |Bool]]''


----

==== Geometry ====
 These properties capture geometric information of the object.  Geometric properties depend on geometric information of the object, like, e.g., vertices or facets.
----
{{anchor:cone_ambient_dim:}}
  ?  **''CONE_AMBIENT_DIM''**
  :: The dimension of the space in which the cone lives.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:cone_dim:}}
  ?  **''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:
    :''[[..:common#Int |Int]]''


----
{{anchor:epsilon:}}
  ?  **''EPSILON''**
  :: Threshold for zero test for scalar products (e.g. vertex * facet normal)
    ? Type:
    :''[[..:common#Float |Float]]''


----
{{anchor:facets:}}
  ?  **''FACETS''**
  :: Facets of the cone, encoded as inequalities. All vectors in this section must be non-zero. Dual to ''[[..:polytope:Cone#RAYS |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 ''[[..:polytope:Cone#FACETS |FACETS]]'' appears only in conjunction with the property ''[[..:polytope:Cone#LINEAR_SPAN |LINEAR_SPAN]]'', or ''[[..:polytope:Polytope#AFFINE_HULL |AFFINE_HULL]]'', respectively. The specification of the property ''[[..:polytope:Cone#FACETS |FACETS]]'' requires the specification of ''[[..:polytope:Cone#LINEAR_SPAN |LINEAR_SPAN]]'', or ''[[..:polytope:Polytope#AFFINE_HULL |AFFINE_HULL]]'', respectively, and vice versa.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:facets_thru_input_rays:}}
  ?  **''FACETS_THRU_INPUT_RAYS''**
  :: Transposed to ''[[..:polytope:Cone#INPUT_RAYS_IN_FACETS |INPUT_RAYS_IN_FACETS]]''. Notice that this is a temporary property; it will not be stored in any file.
    ? Type:
    :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:full_dim:}}
  ?  **''FULL_DIM''**
  :: ''[[..:polytope:Cone#CONE_AMBIENT_DIM |CONE_AMBIENT_DIM]]'' and ''[[..:polytope:Cone#CONE_DIM |CONE_DIM]]'' coincide.  Notice that this makes sense also for the derived Polytope class.
    ? Type:
    :''[[..:common#Bool |Bool]]''


----
{{anchor:inequalities_thru_rays:}}
  ?  **''INEQUALITIES_THRU_RAYS''**
  :: transposed ''[[..:polytope:Cone#RAYS_IN_INEQUALITIES |RAYS_IN_INEQUALITIES]]'' Notice that this is a temporary property; it will not be stored in any file.
    ? Type:
    :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:input_rays_in_facets:}}
  ?  **''INPUT_RAYS_IN_FACETS''**
  :: [[..:polytope:Cone#INPUT_RAYS |Input ray]]-[[..:polytope:Cone#FACETS |facet]] incidence matrix, with rows corresponding to [[..:polytope:Cone#FACETS |facet]] and columns to [[..:polytope:Cone#INPUT_RAYS |input rays]]. Input_rays and facets are numbered from 0 to ''[[..:polytope:Cone#N_INPUT_RAYS |N_INPUT_RAYS]]''-1 rsp. ''[[..:polytope:Cone#N_FACETS |N_FACETS]]''-1, according to their order in ''[[..:polytope:Cone#INPUT_RAYS |INPUT_RAYS]]'' rsp. ''[[..:polytope:Cone#FACETS |FACETS]]''.
    ? Type:
    :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:lineality_dim:}}
  ?  **''LINEALITY_DIM''**
  :: Dimension of the ''[[..:polytope:Cone#LINEALITY_SPACE |LINEALITY_SPACE]]'' (>0 in the non-POINTED case)
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:lineality_space:}}
  ?  **''LINEALITY_SPACE''**
  :: Basis of the linear subspace orthogonal to all ''[[..:polytope:Cone#INEQUALITIES |INEQUALITIES]]'' and ''[[..:polytope:Cone#EQUATIONS |EQUATIONS]]'' All vectors in this section must be non-zero. The property ''[[..:polytope:Cone#LINEALITY_SPACE |LINEALITY_SPACE]]'' appears only in conjunction with the property ''[[..:polytope:Cone#RAYS |RAYS]]'', or ''[[..:polytope:SchlegelDiagram#VERTICES |VERTICES]]'', respectively. The specification of the property ''[[..:polytope:Cone#RAYS |RAYS]]'' or ''[[..:polytope:SchlegelDiagram#VERTICES |VERTICES]]'' requires the specification of ''[[..:polytope:Cone#LINEALITY_SPACE |LINEALITY_SPACE]]'', and vice versa.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:linear_span:}}
  ?  **''LINEAR_SPAN''**
  :: Dual basis of the linear span of the cone. All vectors in this section must be non-zero. The property ''[[..:polytope:Cone#LINEAR_SPAN |LINEAR_SPAN]]'' appears only in conjunction with the property ''[[..:polytope:Cone#FACETS |FACETS]]''. The specification of the property ''[[..:polytope:Cone#FACETS |FACETS]]'' requires the specification of ''[[..:polytope:Cone#LINEAR_SPAN |LINEAR_SPAN]]'', or ''[[..:polytope:Polytope#AFFINE_HULL |AFFINE_HULL]]'', respectively, and vice versa.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:n_equations:}}
  ?  **''N_EQUATIONS''**
  :: The number of ''[[..:polytope:Cone#EQUATIONS |EQUATIONS]]''.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:n_input_lineality:}}
  ?  **''N_INPUT_LINEALITY''**
  :: The number of ''[[..:polytope:Cone#INPUT_LINEALITY |INPUT_LINEALITY]]''.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:n_input_rays:}}
  ?  **''N_INPUT_RAYS''**
  :: The number of ''[[..:polytope:Cone#INPUT_RAYS |INPUT_RAYS]]''.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:one_ray:}}
  ?  **''ONE_RAY''**
  :: A ray of a pointed cone.
    ? Type:
    :''[[..:common#Vector |Vector]]<Scalar>''


----
{{anchor:pointed:}}
  ?  **''POINTED''**
  :: True if the cone does not contain a non-trivial linear subspace.
    ? Type:
    :''[[..:common#Bool |Bool]]''


----
{{anchor:positive:}}
  ?  **''POSITIVE''**
  :: True if all ''[[..:polytope:Cone#RAYS |RAYS]]'' of the cone have non-negative coordinates, that is, if the pointed part of the cone lies entirely in the positive orthant.
    ? Type:
    :''[[..:common#Bool |Bool]]''


----
{{anchor:rays:}}
  ?  **''RAYS''**
  :: Rays of the cone. No redundancies are allowed. All vectors in this section must be non-zero. The property ''[[..:polytope:Cone#RAYS |RAYS]]'' appears only in conjunction with the property ''[[..:polytope:Cone#LINEALITY_SPACE |LINEALITY_SPACE]]''. The specification of the property ''[[..:polytope:Cone#RAYS |RAYS]]'' requires the specification of ''[[..:polytope:Cone#LINEALITY_SPACE |LINEALITY_SPACE]]'', and vice versa.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:rays_in_inequalities:}}
  ?  **''RAYS_IN_INEQUALITIES''**
  :: Ray-inequality incidence matrix, with rows corresponding to facets and columns to rays. Rays and inequalities are numbered from 0 to ''[[..:polytope:Cone#N_RAYS |N_RAYS]]''-1 rsp. number of ''[[..:polytope:Cone#INEQUALITIES |INEQUALITIES]]''-1, according to their order in ''[[..:polytope:Cone#RAYS |RAYS]]'' rsp. ''[[..:polytope:Cone#INEQUALITIES |INEQUALITIES]]''.
    ? Type:
    :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:ray_separators:}}
  ?  **''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:
    :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:rel_int_point:}}
  ?  **''REL_INT_POINT''**
  :: A point in the relative interior of the cone.
    ? Type:
    :''[[..:common#Vector |Vector]]<Scalar>''


----
{{anchor:trivial:}}
  ?  **''TRIVIAL''**
  :: True if the only valid point in the cone is the unique non-sensical point (0,...,0)
    ? Type:
    :''[[..:common#Bool |Bool]]''


----

==== Lattice points in cones ====
 These properties capture information that depends on the lattice structure of the cone. polymake always works with the integer lattice. 
----
{{anchor:cone_toric_ideal:}}
  ?  **''CONE_TORIC_IDEAL''**
  :: The ideal whose vanishing set is the affine toric variety given by the cone. In other words the rows of its ''[[..:ideal:Ideal#BINOMIAL_GENERATORS |BINOMIAL_GENERATORS]]'' give the relations between the Hilbert basis elements.
    ? Type:
    :''[[..:ideal:Ideal |Ideal]]''
    ? depends on extension:
    : [[:external_software|4ti2]]


----
{{anchor:degree_one_generators:}}
  ?  **''DEGREE_ONE_GENERATORS''**
  :: Elements of the ''[[..:polytope:Cone#HILBERT_BASIS |HILBERT_BASIS]]'' for the cone of degree 1 with respect to the ''[[..:polytope:Cone#MONOID_GRADING |MONOID_GRADING]]''.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<[[..:common#Integer |Integer]],[[..:common#NonSymmetric |NonSymmetric]]>''


----
{{anchor:gorenstein_cone:}}
  ?  **''GORENSTEIN_CONE''**
  :: A cone is [[..:polytope:Cone#GORENSTEIN_CONE |Gorenstein]] if it is [[..:polytope:Cone#Q_GORENSTEIN_CONE |Q-Gorenstein]] with index one
    ? Type:
    :''[[..:common#Bool |Bool]]''


----
{{anchor:hilbert_basis_generators:}}
  ?  **''HILBERT_BASIS_GENERATORS''**
  :: Generators for the ''[[..:polytope:Cone#HILBERT_BASIS |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 ''[[..:polytope:Cone#RAYS |RAYS]]'' or ''[[..:polytope:Cone#INPUT_RAYS |INPUT_RAYS]]''       it will be if the cone is pointed (the polytope is bounded)
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Matrix |Matrix]]<[[..:common#Integer |Integer]],[[..:common#NonSymmetric |NonSymmetric]]%%>>%%''
    ? depends on extension:
    : [[:external_software|4ti2 (unbounded) or libnormaliz (bounded)]]


----
{{anchor:hilbert_series:}}
  ?  **''HILBERT_SERIES''**
  :: Hilbert series of the monoid, given by the intersection of the cone with the lattice Z^d with respect to the ''[[..:polytope:Cone#MONOID_GRADING |MONOID_GRADING]]''
    ? Type:
    :''[[..:common#RationalFunction |RationalFunction]]<[[..:common#Rational |Rational]],[[..:common#Int |Int]]>''
    ? depends on extension:
    : [[:external_software|libnormaliz]]


----
{{anchor:homogeneous:}}
  ?  **''HOMOGENEOUS''**
  :: True if the primitive generators of the rays lie on an affine hyperplane in the span of the rays.
    ? Type:
    :''[[..:common#Bool |Bool]]''


----
{{anchor:h_star_polynomial:}}
  ?  **''H_STAR_POLYNOMIAL''**
  :: The Hilbert polynomial, the h^*-polynomial for lattice polytopes. The degree is at most on less then the h^*-vector.
    ? Type:
    :''[[..:common#UniPolynomial |UniPolynomial]]<[[..:common#Rational |Rational]],[[..:common#Int |Int]]>''
    ? depends on extension:
    : [[:external_software|latte or libnormaliz]]


----
{{anchor:h_star_vector:}}
  ?  **''H_STAR_VECTOR''**
  :: The coefficients of the Hilbert polynomial, the h^*-polynomial for lattice polytopes, with respect to the ''[[..:polytope:Cone#MONOID_GRADING |MONOID_GRADING]]'', starting at the constant coefficient. For lattice polytopes the length of this vector is ''[[..:polytope:Cone#CONE_DIM |CONE_DIM]]''. In general the length is one less than the degree of the denominator of the ''[[..:polytope:Cone#HILBERT_SERIES |HILBERT_SERIES]]''.
    ? Type:
    :''[[..:common#Vector |Vector]]<[[..:common#Integer |Integer]]>''
    ? depends on extension:
    : [[:external_software|latte or libnormaliz]]


----
{{anchor:monoid_grading:}}
  ?  **''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 ''[[..:polytope:Cone#HOMOGENEOUS |HOMOGENEOUS]]'' cones the affine hyperplane containing the primitive generators     will be used.
    ? Type:
    :''[[..:common#Vector |Vector]]<[[..:common#Integer |Integer]]>''


----
{{anchor:n_hilbert_basis:}}
  ?  **''N_HILBERT_BASIS''**
  :: The number of elements of the ''[[..:polytope:Cone#HILBERT_BASIS |HILBERT_BASIS]]''.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:q_gorenstein_cone:}}
  ?  **''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:
    :''[[..:common#Bool |Bool]]''


----
{{anchor:q_gorenstein_cone_index:}}
  ?  **''Q_GORENSTEIN_CONE_INDEX''**
  :: If a cone is [[..:polytope:Cone#Q_GORENSTEIN_CONE |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:
    :''[[..:common#Int |Int]]''


----
{{anchor:smooth_cone:}}
  ?  **''SMOOTH_CONE''**
  :: A cone is __smooth__ if the primitive generators are part of a lattice basis.
    ? Type:
    :''[[..:common#Bool |Bool]]''


----

==== Symmetry ====
 These properties capture information of the object that is concerned with the action of permutation groups.
----
{{anchor:group:}}
  ?  **''GROUP''**
  ::
    ? Type:
    :''[[..:group:Group |Group]]''
    ? Methods of GROUP:
    :
      ?  **''REPRESENTATIVE_INEQUALITIES()''**
      :: explicit representatives of equivalence classes of [[..:polytope:Cone#INEQUALITIES |INEQUALITIES]] under a group action
        ? Returns:
        :''[[..:common#Matrix |Matrix]]''
      ?  **''REPRESENTATIVE_INPUT_RAYS()''**
      :: explicit representatives of equivalence classes of ''[[..:polytope:Cone#INPUT_RAYS |INPUT_RAYS]]'' under a group action
        ? Returns:
        :''[[..:common#Matrix |Matrix]]''
    ? Properties of GROUP:
    :
      ?  **''BITSET_ACTION''**
      ::
        ? Type:
        :''[[..:group:ImplicitActionOnSets |ImplicitActionOnSets]]''
      ?  **''MATRIX_ACTION''**
      ::
        ? Type:
        :''[[..:group:MatrixActionOnVectors |MatrixActionOnVectors]]<Scalar>''
        ? Properties of MATRIX_ACTION:
        :
          ?  **''RAYS_ORBITS''**
          :: Alias for property ''[[..:group:MatrixActionOnVectors#VECTORS_ORBITS |VECTORS_ORBITS]]''.
            ? Type:
            :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''
      ?  **''REPRESENTATIVE_BOUNDARY_RIDGE_SIMPLICES''**
      :: One representative for each orbit of boundary ridge simplices
        ? Type:
        :''[[..:common#Array |Array]]<[[..:common#Bitset |Bitset]]<[[..:common#Int |Int]]%%>>%%''
      ?  **''REPRESENTATIVE_FACETS''**
      ::
        ? Type:
        :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''
      ?  **''REPRESENTATIVE_INTERIOR_RIDGE_SIMPLICES''**
      :: One representative for each orbit of interior ridge simplices
        ? Type:
        :''[[..:common#Array |Array]]<[[..:common#Bitset |Bitset]]<[[..:common#Int |Int]]%%>>%%''
      ?  **''REPRESENTATIVE_MAX_BOUNDARY_SIMPLICES''**
      :: One representative for each orbit of maximal-dimensional boundary simplices
        ? Type:
        :''[[..:common#Array |Array]]<[[..:common#Bitset |Bitset]]<[[..:common#Int |Int]]%%>>%%''
      ?  **''REPRESENTATIVE_MAX_INTERIOR_SIMPLICES''**
      :: One representative for each orbit of maximal-dimensional interior simplices
        ? Type:
        :''[[..:common#Array |Array]]<[[..:common#Bitset |Bitset]]<[[..:common#Int |Int]]%%>>%%''
      ?  **''REPRESENTATIVE_RAYS''**
      ::
        ? Type:
        :''[[..:common#Matrix |Matrix]]<Scalar,[[..:common#NonSymmetric |NonSymmetric]]>''


----

==== Triangulation and volume ====
 These properties collect information about triangulations of the object and properties usually computed from such, as the volume.
----
{{anchor:triangulation:}}
  ?  **''TRIANGULATION''**
  ::
    ? Type:
    :''[[..:topaz:GeometricSimplicialComplex |GeometricSimplicialComplex]]<Scalar>''
    ? Properties of TRIANGULATION:
    :
      ?  **''BOUNDARY''**
      ::
        ? derived from:
        : ''[[..:topaz:GeometricSimplicialComplex |GeometricSimplicialComplex]]''
        ? Type:
        :''[[..:topaz:GeometricSimplicialComplex |GeometricSimplicialComplex]]''
        ? Properties of BOUNDARY:
        :
          ?  **''FACET_TRIANGULATIONS''**
          :: For each facet the set of simplex indices of ''[[..:polytope:Cone#BOUNDARY |BOUNDARY]]'' that triangulate it.
            ? Type:
            :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''
      ?  **''REFINED_SPLITS''**
      :: The splits that are coarsenings of the current ''[[..:polytope:Cone#TRIANGULATION |TRIANGULATION]]''. If the triangulation is regular these form the unique split decomposition of the corresponding weight function.
        ? Type:
        :''[[..:common#Set |Set]]<[[..:common#Int |Int]]>''
      ?  **''WEIGHTS''**
      :: Weight vector to construct a regular ''[[..:polytope:Cone#TRIANGULATION |TRIANGULATION]]''. Must be generic.
        ? Type:
        :''[[..:common#Vector |Vector]]<Scalar>''


----
{{anchor:triangulation_int:}}
  ?  **''TRIANGULATION_INT''**
  :: Conceptually, similar to ''[[..:polytope:Cone#TRIANGULATION |TRIANGULATION]]'', but using ''[[..:polytope:Cone#INPUT_RAYS |INPUT_RAYS]]''. However, here we use a small object type.  The main reason for the existence of this property (in this form) is the [[/polytope/preferences/Convex hull computation/beneath_beyond]] algorithm, which automatically produces this data as a by-product of the conversion from ''[[..:polytope:Cone#INPUT_RAYS |INPUT_RAYS]]'' to ''[[..:polytope:Cone#FACETS |FACETS]]''.  And that data is too valuable to throw away.  Use big objects of type ''[[..:polytope:VectorConfiguration |VectorConfiguration]]'' if you want to work with triangulations using redundant points.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''


----

==== Visualization ====
 These properties are for visualization.
----
{{anchor:coordinate_labels:}}
  ?  **''COORDINATE_LABELS''**
  :: Unique names assigned to the coordinate directions, analogous to ''[[..:polytope:Cone#RAY_LABELS |RAY_LABELS]]''. For Polytopes this should contain "inhomog_var" for the homogenization coordinate and this will be added automatically if necessary and ''[[..:polytope:Cone#CONE_AMBIENT_DIM |CONE_AMBIENT_DIM]]'' can be computed.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#String |String]]>''


----
{{anchor:facet_labels:}}
  ?  **''FACET_LABELS''**
  :: Unique names assigned to the ''[[..:polytope:Cone#FACETS |FACETS]]'', analogous to ''[[..:polytope:Cone#RAY_LABELS |RAY_LABELS]]''.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#String |String]]>''


----
{{anchor:ftr_cyclic_normal:}}
  ?  **''FTR_CYCLIC_NORMAL''**
  :: Reordered transposed ''[[..:polytope:Cone#RAYS_IN_FACETS |RAYS_IN_FACETS]]''. Dual to ''[[..:polytope:Cone#RIF_CYCLIC_NORMAL |RIF_CYCLIC_NORMAL]]''.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Array |Array]]<[[..:common#Int |Int]]%%>>%%''


----
{{anchor:inequality_labels:}}
  ?  **''INEQUALITY_LABELS''**
  :: Unique names assigned to the ''[[..:polytope:Cone#INEQUALITIES |INEQUALITIES]]'', analogous to ''[[..:polytope:Cone#RAY_LABELS |RAY_LABELS]]''.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#String |String]]>''


----
{{anchor:input_ray_labels:}}
  ?  **''INPUT_RAY_LABELS''**
  :: Unique names assigned to the ''[[..:polytope:Cone#INPUT_RAYS |INPUT_RAYS]]'', analogous to ''[[..:polytope:Cone#RAY_LABELS |RAY_LABELS]]''.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#String |String]]>''


----
{{anchor:labeled_facets:}}
  ?  **''LABELED_FACETS''**
  :: Print ''[[..:polytope:Cone#RAYS_IN_FACETS |RAYS_IN_FACETS]]'' using ''[[..:polytope:Cone#RAY_LABELS |RAY_LABELS]]''
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#String |String]]>''


----
{{anchor:neighbor_facets_cyclic_normal:}}
  ?  **''NEIGHBOR_FACETS_CYCLIC_NORMAL''**
  :: Reordered ''[[..:polytope:Cone#DUAL_GRAPH |DUAL_GRAPH]]'' for 3d-cones. The neighbor facets are listed in the order corresponding to ''[[..:polytope:Cone#RIF_CYCLIC_NORMAL |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:
    :''[[..:common#Array |Array]]<[[..:common#Array |Array]]<[[..:common#Int |Int]]%%>>%%''


----
{{anchor:neighbor_rays_cyclic_normal:}}
  ?  **''NEIGHBOR_RAYS_CYCLIC_NORMAL''**
  :: Reordered ''[[..:polytope:Cone#GRAPH |GRAPH]]''. Dual to ''[[..:polytope:Cone#NEIGHBOR_FACETS_CYCLIC_NORMAL |NEIGHBOR_FACETS_CYCLIC_NORMAL]]''.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Array |Array]]<[[..:common#Int |Int]]%%>>%%''


----
{{anchor:ray_labels:}}
  ?  **''RAY_LABELS''**
  :: Unique names assigned to the ''[[..:polytope:Cone#RAYS |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:
    :''[[..:common#Array |Array]]<[[..:common#String |String]]>''


----
{{anchor:rif_cyclic_normal:}}
  ?  **''RIF_CYCLIC_NORMAL''**
  :: Reordered ''[[..:polytope:Cone#RAYS_IN_FACETS |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:
    :''[[..:common#Array |Array]]<[[..:common#Array |Array]]<[[..:common#Int |Int]]%%>>%%''


----
===== Methods =====

==== Backward compatibility ====
 These methods are provided for backward compatibility with older versions of polymake only. They should not be used in new code.
----
{{anchor:diameter:}}
  ?  **''DIAMETER()''**
  :: The diameter of the ''[[..:polytope:Cone#GRAPH |GRAPH]]'' of the cone
    ? Returns:
    :''[[..:common#Int |Int]]''


----
{{anchor:dual_diameter:}}
  ?  **''DUAL_DIAMETER()''**
  :: The diameter of the ''[[..:polytope:Cone#DUAL_GRAPH |DUAL_GRAPH]]''
    ? Returns:
    :''[[..:common#Int |Int]]''


----
{{anchor:dual_triangle_free:}}
  ?  **''DUAL_TRIANGLE_FREE()''**
  :: True if the ''[[..:polytope:Cone#DUAL_GRAPH |DUAL_GRAPH]]'' contains no triangle
    ? Returns:
    :''[[..:common#Bool |Bool]]''


----
{{anchor:triangle_free:}}
  ?  **''TRIANGLE_FREE()''**
  :: True if the ''[[..:polytope:Cone#GRAPH |GRAPH]]'' contains no triangle
    ? Returns:
    :''[[..:common#Bool |Bool]]''


----

==== 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.
----
{{anchor:connectivity:}}
  ?  **''CONNECTIVITY()''**
  :: Connectivity of the ''[[..:polytope:Cone#GRAPH |GRAPH]]'' this is the minimum number of nodes that have to be removed from the ''[[..:polytope:Cone#GRAPH |GRAPH]]'' to make it disconnected
    ? Returns:
    :''[[..:common#Int |Int]]''


----
{{anchor:dual_connectivity:}}
  ?  **''DUAL_CONNECTIVITY()''**
  :: Connectivity of the ''[[..:polytope:Cone#DUAL_GRAPH |DUAL_GRAPH]]'' this is the minimum number of nodes that have to be removed from the ''[[..:polytope:Cone#DUAL_GRAPH |DUAL_GRAPH]]'' to make it disconnected
    ? Returns:
    :''[[..:common#Int |Int]]''


----
{{anchor:dual_even:}}
  ?  **''DUAL_EVEN()''**
  :: True if the ''[[..:polytope:Cone#DUAL_GRAPH |DUAL_GRAPH]]'' is bipartite
    ? Returns:
    :''[[..:common#Bool |Bool]]''


----
{{anchor:even:}}
  ?  **''EVEN()''**
  :: True if the ''[[..:polytope:Cone#GRAPH |GRAPH]]'' is bipartite
    ? Returns:
    :''[[..:common#Bool |Bool]]''


----
{{anchor:facet_degrees:}}
  ?  **''FACET_DEGREES()''**
  :: Facet degrees of the polytope. The __degree__ of a facet is the number of adjacent facets.
    ? Returns:
    :''[[..:common#Vector |Vector]]<[[..:common#Int |Int]]>''


----
{{anchor:n_flags:}}
  ?  **''N_FLAGS([[..:common#Int |Int]] type ...)''**
  :: Determine the number of flags of a given type. //type// must belong to {0,...,''[[..:polytope:Cone#COMBINATORIAL_DIM |COMBINATORIAL_DIM]]''-1}. Example: "N_FLAGS(0,3,4)" determines the entry f<sub>034</sub> of the flag vector.
    ? Parameters:
    :: ''[[..:common#Int |Int]]'' ''type ...'': flag type
    ? Returns:
    :''[[..:common#Int |Int]]''


----
{{anchor:vertex_degrees:}}
  ?  **''VERTEX_DEGREES()''**
  :: Ray degrees of the cone
    ? Returns:
    :''[[..:common#Vector |Vector]]<[[..:common#Int |Int]]>''


----
{{anchor:faces_of_dim:}}
  ?  **''faces_of_dim([[..:polytope:Cone |Cone]] c)''**
  :: Output the faces of a given dimension
    ? Parameters:
    :: ''[[..:polytope:Cone |Cone]]'' ''c'': the input cone
    ? Returns:
    :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''


----

==== Geometry ====
 These methods capture geometric information of the object.  Geometric properties depend on geometric information of the object, like, e.g., vertices or facets.
----
{{anchor:ambient_dim:}}
  ?  **''AMBIENT_DIM()''**
  :: returns the dimension of the ambient space of the cone
    ? Returns:
    :''[[..:common#Int |Int]]''


----
{{anchor:dim:}}
  ?  **''DIM()''**
  :: returns the geometric dimension of the cone (including the lineality space) for the dimension of the pointed part ask for ''[[..:polytope:Cone#COMBINATORIAL_DIM |COMBINATORIAL_DIM]]''
    ? Returns:
    :''[[..:common#Int |Int]]''


----
{{anchor:contains:}}
  ?  **''contains([[..:polytope:Cone |Cone]]<Scalar> C_in)''**
  :: checks whether a given cone is containeed in another
    ? Parameters:
    :: ''[[..:polytope:Cone |Cone]]<Scalar>'' ''C_in''
    ? Returns:
    :''[[..:common#Bool |Bool]]''
  ?  **''contains([[..:common#Vector |Vector]]<Scalar> v)''**
  :: checks whether a given point is contained in a cone
    ? Parameters:
    :: ''[[..:common#Vector |Vector]]<Scalar>'' ''v'': point
    ? Returns:
    :''[[..:common#Bool |Bool]]''


----
{{anchor:contains_in_interior:}}
  ?  **''contains_in_interior([[..:common#Vector |Vector]]<Scalar> v)''**
  :: checks whether a given point is contained in the strict interior of a cone
    ? Parameters:
    :: ''[[..:common#Vector |Vector]]<Scalar>'' ''v'': point
    ? Returns:
    :''[[..:common#Bool |Bool]]''


----

==== Lattice points in cones ====
 These methods capture information that depends on the lattice structure of the cone. polymake always works with the integer lattice. 
----
{{anchor:hilbert_basis:}}
  ?  **''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:
    :''[[..:common#Matrix |Matrix]]<[[..:common#Integer |Integer]]>''


----

==== Topology ====
 The following methods compute topological invariants.
----
{{anchor:dual_graph_signature:}}
  ?  **''DUAL_GRAPH_SIGNATURE()''**
  :: Difference of the black and white nodes if the ''[[..:polytope:Cone#DUAL_GRAPH |DUAL_GRAPH]]'' is ''[[..:graph:Graph#BIPARTITE |BIPARTITE]]''. Otherwise -1.
    ? Returns:
    :''[[..:common#Int |Int]]''


----
{{anchor:graph_signature:}}
  ?  **''GRAPH_SIGNATURE()''**
  :: Difference of the black and white nodes if the ''[[..:polytope:Cone#GRAPH |GRAPH]]'' is ''[[..:graph:Graph#BIPARTITE |BIPARTITE]]''. Otherwise -1.
    ? Returns:
    :''[[..:common#Int |Int]]''


----

==== Visualization ====
 These methods are for visualization.
----
{{anchor:visual:}}
  ?  **''VISUAL()''**
  :: Visualizes the cone, intersected with the unit ball.
    ? Options:
    : option list ''[[..:common#Visual_Polygons_decorations |Visual::Polygons::decorations]]''
    : option list ''[[..:common#geometric_options_linear |geometric_options_linear]]''
    ? Returns:
    :''[[..:polytope:Visual_Cone |Visual::Cone]]''


----
{{anchor:visual_dual_face_lattice:}}
  ?  **''VISUAL_DUAL_FACE_LATTICE()''**
  :: Visualize the dual [[..:polytope:Cone#HASSE_DIAGRAM |face lattice]] of a cone as a multi-layer graph.
    ? Options:
    : 
    :: ''[[..:common#Int |Int]]'' ''seed'': random seed value for the node placement
    : option list ''[[..:graph#Visual_Lattice_decorations |Visual::Lattice::decorations]]''
    ? Returns:
    :''[[..:graph:Visual_Lattice |Visual::Lattice]]''


----
{{anchor:visual_face_lattice:}}
  ?  **''VISUAL_FACE_LATTICE()''**
  :: Visualize the ''[[..:polytope:Cone#HASSE_DIAGRAM |HASSE_DIAGRAM]]'' of a cone as a multi-layer graph.
    ? Options:
    : 
    :: ''[[..:common#Int |Int]]'' ''seed'': random seed value for the node placement
    : option list ''[[..:graph#Visual_Lattice_decorations |Visual::Lattice::decorations]]''
    ? Returns:
    :''[[..:graph:Visual_Lattice |Visual::Lattice]]''


----