Differences

This shows you the differences between two versions of the page.

--- playground:playground [2019/03/03 18:17] – oroehrig
+++ playground:playground [2020/05/21 12:24] (current) – removed benmuell
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-----
-==== Input Properties ====
- These properties are for input only. They allow redundant information.
-=== EQUATIONS ===
-| Type: /common/property_types/Algebraic Types/Matrix |
-Equations that hold for all [[/polytope/objects/Cone/properties/Input property/INPUT_RAYS]] of the cone.
- All vectors in this section must be non-zero.
- Input section only.  Ask for [[/polytope/objects/Cone/properties/Geometry/LINEAR_SPAN]] if you want to see an irredundant description of the linear span.
-=== INEQUALITIES ===
-Type: /common/property_types/Algebraic Types/Matrix\\  Inequalities giving rise to the cone; redundancies are allowed.
- All vectors in this section must be non-zero.
- Dual to [[/polytope/objects/Cone/properties/Input property/INPUT_RAYS]].
- Input section only.  Ask for [[/polytope/objects/Cone/properties/Geometry/FACETS]] if you want to compute an H-representation from a V-representation.
-=== INPUT_LINEALITY ===
-Type: /common/property_types/Algebraic Types/Matrix\\  (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/objects/Cone/properties/Input property/EQUATIONS]].
- Input section only.  Ask for [[/polytope/objects/Cone/properties/Geometry/LINEALITY_SPACE]] if
- you want to compute a V-representation from an H-representation.
-=== INPUT_RAYS ===
-Type: /common/property_types/Algebraic Types/Matrix\\  (Non-homogenous) vectors whose positive span form the cone; redundancies are allowed.
- Dual to [[/polytope/objects/Cone/properties/Input property/INEQUALITIES]]. All vectors in the input must be non-zero.
- Input section only.  Ask for [[/polytope/objects/Cone/properties/Geometry/RAYS]] if you want to compute a V-representation from an H-representation.
-----
-==== 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.
-=== F2_VECTOR ===
-Type: /common/property_types/Algebraic Types/Matrix\\  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/objects/Cone/properties/Combinatorics/F_VECTOR]].
-=== ESSENTIALLY_GENERIC ===
-Type: /common/property_types/Basic Types/Bool\\  All intermediate polytopes (with respect to the given insertion order) in the beneath-and-beyond algorithm are simplicial.
- We have the implications: [[/polytope/objects/Cone/properties/Geometry/RAYS]] in general position => ESSENTIALLY_GENERIC => [[/polytope/objects/Cone/properties/Combinatorics/SIMPLICIAL]]
-=== EXCESS_RAY_DEGREE ===
-Type: /common/property_types/Basic Types/Int\\  Measures the deviation of the cone from being simple in terms of the [[/polytope/objects/Cone/properties/Combinatorics/GRAPH]].
-=== COCIRCUIT_EQUATIONS ===
-Type: /common/property_types/Algebraic Types/SparseMatrix\\  A matrix whose rows contain the cocircuit equations of P. The columns correspond to the [[/polytope/objects/Cone/properties/Combinatorics/MAX_INTERIOR_SIMPLICES]].
-=== RAY_SIZES ===
-Type: /common/property_types/Basic Types/Array\\  Number of incident facets for each ray.
-=== FLAG_VECTOR ===
-Type: /common/property_types/Algebraic Types/Vector\\  Condensed form of the flag vector, containing all entries indexed by sparse sets in {0, ..., [[/polytope/objects/Cone/properties/Combinatorics/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/objects/Cone/methods/Combinatorics/N_FLAGS]], to extend.
-=== FACET_SIZES ===
-Type: /common/property_types/Basic Types/Array\\  Number of incident rays for each facet.
-=== RAYS_IN_RIDGES ===
-Type: /common/property_types/Set Types/IncidenceMatrix\\  Ray-ridge incidence matrix, with rows corresponding to ridges and columns
- to rays. Rays and ridges are numbered from 0 to [[/polytope/objects/Cone/properties/Combinatorics/N_RAYS]]-1 rsp.
- [[/polytope/objects/Cone/properties/Geometry/N_RIDGES]]-1, according to their order in [[/polytope/objects/Cone/properties/Geometry/RAYS]] rsp. [[/polytope/objects/Symmetry/SymmetrizedCocircuitEquations/properties/RIDGES]].
-=== SIMPLICIAL_CONE ===
-Type: /common/property_types/Basic Types/Bool\\  True if the cone is simplicial.
-=== SIMPLICIAL ===
-Type: /common/property_types/Basic Types/Bool\\  True if the facets of the cone are simplicial.
-=== FOLDABLE_COCIRCUIT_EQUATIONS ===
-Type: /common/property_types/Algebraic Types/SparseMatrix\\  A matrix whose rows contain the foldable cocircuit equations of P.  The columns correspond to 2 * [[/polytope/objects/Cone/properties/Combinatorics/MAX_INTERIOR_SIMPLICES]].
- col 0 = 0, col 1 = first simplex (black copy), col 2 = first simplex (white copy), col 3 = second simplex (black copy), ...
-=== GRAPH ===
-Type: /graph/objects/Combinatorics/Graph\\  Vertex-edge graph obtained by intersecting the cone with a transversal hyperplane.
-=== RAYS_IN_FACETS ===
-Type: /common/property_types/Set Types/IncidenceMatrix\\  Ray-facet incidence matrix, with rows corresponding to facets and columns
- to rays. Rays and facets are numbered from 0 to [[/polytope/objects/Cone/properties/Combinatorics/N_RAYS]]-1 rsp.
- [[/polytope/objects/Cone/properties/Geometry/N_FACETS]]-1, according to their order in [[/polytope/objects/Cone/properties/Geometry/RAYS]] rsp. [[/polytope/objects/Cone/properties/Geometry/FACETS]].
-=== N_RAY_FACET_INC ===
-Type: /common/property_types/Basic Types/Int\\  Number of pairs of incident vertices and facets.
-=== INTERIOR_RIDGE_SIMPLICES ===
-Type: /common/property_types/Basic Types/Array\\  The (//d//-1)-dimensional simplices in the interior.
-=== HASSE_DIAGRAM ===
-Type: /polytope/objects/Cone/properties/Combinatorics/HASSE_DIAGRAM\\  The face lattice of the cone organized as a directed graph.
- Top and bottom nodes represent the whole cone and the empty face.
- Every other node corresponds to some proper face of the cone.
-=== FACETS_THRU_RAYS ===
-Type: /common/property_types/Set Types/IncidenceMatrix\\  Transposed to [[/polytope/objects/Cone/properties/Combinatorics/RAYS_IN_FACETS]].
- Notice that this is a temporary property; it will not be stored in any file.
-=== SELF_DUAL ===
-Type: /common/property_types/Basic Types/Bool\\  True if the cone is self-dual.
-=== N_RAYS ===
-Type: /common/property_types/Basic Types/Int\\  The number of [[/polytope/objects/Cone/properties/Geometry/RAYS]]
-=== DUAL_GRAPH ===
-Type: /graph/objects/Combinatorics/Graph\\  Facet-ridge graph. Dual to [[/polytope/objects/Cone/properties/Combinatorics/GRAPH]].
-=== MAX_INTERIOR_SIMPLICES ===
-Type: /common/property_types/Basic Types/Array\\  The interior //d//-dimensional simplices of a cone of combinatorial dimension //d//
-=== EXCESS_FACET_DEGREE ===
-Type: /common/property_types/Basic Types/Int\\  Measures the deviation of the cone from being simple in terms of the [[/polytope/objects/Cone/properties/Combinatorics/DUAL_GRAPH]].
-=== SIMPLE ===
-Type: /common/property_types/Basic Types/Bool\\  True if the facets of the cone are simple. Dual to [[/polytope/objects/Cone/properties/Combinatorics/SIMPLICIAL]].
-=== MAX_BOUNDARY_SIMPLICES ===
-Type: /common/property_types/Basic Types/Array\\  The boundary (//d//-1)-dimensional simplices of a cone of combinatorial dimension //d//
-=== COMBINATORIAL_DIM ===
-Type: /common/property_types/Basic Types/Int\\  Combinatorial dimension
- This is the dimension all combinatorial properties of the cone
- like e.g. [[/polytope/objects/Cone/properties/Combinatorics/RAYS_IN_FACETS]] or the [[/polytope/objects/Cone/properties/Combinatorics/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.
-=== F_VECTOR ===
-Type: /common/property_types/Algebraic Types/Vector\\  The vector counting the number of faces (`f<sub>k</sub>` is the number of `(k+1)`-faces).
-----
-==== Lattice points in cones ====
- These properties capture information that depends on the lattice structure of the cone.
- polymake always works with the integer lattice.
-=== H_STAR_VECTOR ===
-Type: /common/property_types/Algebraic Types/Vector\\  The coefficients of the Hilbert polynomial,
- the h^*-polynomial for lattice polytopes,
- with respect to the [[/polytope/objects/Cone/properties/Lattice points in cones/MONOID_GRADING]]
- starting at the constant coefficient.
- For lattice polytopes the length of this vector is [[/polytope/objects/Cone/properties/Geometry/CONE_DIM]].
- In general the length is one less than the degree of the
- denominator of the [[/polytope/objects/Cone/properties/Lattice points in cones/HILBERT_SERIES]].
-=== MONOID_GRADING ===
-Type: /common/property_types/Algebraic Types/Vector\\  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/objects/Cone/properties/Lattice points in cones/HOMOGENEOUS]] cones the affine hyperplane containing the primitive generators
-     will be used.
-=== N_HILBERT_BASIS ===
-Type: /common/property_types/Basic Types/Int\\  The number of elements of the [[/polytope/objects/Cone/methods/Lattice points in cones/HILBERT_BASIS]].
-=== DEGREE_ONE_GENERATORS ===
-Type: /common/property_types/Algebraic Types/Matrix\\  Elements of the [[/polytope/objects/Cone/methods/Lattice points in cones/HILBERT_BASIS]] for the cone of degree 1 with respect
- to the [[/polytope/objects/Cone/properties/Lattice points in cones/MONOID_GRADING]].
-=== HILBERT_BASIS_GENERATORS ===
-Type: /common/property_types/Basic Types/Array\\  Generators for the [[/polytope/objects/Cone/methods/Lattice points in cones/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/objects/Cone/properties/Geometry/RAYS]] or [[/polytope/objects/Cone/properties/Input property/INPUT_RAYS]]
-       it will be if the cone is pointed (the polytope is bounded)
-=== GORENSTEIN_CONE ===
-Type: /common/property_types/Basic Types/Bool\\  A cone is [[/polytope/objects/Cone/properties/Lattice points in cones/GORENSTEIN_CONE|Gorenstein]] if it is [[/polytope/objects/Cone/properties/Lattice points in cones/Q_GORENSTEIN_CONE|Q-Gorenstein]] with index one
-=== SMOOTH_CONE ===
-Type: /common/property_types/Basic Types/Bool\\  A cone is __smooth__ if the primitive generators are part of a lattice basis.
-=== HOMOGENEOUS ===
-Type: /common/property_types/Basic Types/Bool\\  True if the primitive generators of the rays lie on an affine hyperplane in the span of the rays.
-=== HILBERT_SERIES ===
-Type: /common/property_types/Algebraic Types/RationalFunction\\  Hilbert series of the monoid, given by the intersection of the cone with
- the lattice Z^d with respect to the [[/polytope/objects/Cone/properties/Lattice points in cones/MONOID_GRADING]]
-=== Q_GORENSTEIN_CONE_INDEX ===
-Type: /common/property_types/Basic Types/Int\\  If a cone is [[/polytope/objects/Cone/properties/Lattice points in cones/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.
-=== Q_GORENSTEIN_CONE ===
-Type: /common/property_types/Basic Types/Bool\\  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).
-----
-==== Geometry ====
- These properties capture geometric information of the object.
- Geometric properties depend on geometric information of the object, like, e.g., vertices or facets.
-=== N_EQUATIONS ===
-Type: /common/property_types/Basic Types/Int\\  The number of [[/polytope/objects/Cone/properties/Input property/EQUATIONS]].
-=== LINEALITY_SPACE ===
-Type: /common/property_types/Algebraic Types/Matrix\\  Basis of the linear subspace orthogonal to all [[/polytope/objects/Cone/properties/Input property/INEQUALITIES]] and [[/polytope/objects/Cone/properties/Input property/EQUATIONS]]
- All vectors in this section must be non-zero.
- The property [[/polytope/objects/Cone/properties/Geometry/LINEALITY_SPACE]] appears only in conjunction with the property [[/polytope/objects/Cone/properties/Geometry/RAYS]], or [[/polytope/objects/SchlegelDiagram/properties/VERTICES]], respectively.
- The specification of the property [[/polytope/objects/Cone/properties/Geometry/RAYS]] or [[/polytope/objects/SchlegelDiagram/properties/VERTICES]] requires the specification of [[/polytope/objects/Cone/properties/Geometry/LINEALITY_SPACE]], and vice versa.
-=== REL_INT_POINT ===
-Type: /common/property_types/Algebraic Types/Vector\\  A point in the relative interior of the cone.
-=== POINTED ===
-Type: /common/property_types/Basic Types/Bool\\  True if the cone does not contain a non-trivial linear subspace.
-=== INPUT_RAYS_IN_FACETS ===
-Type: /common/property_types/Set Types/IncidenceMatrix\\  [[/polytope/objects/Cone/properties/Input property/INPUT_RAYS|Input ray]]-[[/polytope/objects/Cone/properties/Geometry/FACETS|facet]] incidence matrix, with rows corresponding to [[/polytope/objects/Cone/properties/Geometry/FACETS|facet]] and columns
- to [[/polytope/objects/Cone/properties/Input property/INPUT_RAYS|input rays]]. Input_rays and facets are numbered from 0 to [[/polytope/objects/Cone/properties/Geometry/N_INPUT_RAYS]]-1 rsp.
- [[/polytope/objects/Cone/properties/Geometry/N_FACETS]]-1, according to their order in [[/polytope/objects/Cone/properties/Input property/INPUT_RAYS]]
- rsp. [[/polytope/objects/Cone/properties/Geometry/FACETS]].
-=== CONE_AMBIENT_DIM ===
-Type: /common/property_types/Basic Types/Int\\  The dimension of the space in which the cone lives.
-=== TRIVIAL ===
-Type: /common/property_types/Basic Types/Bool\\  True if the only valid point in the cone is the unique non-sensical point (0,...,0)
-=== LINEALITY_DIM ===
-Type: /common/property_types/Basic Types/Int\\  Dimension of the [[/polytope/objects/Cone/properties/Geometry/LINEALITY_SPACE]] (>0 in the non-POINTED case)
-=== EPSILON ===
-Type: /common/property_types/Basic Types/Float\\  Threshold for zero test for scalar products (e.g. vertex * facet normal)
-=== RAYS ===
-Type: /common/property_types/Algebraic Types/Matrix\\  Rays of the cone. No redundancies are allowed.
- All vectors in this section must be non-zero.
- The property [[/polytope/objects/Cone/properties/Geometry/RAYS]] appears only in conjunction with the property [[/polytope/objects/Cone/properties/Geometry/LINEALITY_SPACE]].
- The specification of the property [[/polytope/objects/Cone/properties/Geometry/RAYS]] requires the specification of [[/polytope/objects/Cone/properties/Geometry/LINEALITY_SPACE]], and vice versa.
-=== N_INPUT_LINEALITY ===
-Type: /common/property_types/Basic Types/Int\\  The number of [[/polytope/objects/Cone/properties/Input property/INPUT_LINEALITY]].
-=== CONE_DIM ===
-Type: /common/property_types/Basic Types/Int\\  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.
-=== RAYS_IN_INEQUALITIES ===
-Type: /common/property_types/Set Types/IncidenceMatrix\\  Ray-inequality incidence matrix, with rows corresponding to facets and columns
- to rays. Rays and inequalities are numbered from 0 to [[/polytope/objects/Cone/properties/Combinatorics/N_RAYS]]-1 rsp.
- number of [[/polytope/objects/Cone/properties/Input property/INEQUALITIES]]-1, according to their order in [[/polytope/objects/Cone/properties/Geometry/RAYS]]
- rsp. [[/polytope/objects/Cone/properties/Input property/INEQUALITIES]].
-=== POSITIVE ===
-Type: /common/property_types/Basic Types/Bool\\  True if all [[/polytope/objects/Cone/properties/Geometry/RAYS]] of the cone have non-negative coordinates,
- that is, if the pointed part of the cone lies entirely in the positive orthant.
-=== N_INPUT_RAYS ===
-Type: /common/property_types/Basic Types/Int\\  The number of [[/polytope/objects/Cone/properties/Input property/INPUT_RAYS]].
-=== FACETS_THRU_INPUT_RAYS ===
-Type: /common/property_types/Set Types/IncidenceMatrix\\  Transposed to [[/polytope/objects/Cone/properties/Geometry/INPUT_RAYS_IN_FACETS]].
- Notice that this is a temporary property; it will not be stored in any file.
-=== FACETS ===
-Type: /common/property_types/Algebraic Types/Matrix\\  Facets of the cone, encoded as inequalities.
- All vectors in this section must be non-zero.
- Dual to [[/polytope/objects/Cone/properties/Geometry/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/objects/Cone/properties/Geometry/FACETS]] appears only in conjunction with the property [[/polytope/objects/Cone/properties/Geometry/LINEAR_SPAN]], or [[/polytope/objects/PointConfiguration/properties/Geometry/AFFINE_HULL]], respectively.
- The specification of the property [[/polytope/objects/Cone/properties/Geometry/FACETS]] requires the specification of [[/polytope/objects/Cone/properties/Geometry/LINEAR_SPAN]],
- or [[/polytope/objects/PointConfiguration/properties/Geometry/AFFINE_HULL]], respectively, and vice versa.
-=== INEQUALITIES_THRU_RAYS ===
-Type: /common/property_types/Set Types/IncidenceMatrix\\  transposed [[/polytope/objects/Cone/properties/Geometry/RAYS_IN_INEQUALITIES]]
- Notice that this is a temporary property; it will not be stored in any file.
-=== ONE_RAY ===
-Type: /common/property_types/Algebraic Types/Vector\\  A ray of a pointed cone.
-=== LINEAR_SPAN ===
-Type: /common/property_types/Algebraic Types/Matrix\\  Dual basis of the linear span of the cone.
- All vectors in this section must be non-zero.
- The property [[/polytope/objects/Cone/properties/Geometry/LINEAR_SPAN]] appears only in conjunction with the property [[/polytope/objects/Cone/properties/Geometry/FACETS]].
- The specification of the property [[/polytope/objects/Cone/properties/Geometry/FACETS]] requires the specification of [[/polytope/objects/Cone/properties/Geometry/LINEAR_SPAN]],
- or [[/polytope/objects/PointConfiguration/properties/Geometry/AFFINE_HULL]], respectively, and vice versa.
-=== FULL_DIM ===
-Type: /common/property_types/Basic Types/Bool\\  [[/polytope/objects/Cone/properties/Geometry/CONE_AMBIENT_DIM]] and [[/polytope/objects/Cone/properties/Geometry/CONE_DIM]] coincide.  Notice that this makes sense also for the derived Polytope class.
-=== RAY_SEPARATORS ===
-Type: /common/property_types/Algebraic Types/Matrix\\  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.
-=== N_RIDGES ===
-Type: /common/property_types/Basic Types/Int\\  The number of [[/polytope/objects/Symmetry/SymmetrizedCocircuitEquations/properties/RIDGES]].
-=== N_FACETS ===
-Type: /common/property_types/Basic Types/Int\\  The number of [[/polytope/objects/Cone/properties/Geometry/FACETS]].
-----
-==== Symmetry ====
- These properties capture information of the object that is concerned with the action of permutation groups.
-=== GROUP ===
-Type: /polytope/objects/Cone/properties/Symmetry/GROUP\\ **Properties of GROUP**:\\ **Symmetry**\\  These properties capture information of the object that is concerned with the action of permutation groups.
-  * **REPRESENTATIVE_MAX_INTERIOR_SIMPLICES**\\ Type: /common/property_types/Basic Types/Array\\  One representative for each orbit of maximal-dimensional interior simplices
-  * **REPRESENTATIVE_INTERIOR_RIDGE_SIMPLICES**\\ Type: /common/property_types/Basic Types/Array\\  One representative for each orbit of interior ridge simplices
-  * **BITSET_ACTION**\\ Type: /group/objects/ImplicitActionOnSets  * **REPRESENTATIVE_MAX_BOUNDARY_SIMPLICES**\\ Type: /common/property_types/Basic Types/Array\\  One representative for each orbit of maximal-dimensional boundary simplices
-  * **MATRIX_ACTION**\\ Type: /polytope/objects/Cone/properties/Symmetry/GROUP/properties/Symmetry/MATRIX_ACTION\\ **Properties of MATRIX_ACTION**:\\ **Symmetry**\\  These properties capture information of the object that is concerned with the action of permutation groups.
-    * **RAYS_ORBITS**\\ Type: /common/property_types/Basic Types/Array\\  Alias for property [[/group/objects/MatrixActionOnVectors/properties/Symmetry/VECTORS_ORBITS]].
-  * **REPRESENTATIVE_RAYS**\\ Type: /common/property_types/Algebraic Types/Matrix  * **REPRESENTATIVE_BOUNDARY_RIDGE_SIMPLICES**\\ Type: /common/property_types/Basic Types/Array\\  One representative for each orbit of boundary ridge simplices
-  * **REPRESENTATIVE_FACETS**\\ Type: /common/property_types/Algebraic Types/Matrix
-----
-==== Visualization ====
- These properties are for visualization.
-=== FACET_LABELS ===
-Type: /common/property_types/Basic Types/Array\\  Unique names assigned to the [[/polytope/objects/Cone/properties/Geometry/FACETS]], analogous to [[/polytope/objects/Cone/properties/Visualization/RAY_LABELS]].
-=== INPUT_RAY_LABELS ===
-Type: /common/property_types/Basic Types/Array\\  Unique names assigned to the [[/polytope/objects/Cone/properties/Input property/INPUT_RAYS]], analogous to [[/polytope/objects/Cone/properties/Visualization/RAY_LABELS]].
-=== NEIGHBOR_RAYS_CYCLIC_NORMAL ===
-Type: /common/property_types/Basic Types/Array\\  Reordered [[/polytope/objects/Cone/properties/Combinatorics/GRAPH]]. Dual to [[/polytope/objects/Cone/properties/Visualization/NEIGHBOR_FACETS_CYCLIC_NORMAL]].
-=== COORDINATE_LABELS ===
-Type: /common/property_types/Basic Types/Array\\  Unique names assigned to the coordinate directions, analogous to [[/polytope/objects/Cone/properties/Visualization/RAY_LABELS]].
- For Polytopes this should contain "inhomog_var" for the homogenization coordinate and this will
- be added automatically if necessary and [[/polytope/objects/Cone/properties/Geometry/CONE_AMBIENT_DIM]] can be computed.
-=== RAY_LABELS ===
-Type: /common/property_types/Basic Types/Array\\  Unique names assigned to the [[/polytope/objects/Cone/properties/Geometry/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.
-=== FTR_CYCLIC_NORMAL ===
-Type: /common/property_types/Basic Types/Array\\  Reordered transposed [[/polytope/objects/Cone/properties/Combinatorics/RAYS_IN_FACETS]]. Dual to [[/polytope/objects/Cone/properties/Visualization/RIF_CYCLIC_NORMAL]].
-=== NEIGHBOR_FACETS_CYCLIC_NORMAL ===
-Type: /common/property_types/Basic Types/Array\\  Reordered [[/polytope/objects/Cone/properties/Combinatorics/DUAL_GRAPH]] for 3d-cones.
- The neighbor facets are listed in the order corresponding to [[/polytope/objects/Cone/properties/Visualization/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.
-=== INEQUALITY_LABELS ===
-Type: /common/property_types/Basic Types/Array\\  Unique names assigned to the [[/polytope/objects/Cone/properties/Input property/INEQUALITIES]], analogous to [[/polytope/objects/Cone/properties/Visualization/RAY_LABELS]].
-=== RIF_CYCLIC_NORMAL ===
-Type: /common/property_types/Basic Types/Array\\  Reordered [[/polytope/objects/Cone/properties/Combinatorics/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.
-----
-==== Triangulation and volume ====
- These properties collect information about triangulations of the object and properties usually computed from such, as the volume.
-=== TRIANGULATION_INT ===
-Type: /common/property_types/Basic Types/Array\\  Conceptually, similar to [[/polytope/objects/Cone/properties/Triangulation and volume/TRIANGULATION]], but using [[/polytope/objects/Cone/properties/Input property/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/objects/Cone/properties/Input property/INPUT_RAYS]] to [[/polytope/objects/Cone/properties/Geometry/FACETS]].  And that data is too valuable
- to throw away.  Use big objects of type [[/polytope/objects/Geometry/VectorConfiguration]] if you want to work with
- triangulations using redundant points.
-=== TRIANGULATION ===
-Type: /polytope/objects/Cone/properties/Triangulation and volume/TRIANGULATION\\  Some triangulation of the cone using only its [[/polytope/objects/Cone/properties/Geometry/RAYS]].
-\\ **Properties of TRIANGULATION**:\\ **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.
-  * **REFINED_SPLITS**\\ Type: /common/property_types/Set Types/Set\\  The splits that are coarsenings of the current [[/polytope/objects/Cone/properties/Triangulation and volume/TRIANGULATION]].
- If the triangulation is regular these form the unique split decomposition of
- the corresponding weight function.
-  * **BOUNDARY**\\ Type: /polytope/objects/Cone/properties/Triangulation and volume/TRIANGULATION/properties/Combinatorics/BOUNDARY\\ Augmented subobject [[topaz::SimplicialComplex::BOUNDARY
-\\ **Properties of BOUNDARY**:\\ **no category**\\     * **FACET_TRIANGULATIONS**\\ Type: /common/property_types/Basic Types/Array\\  For each facet the set of simplex indices of [[/polytope/objects/Cone/properties/Triangulation and volume/TRIANGULATION/properties/Combinatorics/BOUNDARY]] that triangulate it.
-**Geometry**\\  These properties capture geometric information of the object.
- Geometric properties depend on geometric information of the object, like, e.g., vertices or facets.
-  * **WEIGHTS**\\ Type: /common/property_types/Algebraic Types/Vector\\  Weight vector to construct a regular [[/polytope/objects/Cone/properties/Triangulation and volume/TRIANGULATION]].
- Must be generic.