application: polytope
This is the historically first application, and the largest one.
It deals with convex pointed polyhedra. It allows to define a polyhedron either as a convex hull of a point set, an intersection of halfspaces, or as an incidence matrix without any embedding. Then you can ask for a plenty of its (especially combinatorial) properties, construct new polyhedra by modifying it, or study the behavior of the objective functions.
There is a wide range of visualization methods for polyhedra, even for dimensions > 4 and purely combinatorial descriptions, including interfaces to interactive geometry viewers (such as JavaView or geomview), generating PostScript drawings and povray scene files.
imports from: common, graph
uses: topaz
Objects
- Cone
A polyhedral cone, not necessarily pointed. Note that in contrast to the vertices of a polytope, the RAYS are given in affine coordinates.
Properties of Cone
- Basic propertiesCOMBINATORIAL_DIM: common::Int
the combinatorial dimension defined by the ray-(2-face)-lattice equals one less than the dimension of the pointed part of the cone equals pointed part of a polytope
CONE_DIM: common::IntDimension 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.
COORDINATE_LABELS: common::Array<String>Unique names assigned to the coordinate directions, analogous to RAY_LABELS.
EQUATIONS: common::Matrix<Scalar>Equations that hold for all INPUT_RAYS of the cone.
Input section only. Ask for LINEAR_SPAN if you want to see an irredundant description of the linear span.
FULL_DIM: common::BoolCONE_AMBIENT_DIM and CONE_DIM coincide. Notice that this makes sense also for the derived Polytope class.
INEQUALITIES: common::Matrix<Scalar>Inequalities giving rise to the cone; redundancies are allowed. Dual to INPUT_RAYS.
Input section only. Ask for FACETS if you want to compute an H-representation from a V-representation.
INPUT_LINEALITY: common::Matrix<Scalar>(Non-homogenous) vectors whose linear span defines a subset of the lineality space of the cone; redundancies are allowed. Dual to EQUATIONS.
Input section only. Ask for LINEALITY_SPACE if you want to compute a V-representation from an H-representation.
INPUT_RAYS: common::Matrix<Scalar>(Non-homogenous) vectors whose positive span form the cone; redundancies are allowed. Dual to INEQUALITIES.
Input section only. Ask for RAYS if you want to compute a V-representation from an H-representation.
LINEALITY_SPACE: common::Matrix<Scalar>Basis of the linear subspace orthogonal to all INEQUALITIES and EQUATIONS
POSITIVE: common::BoolTrue 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. FIXME should we set POSITIVE = false for non-pointed cones?
RAY_LABELS: common::Array<String>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.
RAY_SEPARATORS: common::Matrix<Scalar>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.
- CombinatoricsESSENTIALLY_GENERIC: common::Bool
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
F2_VECTOR: common::Matrix<Integer>The vector counting the number of incidences between pairs of faces. `fik` is the number of incident pairs of `(i+1)`-faces and `(k+1)`-faces. The main diagonal contains the F_VECTOR.
FACETS_THRU_INPUT_RAYS: common::IncidenceMatrix<NonSymmetric>transposed INPUT_RAYS_IN_FACETS Notice that this is a temporary property; it will not be stored in any file.
FACETS_THRU_RAYS: common::IncidenceMatrix<NonSymmetric>transposed RAYS_IN_FACETS Notice that this is a temporary property; it will not be stored in any file.
FLAG_VECTOR: common::Vector<Integer>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.
F_VECTOR: common::Vector<Integer>The vector counting the number of faces (`fk` is the number of `(k+1)`-faces).
GRAPH: graph::Graph<Undirected>Vertex-edge graph obtained by intersecting the cone with a transversal hyperplane.
Properties of GRAPH
- EDGE_DIRECTIONS: common::EdgeMap<Undirected, Vector<Scalar>>
Difference of the vertices for each edge (only defined up to signs).
INEQUALITIES_THRU_RAYS: common::IncidenceMatrix<NonSymmetric>transposed RAYS_IN_INEQUALITIES Notice that this is a temporary property; it will not be stored in any file.
INPUT_RAYS_IN_FACETS: common::IncidenceMatrix<NonSymmetric>Input_ray-facet incidence matrix, with rows corresponding to facets 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.
RAYS_IN_INEQUALITIES: common::IncidenceMatrix<NonSymmetric>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. N_INEQUALITIES-1, according to their order in RAYS rsp. INEQUALITIES.
- Triangulation and volumeTRIANGULATION: topaz::SimplicialComplex
Some triangulation of the polytope using only its vertices. Each line contains indices from VERTICES comprising a simplex.
Properties of TRIANGULATION
- FACET_TRIANGULATIONS: common::Array<Set<Int>>
For each facet the set of simplex indices of BOUNDARY that triangulate it.
- GKZ_VECTOR: common::Vector<Scalar>
GKZ-vector
See Chapter 7 in Gelfand, Kapranov, and Zelevinsky:Discriminants, Resultants and Multidimensional Determinants, Birkhäuser 1994 - REFINED_SPLITS: common::Set<Int>
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.
User Methods of Cone
- AMBIENT_DIM ()
FIXME why do the following clients use different methods to determine their type?
- DIM ()
UNDOCUMENTED
- Backward compatibilityDUAL_DIAMETER ()
The diameter of the DUAL_GRAPH
DUAL_TRIANGLE_FREE ()True if the DUAL_GRAPH contains no triangle
TRIANGLE_FREE ()True if the GRAPH contains no triangle
- CombinatoricsCONNECTIVITY ()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
DUAL_EVEN ()True if the DUAL_GRAPH is bipartite
FACET_DEGREES () → Vector<Int>Facet degrees of the polytope. The degree of a facet is the number of adjacent facets.
Returns
Vector<Int> - in the same order as FACETSN_FLAGS (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 typeVERTEX_DEGREES () → Vector<Int>Ray degrees of the cone
Returns
Vector<Int> - in the same order as RAYS - TopologyDUAL_GRAPH_SIGNATURE ()
Difference of the black and white nodes if the DUAL_GRAPH is BIPARTITE. Otherwise -1.
Permutations of Cone
- FacetPermUNDOCUMENTED
Properties of FacetPerm
- VertexPermUNDOCUMENTED
Properties of VertexPerm
derived from: ConeCone<Float>Properties of Cone<Float>
derived from: ConeCone<Rational>Properties of Cone<Rational>
- Q_GORENSTEIN_CONE: common::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).
- Q_GORENSTEIN_CONE_INDEX: common::Int
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.
- Basic propertiesHOMOGENEOUS: common::Bool
true if the primitive generators of the rays lie on an affine hyperplane in the span of the rays
- GeometricGraphderived from: graph::Graph<Undirected>
An undirected graph with given node coordinates and a bounding box
Properties of GeometricGraph
- BOUNDING_BOX: common::Matrix<Rational>
Since a Voronoi polyhedron is unbounded it must be artificially bounded for visualization purposes. Allowed is any set of hyperplanes which makes the projection onto the last d-1 coordinates bounded. By default, these are the vertical facets of a suitably scaled cube.
- COORDINATES: common::NodeMap<Undirected, Vector<Rational>>
The coordinates of the nodes of the graph.
- GroebnerBasis
The Groebner basis of the homogeneous toric ideal associated to the polytope, the term order is given in matrix form.
Properties of GroebnerBasis
- TERM_ORDER_MATRIX: common::Matrix<Integer>
The term order in matrix form; if not square, then a tie breaker is used.
- TERM_ORDER_NAME: common::String
A term order by name; allowed acronyms are
lex,deglexanddegrevlex.
- LatticePolytopederived from: Polytope<Rational>
Polytope all of whose vertex coordinates are integral
Properties of LatticePolytope
- EHRHART_POLYNOMIAL_COEFF: common::Vector<Rational>
The coefficients of the Ehrhart polynomial starting at the constant coefficient.
- FACET_VERTEX_LATTICE_DISTANCES: common::Matrix<Integer>
The entry (i,j) equals the lattice distance of vertex j from facet i.
- FACET_WIDTH: common::Integer
The maximal integral width of the polytope with respect to the facet normals.
- FACET_WIDTHS: common::Vector<Integer>
The integral width of the polytope with respect to each facet normal.
- GORENSTEIN: common::Bool
The polytope is Gorenstein if a dilation of the polytope is reflexive up to translation.
- GORENSTEIN_INDEX: common::Integer
If the polytope is Gorenstein then this is the multiple such that the polytope is reflexive.
- GORENSTEIN_VECTOR: common::Vector<Integer>
If the polytope is Gorenstein, then this is the unique interior lattice point in the multiple of the polytope that is reflexive.
- GROEBNER_BASIS: GroebnerBasis
The Groebner basis for the toric ideal associated to the lattice points in the polytope using any term order.
- H_STAR_VECTOR: common::Vector<Integer>
The coefficients of the h^*-polynomial starting at the constant coefficient
- LATTICE_CODEGREE: common::Int
POLYTOPE_DIM+1-LATTICE_DEGREE or the smallest integer k such that k*P has an interior lattice point.
- LATTICE_EMPTY: common::Bool
True if the polytope contains no lattice points other than the vertices.
- NORMAL: common::Bool
The polytope is normal if the cone spanned by P x {1} is generated in height 1.
- SMOOTH: common::Bool
The polytope is smooth if the associated projective variety is smooth; the determinant of the edge directions is +/-1 at every vertex.
- TERMINAL: common::Bool
The polytope is terminal if there is exactly one interior lattice point and all other lattice points are vertices.
- VERY_AMPLE: common::Bool
The polytope is very ample if the Hilbert Basis of the cone spanned by the edge-directions of any vertex lies inside the polytope.
User Methods of LatticePolytope
- Lattice polytopesFACET_POINT_LATTICE_DISTANCES (v) → Vector<Integer>
Vector containing the distances of a given point v from all facets
N_LATTICE_POINTS_IN_DILATION (n)The number of LATTICE_POINTS in the n-th dilation of the polytope
Parameters
Int n dilation factor
Category: OptimizationLinearProgramA linear program specified by a linear or abstract objective function
Properties of LinearProgram
- ABSTRACT_OBJECTIVE: common::Vector<Scalar>
Abstract objective function. Defines a direction for each edge such that each non-empty face has a unique source and a unique sink. The i-th element is the value of the objective function at vertex number i. Only defined for bounded polytopes.
- DIRECTED_GRAPH: graph::Graph<Directed>
Subgraph of Polytope::GRAPH. Consists only of directed arcs along which the value of the objective function increases.
- LINEAR_OBJECTIVE: common::Vector<Scalar>
Linear objective function. In d-space a linear objective function is given by a (d+1)-vector. The first coordinate specifies a constant that is added to the resulting value.
- MAXIMAL_FACE: common::Set<Int>
Indices of vertices at which the maximum of the objective function is attained.
- MAXIMAL_VALUE: Scalar
Maximum value of the objective function. Negated if linear problem is unbounded.
- MAXIMAL_VERTEX: common::Vector<Scalar>
Coordinates of a (possibly not unique) affine vertex at which the maximum of the objective function is attained.
- RANDOM_EDGE_EPL: common::Vector<Rational>
Expected average path length for a simplex algorithm employing "random edge" pivoting strategy.
- Unbounded polyhedraDIRECTED_BOUNDED_GRAPH: graph::Graph<Directed>
Subgraph of BOUNDED_GRAPH. Consists only of directed arcs along which the value of the objective function increases.
User Methods of LinearProgram
- VERTEX_IN_DEGREES ()
Array of in-degrees for all nodes of DIRECTED_GRAPH or numbers of objective decreasing edges at each vertex
- VERTEX_OUT_DEGREES ()
Array of out-degrees for all nodes of DIRECTED_GRAPH or numbers of objective increasing edges at each vertex
- PointConfiguration
The object point configuration also deals with non-convex finite point sets.
Properties of PointConfiguration
- SPLITS: common::Matrix<Scalar>
The splits of the point configuration, i.e., hyperplanes cutting the configuration in two parts such that we have a regular subdivision.
- SPLITS_IN_SUBDIVISION: common::Set<Int>
The splits that are coarsenings of the current POLYTOPAL_SUBDIVISION. If the subdivision is regular these form the unique split decomposition of the corresponding weight function.
- SPLIT_COMPATIBILITY_GRAPH: graph::Graph<Undirected>
Two SPLITS are compatible if the defining hyperplanes do not intersect in the interior of the point configuration. This defines a graph.
- Basic propertiesCONVEX_HULL: Polytope<Scalar>
The polytope being the convex hull of the point configuration.
Properties of CONVEX_HULL
LABELS: common::Array<String>Unique names assigned to the POINTS. If specified, they are shown by visualization tools instead of point indices.
- CombinatoricsGRAPH: graph::Graph<Undirected>
Graph of the point configuration. Two points are adjacent if they lie in a common edge of the CONVEX_HULL.
- Triangulation and volumePOLYTOPAL_SUBDIVISION: common::Array<Set<Int>>
Polytopal Subdivision of the point configuration. Each line contains indices from POINTS comprising a facet of the subdivision.
TRIANGULATION: topaz::SimplicialComplexSome triangulation of the point configuration.
Properties of TRIANGULATION
- FACET_TRIANGULATIONS: common::Array<Set<Int>>
DOC_FIXME: Incomprehensible description! For each facet the set of simplex indices of BOUNDARY that triangulate it.
- GKZ_VECTOR: common::Vector<Scalar>
GKZ-vector
See Chapter 7 in Gelfand, Kapranov, and Zelevinsky:Discriminants, Resultants and Multidimensional Determinants, Birkhäuser 1994 - REFINED_SPLITS: common::Set<Int>
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.
- VisualizationPIF_CYCLIC_NORMAL: common::Array<List<Int>>
Polytope::VIF_CYCLIC_NORMAL of the CONVEX_HULL, but with the indices form POINTS instead of Polytope::VERTICES
User Methods of PointConfiguration
- VISUAL ()
UNDOCUMENTED
Options
option list: Visual::Polygons::decorations - Triangulation and volumeTRIANGULATION_SIGNS () → Array<int>
For each simplex in the TRIANGULATION, this contains the sign of the determinant of its coordinate matrix, which is the orientation of the simplex.
Returns
Array<int> - VisualizationVISUAL () → Visual::PointConfiguration
Visualize a point configuration.
VISUAL_POINTS () → Visual::PointSetVisualize the POINTS of a point configuration.
Options
option list: Visual::Polygons::decorations Returns
Visual::PointSet
- Polytopederived from: Cone
A bounded or unbounded pointed polyhedron. Note that a pointed polyhedron is projectively equivalent to a polytope. Scalar is the numeric data type used for the coordinates.
Properties of Polytope
- SPLITS: common::Matrix<Scalar>
The splits of the polytope, i.e., hyperplanes cutting the polytope in two parts such that we have a regular subdivision.
- SPLITS_IN_SUBDIVISION: common::Set<Int>
The splits that are coarsenings of the current POLYTOPAL_SUBDIVISION. If the subdivision is regular these form the unique split decomposition of the corresponding weight function.
- SPLIT_COMPATIBILITY_GRAPH: graph::Graph<Undirected>
Two SPLITS are compatible if the defining hyperplanes do not intersect in the interior of the polytope. This defines a graph.
- Basic propertiesAFFINE_HULL: common::Matrix<Scalar>
Dual basis of the affine hull of the polyhedron. Alias for property LINEAR_SPAN.
MINIMAL_VERTEX_ANGLE: common::FloatThe minimal angle between any two vertices (seen from the VERTEX_BARYCENTER).
POINTS: common::Matrix<Scalar>Points such that the polyhedron is their convex hull. Redundancies are allowed. The vector (x0, x1, ... xd) represents a point in d-space given in homogeneous coordinates. Affine points are identified by x0 > 0. Points with x0 = 0 can be interpreted as rays.
polymake automatically normalizes each coordinate vector, dividing them by the first non-zero element. The clients and rule subroutines can always assume that x0 is either 0 or 1. Dual to INEQUALITIES.
Input section only. Ask for VERTICES if you want to compute a V-representation from an H-representation. Alias for property INPUT_RAYS.
STEINER_POINTS: common::Matrix<Scalar>A weighted inner point depending on the outer angle called Steiner point for all faces of dimensions 2 to d.
VALID_POINT: common::Vector<Scalar>Some point belonging to the polyhedron. Alias for property VALID_RAY.
VERTEX_BARYCENTER: common::Vector<Scalar>The center of gravity of the vertices of a bounded polytope.
VERTEX_LABELS: common::Array<String>Unique names assigned to the VERTICES. If specified, they are shown by visualization tools instead of vertex indices.
For a polytope build from scratch, you should create this property by yourself, either manually in a text editor, or with a client program.
If you build a polytope with a construction function taking some other input polytope(s), you can create the labels automatically if you call the function with a relabel option. The exact format of the labels is dependent on the construction, and is described in the corresponding help topic.
VERTEX_NORMALS: common::Matrix<Scalar>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. Alias for property RAY_SEPARATORS.
ZONOTOPE_INPUT_VECTORS: common::Matrix<Scalar>DOC_FIXME: Incomprehensible! Contains the vector configuration for which a zonotope can be built.
- CombinatoricsALTSHULER_DET: common::Integer
Let M be the vertex-facet incidence matrix, then the Altshulter determinant is defined as max{det(M ∗ MT), det(MT ∗ M)}.
DUAL_BOUNDED_H_VECTOR: common::Vector<Integer>h-vector of the bounded subcomplex, defined for not necessarily bounded polyhedra which are simple (as polyhedra, i.e., VERTEX_DEGREES on the FAR_FACE do not matter). Coincides with the reverse h-vector of the dual simplicial ball. Note that this vector will usually start with a number of zero entries.
DUAL_H_VECTOR: common::Vector<Integer>h-vector, defined via recursion on the face lattice of a polytope. Coincides for simple polytopes with the combinatorial definition of the h-vector via abstract objective functions.
F2_VECTOR: common::Matrix<Integer>fik is the number of incident pairs of i-faces and k-faces; the main diagonal contains the F_VECTOR.
FACETS_THRU_VERTICES: common::IncidenceMatrix<NonSymmetric>transposed VERTICES_IN_FACETS Notice that this is a temporary property; it will not be stored in any file. Alias for property FACETS_THRU_RAYS.
G_VECTOR: common::Vector<Integer>(Toric) g-vector, defined via the (generalized) h-vector as gi = hi - hi-1.
H_VECTOR: common::Vector<Integer>h-vector, defined via recursion on the face lattice of a polytope. Coincides for simplicial polytopes with the combinatorial definition of the h-vector via shellings
N_VERTEX_FACET_INC: common::IntNumber of pairs of incident vertices and facets. Alias for property N_RAY_FACET_INC.
SUBRIDGE_SIZES: common::Map<Int, Int>Lists for each occurring size (= number of incident facets or ridges) of a subridge how many there are.
TWO_FACE_SIZES: common::Map<Int, Int>Lists for each occurring size (= number of incident vertices or edges) of a 2-face how many there are.
VERTEX_SIZES: common::Array<Int>Number of incident facets for each vertex. Alias for property RAY_SIZES.
VERTICES_IN_FACETS: common::IncidenceMatrix<NonSymmetric>Vertex-facet incidence matrix, with rows corresponding to facets and columns to vertices. Vertices and facets are numbered from 0 to N_VERTICES-1 rsp. N_FACETS-1, according to their order in VERTICES rsp. FACETS.
This property is at the core of all combinatorial properties. It has the following semantics: (1) The combinatorics of an unbounded and pointed polyhedron is defined to be the combinatorics of the projective closure. (2) The combiantorics of an unbounded polyhedron which is not pointed is defined to be the combinatorics of the quotient modulo the lineality space. Therefore: VERTICES_IN_FACETS and each other property which is grouped under "Combinatorics" always refers to some polytope. Alias for property RAYS_IN_FACETS.
- Convex hull computationINEQUALITIES_THRU_VERTICES: common::IncidenceMatrix<NonSymmetric>
transposed VERTICES_IN_INEQUALITIES Alias for property INEQUALITIES_THRU_RAYS.
POINTS_IN_FACETS: common::IncidenceMatrix<NonSymmetric>Similar to VERTICES_IN_FACETS, but with columns corresponding to POINTS instead of VERTICES. This property is a byproduct of convex hull computation algorithms. It is discarded as soon as VERTICES_IN_FACETS is computed. Alias for property INPUT_RAYS_IN_FACETS.
VERTICES_IN_INEQUALITIES: common::IncidenceMatrix<NonSymmetric>Similar to VERTICES_IN_FACETS, but with rows corresponding to INEQUALITIES instead of FACETS. This property is a byproduct of convex hull computation algorithms. It is discarded as soon as VERTICES_IN_FACETS is computed. Alias for property RAYS_IN_INEQUALITIES.
- PolarizationFAR_HYPERPLANE: common::Vector<Scalar>
Valid strict inequality for all affine points of the polyhedron.
- Triangulation and volumePOLYTOPAL_SUBDIVISION: common::Array<Set<Int>>
Polytopal Subdivision of the polytope using only its vertices. Each line contains indices from VERTICES comprising a facet of the subdivision.
- Unbounded polyhedraBOUNDED_COMPLEX: common::IncidenceMatrix<NonSymmetric>
Bounded subcomplex. FIXME: type should be a newly defined "PolyhedralComplex"
BOUNDED_GRAPH: graph::Graph<Undirected>Graph of the bounded subcomplex.
Properties of BOUNDED_GRAPH
- EDGE_COLORS: common::EdgeMap<Undirected, Int>
Each edge indicates the maximal dimension of a bounded face containing it. Mainly used for visualization purposes.
- EDGE_DIRECTIONS: common::EdgeMap<Undirected, Vector<Scalar>>
Difference of the vertices for each edge (only defined up to signs).
BOUNDED_HASSE_DIAGRAM: graph::FaceLatticeHASSE_DIAGRAM constrained to affine vertices Nodes representing the maximal inclusion-independent faces are connected to the top-node regardless of their dimension
TOWARDS_FAR_FACE: common::Vector<Scalar>A linear objective function for which each unbounded edge is increasing; only defined for unbounded polyhedra.
- VisualizationFTV_CYCLIC_NORMAL: common::Array<List<Int>>
Reordered transposed VERTICES_IN_FACETS. Dual to VIF_CYCLIC_NORMAL.
NEIGHBOR_FACETS_CYCLIC_NORMAL: common::Array<List<Int>>Reordered DUAL_GRAPH for 3d-polytopes. The neighbor facets are listed in the order corresponding to VIF_CYCLIC_NORMAL, so that the first two vertices in VIF_CYCLIC_NORMAL make up the ridge to the first neighbor facet and so on.
NEIGHBOR_VERTICES_CYCLIC_NORMAL: common::Array<List<Int>>Reordered GRAPH. Dual to NEIGHBOR_FACETS_CYCLIC_NORMAL.
SCHLEGEL_DIAGRAM: SchlegelDiagram<Scalar>Holds one special projection (the Schlegel diagram) of the polytope.
VIF_CYCLIC_NORMAL: common::Array<List<Int>>Reordered VERTICES_IN_FACETS for 2d and 3d-polytopes. Vertices are listed in the order of their appearance when traversing the facet border counterclockwise seen from outside of the polytope.
For a 2d-polytope (which is a closed polygon), lists all vertices in the border traversing order.
User Methods of Polytope
- AMBIENT_DIM ()
UNDOCUMENTED
- DIM ()
UNDOCUMENTED
- labeled_vertices (label ...) → Set<Int>
Find the vertices by given labels.
- CombinatoricsN_RIDGES ()
The number of ridges (faces of codimension 2) of the polytope equals the number of edges of the DUAL_GRAPH
- FormattingCD_INDEX ()
Prettily print the cd-index given in CD_INDEX_COEFFICIENTS
- Triangulation and volumeTRIANGULATION_INT_SIGNS () → Array<Int>
the orientation of the simplices of TRIANGULATION_INT in the given order of the POINTS
Returns
Array<Int> - +1/-1 array specifying the sign of the determinant of each simplexTRIANGULATION_SIGNS () → Array<Int>For each simplex in the TRIANGULATION, contains the sign of the determinant of its coordinate matrix, telling about its orientation.
Returns
Array<Int> - Unbounded polyhedraBOUNDED_FACETS () → Set<Int>
Indices of FACETS that are bounded.
Returns
Set<Int> BOUNDED_VERTICES () → Set<Int>Indices of VERTICES that are no rays.
Returns
Set<Int> - VisualizationGALE () → Visual::Gale
Generate the Gale diagram of a d-polyhedron with at most d+4 vertices.
Returns
Visual::Gale SCHLEGEL () → Visual::SchlegelDiagramCreate a Schlegel diagram and draw it.
Options
FIXME proj_facet decorations for the Edges of the projection faceoption list: schlegel_init option list: Visual::Wire::decorations Returns
Visual::SchlegelDiagram VISUAL () → Visual::PolytopeVisualize a polytope as a graph (if 1d), or as a solid object (if 2d or 3d), or as a Schlegel diagram (4d).
VISUAL_BOUNDED_GRAPH () → Visual::PolytopeGraphVisualize the BOUNDED_GRAPH of a polyhedron.
Options
Int seed random seed value for the string embedderoption list: Visual::Graph::decorations Returns
Visual::PolytopeGraph VISUAL_DUAL () → Visual::PolygonsOptions
option list: Visual::Polygons::decorations Returns
Visual::Polygons VISUAL_DUAL_FACE_LATTICE () → Visual::PolytopeLatticeVisualize the dual face lattice of a polyhedron as a multi-layer graph.
Options
Int seed random seed value for the node placementoption list: Visual::Lattice::decorations Returns
Visual::PolytopeLattice VISUAL_DUAL_GRAPH () → Visual::GraphVisualize the DUAL_GRAPH of a polyhedron.
Options
Int seed random seed value for the string embedderoption list: Visual::Graph::decorations Returns
Visual::Graph VISUAL_FACE_LATTICE () → Visual::PolytopeLatticeVisualize the HASSE_DIAGRAM of a polyhedron as a multi-layer graph.
Options
Int seed random seed value for the node placementoption list: Visual::Lattice::decorations Returns
Visual::PolytopeLattice VISUAL_GRAPH () → Visual::PolytopeGraphVisualize the GRAPH of a polyhedron.
Options
Int seed random seed value for the string embedderoption list: Visual::Graph::decorations Returns
Visual::PolytopeGraph VISUAL_TRIANGULATION_BOUNDARY () → Visual::PolygonsVisualize the TRIANGULATION_BOUNDARY of the polytope. Obsolete: the preferred procedure is to create a SimplicialComplex using the boundary_complex client of the application <a target="_top" href="../../topaz/model.html">topaz</a> and call its VISUAL method.
Options
option list: Visual::Polygon::decorations Returns
Visual::Polygons
- Polytope<Float>derived from: Polytope
A pointed polyhedron realized in Rd.
Convex hull and related algorithms use floating-point arithmetics. Due to numerical errors inherent to this kind of computations, the resulting combinatorial description can be arbitrarily far away from the truth, or even not correspond to any valid polytope. You have been warned.
None of the standard construction clients produces objects of this type. If you want to get one, create it with the explicit constructor or convert_to.
derived from: PolytopePolytope<Rational>Properties of Polytope<Rational>
- BOUNDARY_LATTICE_POINTS: common::Matrix<Integer>
The lattice points on the boundary of the polytope, including the vertices
- INTERIOR_LATTICE_POINTS: common::Matrix<Integer>
The lattice points strictly in the interior of the polytope
- MINKOWSKI_SUMMAND_CONE: Cone<Rational>
The cone of all Minkowski summands of the polytope P. Up to scaling, a polytope S is a Minkowski summand of P if and only if the edge directions of S are a subset of those of P, and the closing condition around any 2-face of P is preserved. Coordinates of the cone correspond to the rescaled lengths of the edges of the graph of P (in the order given by the property EDGES of the GRAPH of P). The Minkowski cone is defined as the intersection of all equations given by the closing condition around 2-faces with the positive orthant. For more information see e.g.
Klaus Altmann: The versal deformation of an isolated toric Gorenstein singularity - Lattice polytopesLATTICE: common::Bool
This is the defining property: A polytope is lattice if each vertex has integer coordinates.
User Methods of Polytope<Rational>
- Lattice polytopesMINKOWSKI_SUMMAND_COEFF (coeff) → Polytope<Rational>
returns the Minkowski summand of a polytope P given by a coefficient vector to the rays of the MINKOWSKI_SUMMAND_CONE.
Parameters
Vector<Rational> coeff coefficient vector to the rays of the Minkowski summand coneReturns
Polytope<Rational> MINKOWSKI_SUMMAND_POINT (point) → Polytope<Rational>returns the Minkowski summand of a polytope P given by a point in the MINKOWSKI_SUMMAND_CONE.
- PropagatedPolytopederived from: Polytope<Rational>
Polytope propagation means to define a polytope inductively by assigning vectors to arcs of a directed graph. At each node of such a graph a polytope arises as the joint convex hull of the polytopes at the translated sources of the inward pointing arcs.
For details see
Joswig: Polytope Propagation on Graphs.Chapter 6 inPachter/Sturmfels: Algebraic Statistics for Computational Biology, Cambridge 2005.Properties of PropagatedPolytope
- SUM_PRODUCT_GRAPH: graph::Graph<Directed>
Directed graph to define the propagated polytope. There is a (translation) vector assigned to each arc. We assume that this graph is acyclic with a unique sink.
Properties of SUM_PRODUCT_GRAPH
- SchlegelDiagram
A Schlegel diagram of a polytope
Properties of SchlegelDiagram
- FACET_POINT: common::Vector<Scalar>
The intersection point of the projection facet and the view ray.
- ROTATION: common::Matrix<Float>
Rotation matrix making the projection facet coinciding with (0 0 0 -1) We want a negatively oriented coordinate system since the view point lies on the negative side of the facet.
- TRANSFORM: common::Matrix<Scalar>
Matrix of a projective transformation mapping the whole polytope into the FACET The points belonging to this facet stay fixed.
- VERTICES: common::Matrix<Float>
Coordinates in affine 3-space of the vertices which correspond to a 3-dimensional (Schlegel-) projection of a 4-polytope.
User Methods of SchlegelDiagram
- VISUAL () → Visual::SchlegelDiagram
Draw the Schlegel diagram
Options
pm::perl::Hash proj_facet decoration for the edges of the projection faceoption list: Visual::Graph::decorations Returns
Visual::SchlegelDiagram
- TightSpanderived from: Polytope
Bounded subcomplex of an unbounded polyhedron, which is associated with a finite metric space. The tight span is 1-dimensional if and only if the metric is tree-like. In this sense, the tight span captures the deviation of the metric from a tree-like one.
Properties of TightSpan
- Phylogenetic AnalysisTAXA: common::Array<String>
Labels for the rows and columns of the METRIC space. Default TAXA are just consecutive numbers.
User Methods of TightSpan
- VisualizationVISUAL_BOUNDED_GRAPH () → Visual::PolytopeGraph
Visualize the BOUNDED_GRAPH of a tight span.
Options
Int seed random seed value for the string embedderoption list: Visual::Graph::decorations Returns
Visual::PolytopeGraph VISUAL_SPLITS () → Visual::FiniteMetricSpaceVisualize the splits of a finite metric space (that is, a planar image of a tight span). Calls SplitsTree.
Returns
Visual::FiniteMetricSpace VISUAL_TIGHT_SPAN () → Visual::GraphThis is a variation of Polytope::VISUAL_BOUNDED_GRAPH for the special case of a tight span. The tropical vertices are embedded according to the METRIC, the others are hanged in between.
Category: VisualizationVisual::PointConfigurationVisualization of the point configuration.
User Methods of Visual::PointConfiguration
- VisualizationPOLYTOPAL_SUBDIVISION () → Visual::PointConfiguration
Visualize the POLYTOPAL_SUBDIVISION of a point configuration
TRIANGULATION () → Visual::PointConfigurationVisualize the TRIANGULATION of a point configuration
TRIANGULATION_BOUNDARY () → Visual::PointConfigurationDraw the edges of the TRIANGULATION_BOUNDARY. The facets are made transparent.
Category: VisualizationVisual::PolytopeVisualization of a polytope as a graph (if 1d), or as a solid object (if 2d or 3d), or as a Schlegel diagram (4d).
User Methods of Visual::Polytope
- DIRECTED_GRAPH (lp) → Visual::Polytope
Illustrate the behavior of a linear objective function on the polytope. Superpose the drawing with the directed graph induced by the objective function.
- MIN_MAX_FACE (lp) → Visual::Polytope
Illustrate the behavior of a linear objective function on the polytope. Draw the facets contained in MAXIMAL_FACE and MINIMAL_FACE in distinct colors.
Parameters
LinearProgram lp a LinearProgram object attached to the polytope.Options
Color min minimal face decoration (default: yellow vertices and/or facets)Color max maximal face decoration (default: red vertices and/or facets)Returns
Visual::Polytope - VERTEX_COLORS (lp) → Visual::Polytope
Illustrate the behavior of a linear objective function on the polytope. Color the vertices according to the values of the objective function.
Parameters
LinearProgram lp a LinearProgram object attached to the polytopeOptions
Color min minimal vertex color (default: yellow)Color max maximal vertex color (default: red)Returns
Visual::Polytope - VisualizationLATTICE () → Visual::Polytope
Visualize the LATTICE_POINTS of a polytope
LATTICE_COLORED () → Visual::PolytopeVisualize the LATTICE_POINTS of a polytope in different colors (interior / boundary / vertices)
STEINER () → Visual::PolytopeAdd the STEINER_POINTS to the 3-d visualization. The facets become transparent.
TRIANGULATION (TR) → Visual::PolytopeAdd the triangulation to the drawing. The facets of the whole polytope become transparent.
You may specify any triangulation of the current polytope. Per default, the TRIANGULATION property is taken. (Currently there is only one possible alternative triangulation: TRIANGULATION_INT).
Hint: Use the method Method -> Effect -> Explode Group of Geometries of JavaView for better insight in the internal structure.
Parameters
Array<Set<Int>> TR facets of the triangulationOptions
option list: Visual::Polygons::decorations Returns
Visual::Polytope TRIANGULATION_BOUNDARY () → Visual::PolytopeDraw the edges of the Polytope::TRIANGULATION_BOUNDARY. The facets are made transparent.
Category: VisualizationVisual::PolytopeGraphVisualization of the graph of a polyhedron.
User Methods of Visual::PolytopeGraph
- VisualizationDIRECTED_GRAPH (lp) → Visual::PolytopeGraph
Show the growth direction of a linear objective function via arrowed edges.
Parameters
LinearProgram lp a LinearProgram object attached to the polytopeReturns
Visual::PolytopeGraph EDGE_COLORS () → Visual::PolytopeGraphproduce an edge coloring of a bounded graph from local data in the Hasse diagram
Returns
Visual::PolytopeGraph MIN_MAX_FACE (lp) → Visual::PolytopeGraphIllustrate the behavior of a linear objective function on the polytope. The vertices belonging to MINIMAL_FACE and MAXIMAL_FACE are drawn in distinct colors
The spring embedder applies an additional force, which tries to arrange the nodes in the z-axis direction corresponding to the objective function values.
Parameters
LinearProgram lp a LinearProgram object attached to the polytopeOptions
Color min minimal face decoration (default: yellow nodes)Color max maximal face decoration (default: red nodes)Returns
Visual::PolytopeGraph VERTEX_COLORS (lp) → Visual::PolytopeGraphIllustrate the behavior of a linear objective function on the polytope. Color the nodes according to the value the objective function takes on the vertices.
The spring embedder applies an additional force, which tries to arrange the nodes in the z-axis direction corresponding to the objective function values.
Parameters
LinearProgram lp a LinearProgram object attached to the polytope.Options
Color min minimal face color (default: yellow)Color max maximal face color (default: red)Returns
Visual::PolytopeGraph
Category: VisualizationVisual::PolytopeLatticeVisualization of the HASSE_DIAGRAM of a polyhedron as a multi-layer graph..
User Methods of Visual::PolytopeLattice
- MIN_MAX_FACE (lp) → Visual::PolytopeLattice
Illustrate the behavior of a linear objective function on the polytope. Draw the filters of the MAXIMAL_FACE and MINIMAL_FACE in distinct colors.
Parameters
LinearProgram lp a LinearProgram object attached to the polytopeOptions
Color min minimal face decoration (default: yellow border and ingoing edges)Color max maximal face decoration (default: red border and ingoing edges)Returns
Visual::PolytopeLattice
Category: VisualizationVisual::SchlegelDiagramVisualization of the Schlegel diagram of a polytope.
User Methods of Visual::SchlegelDiagram
- STEINER ()
UNDOCUMENTED
Options
option list: Visual::PointSet::decorations - VisualizationCONSTRUCTION () → Visual::SchlegelDiagram
Visualize the construction of a 3D Schlegel diagram, that is, the Viewpoint, the 3-polytope and the projection onto one facet
DIRECTED_GRAPH (lp) → Visual::SchlegelDiagramIllustrate the behavior of a linear objective function on the polytope. Superpose the drawing with the directed graph induced by the objective function.
Parameters
LinearProgram lp a LinearProgram object attached to the polytope.Returns
Visual::SchlegelDiagram MIN_MAX_FACE (lp) → Visual::SchlegelDiagramIllustrate the behavior of a linear objective function on the polytope. The vertices belonging to MINIMAL_FACE and MAXIMAL_FACE are drawn in distinct colors
Parameters
LinearProgram lp a LinearProgram object attached to the polytope.Options
Color min minimal face decoration (default: yellow vertices and/or facets)Color max maximal face decoration (default: red vertices and/or facets)Returns
Visual::SchlegelDiagram SOLID () → Visual::SchlegelDiagramDraw the facets of the Schlegel diagram as polytopes.
TRIANGULATION_BOUNDARY () → Visual::SchlegelDiagramDraw the edges of the TRIANGULATION_BOUNDARY
TRIANGULATION_BOUNDARY_SOLID () → Visual::SchlegelDiagramDraw the boundary simplices of the triangulation as solid tetrahedra
VERTEX_COLORS (lp) → Visual::SchlegelDiagramIllustrate the behavior of a linear objective function on the polytope. Color the vertices according to the values of the objective function.
Parameters
LinearProgram lp a LinearProgram object attached to the polytope.Options
Color min minimal vertex color (default: yellow)Color max maximal vertex color (default: red)Returns
Visual::SchlegelDiagram
- VoronoiDiagramderived from: Polytope<Rational>
For a finite set of SITES S the Voronoi region of each site is the set of points closest (with respect to Euclidean distance) to the given site. All Voronoi regions (and their faces) form a polyhedral complex which is a vertical projection of the boundary complex of an unbounded polyhedron P(S). This way VoronoiDiagram becomes a derived class from Polytope<Rational>.
Properties of VoronoiDiagram
- DELAUNAY_TRIANGULATION: common::Array<Set<Int>>
Delaunay triangulation of the sites. (Delaunay subdivision, non-simplices are triangulated.)
- ITERATED_VORONOI_GRAPH: GeometricGraph
Graph of the joint Voronoi diagram of the SITES and the vertices of Vor(SITES). The coordinates (homogeneous, before projection) are stored as node attributes. The graph is truncated according to the VORONOI_GRAPH.BOUNDING_BOX. For the default BOUNDING_BOX it may happen that some of the iterated Voronoi vertices are truncated. Create new objects of type VoronoiDiagram to produce proper iterated Voronoi diagrams.
- NN_CRUST_GRAPH: graph::Graph<Undirected>
Graph of the nearest neighbor crust, as defined in:
T. K. Dey and P. Kumar: A simple provable algorithm for curve reconstruction.Proc. 10th. Annu. ACM-SIAM Sympos. Discrete Alg., 1999, 893-894.Polygonal reconstruction of a smooth planar curve from a finite set of samples. Sampling rate of <= 1/3 suffices.
- NN_GRAPH: graph::Graph<Undirected>
Graph of the nearest neighbors. This is a subgraph of NN_CRUST_GRAPH.
- SITES: common::Matrix<Rational>
Coordinates of the sites in case the polyhedron is Voronoi. Sites must be pairwise distinct.
- VORONOI_GRAPH: GeometricGraph
Graph of the Voronoi diagram of the SITES. The homogeneous coordinates after projection are stored as node attributes. The graph is truncated according to the BOUNDING_BOX. All vertices of the Voronoi diagram are visible (and represented in the VORONOI_GRAPH) for the default BOUNDING_BOX.
User Methods of VoronoiDiagram
- VisualizationVISUAL_CRUST () → Visual::Container
Draw a Voronoi diagram, its |dual graph and the crust. Use the interactive features of the viewer to select.
Options
option list: Visual::Graph::decorations Returns
Visual::Container VISUAL_NN_CRUST () → Visual::ContainerDraw a Voronoi diagram, its dual graph and the nearest neighbor crust. Use the interactive features of the viewer to select.
Options
option list: Visual::Graph::decorations Returns
Visual::Container VISUAL_VORONOI () → Visual::ContainerDraw a Voronoi diagram and its dual. Use the interactive features of the viewer to select.
Options
option list: Visual::Graph::decorations Returns
Visual::Container
User Functions
- tutte_lifting (G) → Polytope
Let G be a 3-connected planar graph. If the corresponding polytope contains a triangular facet (ie. the graph contains a non- separating cycle of length 3), the client produces a realization in R3.
- Comparingcongruent (P1, P2)
Check whether two given polytopes P1 and P2 are congruent, i.e. whether there is an affine isomorphism between them that is induced by a (possibly scaled) orthogonal matrix. Returns the scale factor, or 0 if the polytopes are not congruent.
We are using the reduction of the congruence problem (for arbitrary point sets) to the graph isomorphism problem due to:
Akutsu, T.: On determining the congruence of point sets in `d` dimensions.Comput. Geom. Theory Appl. 9, 247--256 (1998), no. 4equal_polyhedra (P1, P2)Tests if the two polyhedra P1 and P2 are equal.
find_facet_vertex_permutations (P1, P2) → Pair<Array<Int>, Array<Int>>Find the permutations of facets and vertices which maps the polyhedron P1 to P2. The facet permutation is the first component of the return value.
Only the combinatorial isomorphism is considered. If the polytopes are not isomorphic, an exception is thrown.
included_polyhedra (P1, P2)Tests if polyhedron P1 is included in polyhedron P2.
isomorphic (P1, P2)Check whether the face lattices of two polytopes are isomorphic. The problem is reduced to graph isomorphism of the vertex-facet incidence graphs.
lattice_isomorphic_smooth_polytopes (P1, P2)Tests whether two smooth lattice polytopes are lattice equivalent by comparing lattice distances between vertices and facets.
Parameters
LatticePolytope P1 LatticePolytope P2 - Consistency checkcheck_inc (points, hyperplanes, sign, verbose)
Check coordinate data. For each pair of vectors from two given matrices their inner product must satisfy the given relation.
check_poly (VIF) → PolytopeTry to check whether a given vertex-facet incidence matrix VIF defines a polytope. Note that a successful certification by check_poly is not sufficient to determine whether an incidence matrix actually defines a polytope. Think of it as a plausibility check.
Parameters
IncidenceMatrix VIF Options
Bool dual transposes the incidence matrixBool verbose prints information about the check.Returns
Polytope - Constructing a conenormal_cone (p, v)
Computes the outer normal cone of p at the vertex v.
Parameters
Polytope p int v vertex number - Coordinate conversionsaffine_float_coords (P) → Matrix<Float>
dehomogenize the vertex coordinates and convert them to Float
convert_to <Coord> (c) → Cone<Coord>Creates a new Cone object with different coordinate type target coordinate type Coord must be specified in angle brackets e.g. $new_cone = convert_to<Coord>($cone)
Type Parameters
Coord target coordinate typeParameters
Cone c the input coneReturns
Cone<Coord> a new cone object or C itself it has the requested typeconvert_to <Coord> (P) → Polytope<Coord>provide a Polytope object with desired coordinate type
Type Parameters
Coord target coordinate typeParameters
Polytope P source objectReturns
Polytope<Coord> P if it already has the requested type, a new object otherwise - Delaunay subdivisions and Voronoi diagramsdelaunay_triangulation (V) → Array<Set<Int>>
Compute the (a) Delaunay triangulation of the given SITES of a VoronoiDiagram V. If the sites are not in general position, the non-triangular facets of the Delaunay subdivision are triangulated (by applying the beneath-beyond algorithm).
voronoi (V) → MatrixCompute the inequalities of the Voronoi polyhedron of a given VoronoiDiagram V. The polyhedron is always unbounded. Introduce artificial cut facets later (e.g., for visualization); this must be done after the vertices have been computed.
- Formattingprint_constraints (P) → bool
Write the FACETS / INEQUALITIES and the AFFINE_HULL / EQUATIONS of a polytope P in a readable way. COORDINATE_LABELS are adopted if present.
- Linear optimizationdgraph (P, LP) → Graph<Directed>
Direct the graph of a polytope P according to a linear or abstract objective function. The maximal and minimal values, which are attained by the objective function, as well as the minimal and the maximal face are written into separate sections.
The option inverse directs the graph with decreasing instead of increasing edges. If the option generic is set, ties will be broken by lexicographical ordering.
Parameters
Polytope P LinearProgram LP Options
Bool inverse inverts the directionBool generic breaks tiesReturns
Graph<Directed> inner_point (points)Compute a true inner point of a convex hull of the given set of points.
Parameters
Matrix points random_edge_epl (G) → Vector<Rational>Computes a vector containing the expected path length to the maximum for each vertex of a directed graph G. The random edge pivot rule is applied.
rand_aof (P, start) → Vector<Rational>Produce a random abstract objective function on a given simple polytope P. It is assumed that the boundary complex of the dual polytope is extendibly shellable. If, during the computation, it turns out that a certain partial shelling cannot be extended, then this is given instead of an abstract objective function. It is possible (but not required) to specify the index of the starting vertex start.
Parameters
Polytope P a simple polytopeInt start the index of the starting vertex; default value: randomOptions
Int seed controls the outcome of the random number generator; fixing a seed number guarantees the same outcome.Returns
Vector<Rational> rel_int_point (P, unbounded, affine_hull)Computes a relative interior point of a polyhedron P and stores it in P->REL_INT_POINT. The unbounded flag needs to be set to true if the polyhedron could be unbounded.
vertex_colors (P, LP) → Array<RGB>Calculate RGB-color-values for each vertex depending on a linear or abstract objective function. Maximal and minimal affine vertices are colored as specified. Far vertices (= rays) orthogonal to the linear function normal vector are white. The colors for other affine vertices are linearly interpolated in the HSV color model.
If the objective function is linear and the corresponding LP problem is unbounded, then the affine vertices that would become optimal after the removal of the rays are painted pale.
Parameters
Polytope P LinearProgram LP Options
RGB min the minimal RGB valueRGB max the maximal RGB valueReturns
Array<RGB> - Metric propertiesall_steiner_points (P) → Matrix
Compute the Steiner points of all faces of a polyhedron P using a randomized approximation of the angles. P must be BOUNDED.
steiner_point (P) → VectorCompute the Steiner point of a polyhedron P using a randomized approximation of the angles.
- Optimizationcore_point_algo (p, optLPvalue) → perl::ListReturn
Algorithm to solve highly symmetric integer linear programs (ILP). It is required that the group of the ILP induces the alternating or symmetric group on the set of coordinate directions. The linear objective function is the vector (0,1,1,..,1).
find_transitive_lp_sol (Inequalities) → perl::ListReturnAlgorithm to solve symmetric linear programs (LP) of the form max ctx , c=(0,1,1,..,1) subject to the inequality system given by Inequalities. It is required that the symmetry group of the LP acts transitively on the coordinate directions.
Parameters
Matrix Inequalities the inequalities describing the feasible regionReturns
perl::ListReturn (optLPsolution,optLPvalue,feasible,max_bounded) - Otherinduced_lattice_basis (p) → Matrix<Integer>
Returns a basis of the affine lattice spanned by the vertices
minimal_vertex_angle (P) → FloatComputes the minimal angle between two vertices of the input polytope P.
primitive (M) → Matrix<Integer>scales each row of the matrix to a primitive integral vector
primitive (v) → Vector<Integer>scales the vector to a primitive inegral vector
primitive_affine (M) → Matrix<Integer>scales the affine part of each row of the matrix to a primitive integral vector
primitive_affine (v) → Vector<Integer>scales the affine part of a vector to a primitive inegral vector
- Polytope Propagationbinary_markov_graph (observation) → PropagatedPolytope
Defines a very simple graph for a polytope propagation related to a Hidden Markov Model. The propagated polytope is always a polygon. For a detailed description see
M. Joswig: Polytope propagation, in: Algebraic statistics and computational biologyby L. Pachter and B. Sturmfels (eds.), Cambridge, 2005.binary_markov_graph (observation)Parameters
String observation encoded as a string of "0" and "1". - Producing a new polyhedron from othersbipyramid (P, z, z_prime)
Make a bipyramid over a polyhedron. The bipyramid is the convex hull of the input polyhedron P and two points (v, z), (v, z_prime) on both sides of the affine span of P. For bounded polyhedra, the apex projections v to the affine span of P coincide with the vertex barycenter of P.
Parameters
Polytope P Rational z distance between the vertex barycenter and the first apex, default value is 1.Rational z_prime distance between the vertex barycenter and the second apex, default value is -z.Options
Bool noc : don't compute the coordinates, purely combinatorial description is produced.Bool relabel copy the vertex labels from the original polytope, label the new vertices with "Apex" and "Apex'".blending (P1, v1, P2, v2) → PolytopeCompute the blending of two polyhedra at simple vertices. This is a slightly less standard construction. A vertex is simple if its vertex figure is a simplex.
Moving a vertex v of a bounded polytope to infinity yields an unbounded polyhedron with all edges through v becoming mutually parallel rays. Do this to both input polytopes P1 and P2 at simple vertices v1 and v2, respectively. Up to an affine transformation one can assume that the orthogonal projections of P1 and P2 in direction v1 and v2, respectively, are mutually congruent.
Any bijection b from the set of edges through v1 to the edges through v2 uniquely defines a way of glueing the unbounded polyhedra to obtain a new bounded polytope, the blending with respect to b. The bijection is specified as a permutation of indices 0 1 2 etc. The default permutation is the identity.
The number of vertices of the blending is the sum of the numbers of vertices of the input polytopes minus 2. The number of facets is the sum of the numbers of facets of the input polytopes minus the dimension.
The resulting polytope is described only combinatorially.
cayley_embedding (P, P_prime, z, z_prime) → PolytopeCreate a Cayley embedding of two polytopes. The vertices of the first polytope P get an extra coordinate z and the vertices of the second polytope P_prime get z_prime.
Default values are z=1 and z_prime=-z.
The option relabel creates an additional section VERTEX_LABELS. The vertices of P inherit the original labels unchanged; the vertex labels of P_prime get a tick (') appended.
cayley_polytope (P_Array) → PolytopeConstruct the cayley polytope of a set of lattice polytopes contained in P_Array which is the convex hull of P1×e1, ..., Pk×ek where e1, ...,ek are the standard unit vectors in Rk. In this representation the last k coordinates always add up to 1. The option proj projects onto the complement of the last coordinate.
Parameters
Array<LatticePolytope> P_Array an array containing the lattice polytopes P1,...,PkOptions
Bool proj Returns
Polytope cells_from_subdivision (P, cells) → PolytopeExtract the given cells of the subdivision of a polyhedron and write their union as a new polyhedron.
cell_from_subdivision (P, cell) → PolytopeExtract the given cell of the subdivision of a polyhedron and write it as a new polyhedron.
conv (P_Array) → PropagatedPolytopeConstruct a new polyhedron as the convex hull of the polyhedra given in P_Array.
edge_middle (P) → PolytopeProduce the convex hull of all edge middle points of some polytope P. The polytope must be BOUNDED.
facet_to_infinity (i) → PolytopeMake an affine transformation such that the i-th facet is transformed to infinity
intersection (C ...) → ConeConstruct a new polyhedron or cone as the intersection of given polyhedra and/or cones. Works only if all AMBIENT_DIM values are equal. If the input contains both cones and polytopes, the output will be a polytope. (In this case the condition on AMBIENT_DIM means that the CONE_AMBIENT_DIM of the cones has to match the POLYTOPE_AMBIENT_DIM of the polytopes.)
join_polytopes (P1, P2) → PolytopeConstruct a new polyhedron as the join of two given ones.
lattice_bipyramid (P, v, v_prime, z, z_prime) → PolytopeMake a lattice bipyramid over a polyhedron. The bipyramid is the convex hull of the input polyhedron P and two points (v, z), (v_prime, z_prime) on both sides of the affine span of P.
Parameters
Polytope P Vector v basis point for the first apexVector v_prime basis for the second apex If v_prime is omitted, v will be used for both apices. If both v and v_prime are omitted, it tries to find two vertices which don't lie in a common facet. If no such vertices can be found or P is a simplex, it uses an interior lattice point as both v and v_prime.Rational z height for the first apex, default value is 1Rational z_prime hieght for the second apex, default value is -zOptions
Bool relabel copy the vertex labels from the original polytope, label the new vertices with "Apex" and "Apex'".Returns
Polytope lattice_pyramid (P, z, v) → PolytopeMake a lattice pyramid over a polyhedron. The pyramid is the convex hull of the input polyhedron P and a point v outside the affine span of P.
mapping_polytope (P1, P2) → PolytopeConstruct a new polytope as the mapping polytope of two polytopes P1 and P2. The mapping polytope is the set of all affine maps from Rp to Rq, that map P1 into P2.
The label of a new facet corresponding to v1 and h1 will have the form "v1*h1".
minkowski_sum (lambda, P1, mu, P2) → PolytopeProduces the polytope lambda*P1+mu*P2, where * and + are scalar multiplication and Minkowski addition, respectively.
minkowski_sum (P1, P2) → PolytopeProduces the Minkowski sum of P1 and P2.
prism (P, z1, z2) → PolytopeMake a prism over a polytope. The prism is the product of the input polytope P and the interval [z1, z2].
Parameters
Polytope P the input polytopeRational z1 the left endpoint of the interval; default value: -1Rational z2 the right endpoint of the interval; default value: -z1Options
Bool noc only combinatorial information is handledBool relabel creates an additional section VERTEX_LABELS; the bottom facet vertices get the labels from the original polytope; the labels of their clones in the top facet get a tick (') appended.Returns
Polytope product (P1, P2) → PolytopeConstruct a new polytope as the product of two given polytopes P1 and P2.
projection (P, indices) → PolytopeOrthogonally project a polyhedron to a coordinate subspace.
The subspace the polyhedron P is projected on is given by the coordinate indices in the set indices. The option revert inverts the coordinate list. The client scans for all coordinate sections and produces proper output from each. If a description in terms of inequalities is found, the client performs Fourier-Motzkin elimination unless the nofm option is set. Setting the nofm option is useful if the corank of the projection is large; in this case the number of inequalities produced grows quickly.
Parameters
Polytope P Array<Int> indices Options
Bool revert inverts the coordinate listBool nofm suppresses Fourier-Motzkin eliminationReturns
Polytope projection_full (P) → PolytopeOrthogonally project a polyhedron to a coordinate subspace such that "redundant" columns are omitted, i.e., the projection becomes full-dimensional without changing the combinatorial type. The client scans for all coordinate sections and produces proper output from each. If a description in terms of inequalities is found, the client performs Fourier-Motzkin elimination unless the nofm option is set. Setting the nofm option is useful if the corank of the projection is large; in this case the number of inequalities produced grows quickly.
pyramid (P, z) → PolytopeMake a pyramid over a polyhedron. The pyramid is the convex hull of the input polyhedron P and a point v outside the affine span of P. For bounded polyhedra, the projection of v to the affine span of P coincides with the vertex barycenter of P.
rand_inner_points (P, n) → PolytopeProduce a polytope with n random points from the input polytope P. Each generated point is a convex linear combination of the input vertices with uniformly distributed random coefficients. Thus, the output points can't in general be expected to be distributed uniformly within the input polytope; cf. unirand for this. The polytope must be BOUNDED.
rand_vert (V, n) → MatrixSelects n random vertices from the set of vertices V. This can be used to produce random polytopes which are neither simple nor simplicial as follows: First produce a simple polytope (for instance at random, by using rand_sphere, rand, or unirand). Then use this client to choose among the vertices at random. Generalizes random 0/1-polytopes.
stack (P, stack_facets) → PolytopeStack a (simplicial or cubical) polytope over one or more of its facets.
For each facet of the polytope P specified in stack_facets, the barycenter of its vertices is lifted along the normal vector of the facet. In the simplicial case, this point is directly added to the vertex set, thus building a pyramid over the facet. In the cubical case, this pyramid is truncated by a hyperplane parallel to the original facet at its half height. This way, the property of being simplicial or cubical is preserved in both cases.
The option lift controls the exact coordinates of the new vertices. It should be a rational number between 0 and 1, which expresses the ratio of the distance between the new vertex and the stacked facet, to the maximal possible distance. When lift=0, the new vertex would lie on the original facet. lift=1 corresponds to the opposite extremal case, where the new vertex hit the hyperplane of some neighbor facet. As an additional restriction, the new vertex can't lie further from the facet as the vertex barycenter of the whole polytope. Alternatively, the option noc (no coordinates) can be specified to produce a pure combinatorial description of the resulting polytope.
Parameters
Polytope P Set<Int> stack_facets the facets to be stacked; A single facet to be stacked is specified by its number. Several facets can be passed in a Set or in an anonymous array of indices: [n1,n2,...] Special keyword All means that all factes are to be stacked.Options
Rational lift controls the exact coordinates of the new vertices; rational number between 0 and 1; default value: 1/2Bool noc produces a pure combinatorial description (in contrast to lift)Bool relabel creates an additional section VERTEX_LABELS; New vertices get labels 'f(FACET_LABEL)' in the simplicial case, and 'f(FACET_LABEL)-NEIGHBOR_VERTEX_LABEL' in the cubical case.Returns
Polytope stellar_all_faces (P, d) → PolytopePerform a stellar subdivision of all proper faces, starting with the facets.
Parameter d specifies the lowest dimension of the faces to be divided. It can also be negative, then treated as the co-dimension. Default is 1, that is, the edges of the polytope.
stellar_indep_faces (P, in_faces) → PolytopePerform a stellar subdivision of the faces in_faces of a polyhedron P.
The faces must have the following property: The open vertex stars of any pair of faces must be disjoint.
truncation (P, trunc_vertices) → PolytopeCut off one or more vertices of a polyhedron.
The exact location of the cutting hyperplane(s) can be controlled by the option cutoff, a rational number between 0 and 1. When cutoff=0, the hyperplane would go through the chosen vertex, thus cutting off nothing. When cutoff=1, the hyperplane touches the nearest neighbor vertex of a polyhedron.
Alternatively, the option noc (no coordinates) can be specified to produce a pure combinatorial description of the resulting polytope, which corresponds to the cutoff factor 1/2.
Parameters
Polytope P Set<Int> trunc_vertices the vertex/vertices to be cut off; A single vertex to be cut off is specified by its number. Several vertices can be passed in a Set or in an anonymous array of indices: [n1,n2,...] Special keyword All means that all vertices are to be cut off.Options
Rational cutoff controls the exact location of the cutting hyperplane(s); rational number between 0 and 1; default value: 1/2Bool noc produces a pure combinatorial description (in contrast to cutoff)Bool relabel creates an additional section VERTEX_LABELS; New vertices get labels of the form 'LABEL1-LABEL2', where LABEL1 is the original label of the truncated vertex, and LABEL2 is the original label of its neighbor.Returns
Polytope unirand (n) → PolytopeProduce a polytope with n random points that are uniformly distributed within the given polytope P. P must be full-dimensional.
vertex_figure (p, n) → PolytopeConstruct the vertex figure of the vertex n of a polyhedron. The vertex figure is dual to a facet of the dual polytope.
Parameters
Polytope p Int n number of the chosen vertexOptions
Rational cutoff controls the exact location of the cutting hyperplane. It should lie between 0 and 1. Value 0 would let the hyperplane go through the chosen vertex, thus degenerating the vertex figure to a single point. Value 1 would let the hyperplane touch the nearest neighbor vertex of a polyhedron. Default value is 1/2.Bool noc skip the coordinates computation, producing a pure combinatorial description.Bool relabel inherit vertex labels from the corresponding neighbor vertices of the original polytope.Returns
Polytope wedge (P, facet, z, z_prime) → PolytopeMake a wedge from a polytope over the given facet. The polytope must be bounded. The inclination of the bottom and top side facet is controlled by z and z_prime, which are heights of the projection of the old vertex barycenter on the bottom and top side facet respectively.
Parameters
Polytope P Int facet the `cutting edge'.Rational z default value is 0.Rational z_prime default value is -z, or 1 if z==0.Options
Bool noc don't compute coordinates, pure combinatorial description is produced.Bool relabel create vertex labels: The bottom facet vertices obtain the labels from the original polytope; the labels of their clones in the top facet get a tick (') appended.Returns
Polytope - Producing from othercut_polytope (G) → Polytope
Cut polytope of an undirected graph.
matching_polytope (G) → PolytopeMatching polytope of an undirected graph.
- Producing from scratchassociahedron (d) → Polytope
Produce a d-dimensional associahedron (or Stasheff polytope). We use the facet description given in Ziegler's book on polytopes, section 9.2.
create_24_cell () → PolytopeCreate the 24-cell polytope.
Returns
Polytope create_600_cell () → PolytopeCreate the 600-cell polytope. Cf. Coxeter, Introduction to Geometry, pp 403-404.
Returns
Polytope cube (d, x_up, x_low) → PolytopeProduce a d-dimensional cube. Regular polytope corresponding to the Coxeter group of type Bd-1 = Cd-1.
The bounding hyperplanes are xi <= x_up and xi >= x_low. Default values: x_up=1, x_low=-x_up or 1 if x_up==0.
cyclic (d, n, start) → PolytopeProduce a d-dimensional cyclic polytope with n points. Prototypical example of a neighborly polytope. Combinatorics completely known due to Gale's evenness criterion. Coordinates are chosen on the moment curve at integer steps from start, or 0 if not specified.
cyclic_caratheodory (d, n) → PolytopeProduce a d-dimensional cyclic polytope with n points. Prototypical example of a neighborly polytope. Combinatorics completely known due to Gale's evenness criterion. Coordinates are chosen on the trigonometric moment curve.
dwarfed_cube (d) → PolytopeProduce a d-dimensional dwarfed cube.
dwarfed_product_polygons (d, s) → PolytopeProduce a d-dimensional dwarfed product of polygons of size s.
goldfarb (d, e, g) → PolytopeProduces a d-dimensional Goldfarb cube if e<1/2 and g<=e/4. The Goldfarb cube is a combinatorial cube and yields a bad example for the Simplex Algorithm using the Shadow Vertex Pivoting Strategy. Here we use the description as a deformed product due to Amenta and Ziegler. For e<1/2 and g=0 we obtain the Klee-Minty cubes.
hypersimplex (k, d) → PolytopeProduce the hypersimplex Δ(k,d), that is the the convex hull of all 0/1-vector in Rd with exactly k 1s. Note that the output is never full-dimensional.
hypertruncated_cube (d, k, lambda) → PolytopeProduce a d-dimensional hypertruncated cube. With symmetric linear objective function (0,1,1,...,1).
knapsack (b) → PolytopeProduce a knapsack polytope defined by one linear inequality (and non-negativity constraints).
k_cyclic (n, s) → PolytopeProduce a (rounded) 2*k-dimensional k-cyclic polytope with n points, where k is the length of the input vector s. Special cases are the bicyclic (k=2) and tricyclic (k=3) polytopes. Only possible in even dimensions.
The parameters s_i can be specified as integer, floating-point, or rational numbers. The coordinates of the i-th point are taken as follows:
cos(s_1 * 2πi/n),sin(s_1 * 2πi/n),...cos(s_k * 2πi/n),sin(s_k * 2πi/n)Warning: Some of the k-cyclic polytopes are not simplicial. Since the components are rounded, this function might output a polytope which is not a k-cyclic polytope!
More information can be found in the following references:
P. Schuchert: "Matroid-Polytope und Einbettungen kombinatorischer Mannigfaltigkeiten",PhD thesis, TU Darmstadt, 1995.Z. Smilansky: "Bi-cyclic 4-polytopes",Isr. J. Math. 70, 1990, 82-92max_GC_rank (d) → PolytopeProduce a d-dimensional polytope of maximal Gomory-Chvatal rank Omega(d/log(d)), integrally infeasible. With symmetric linear objective function (0,1,1..,1). Construction due to Pokutta and Schulz.
multiplex (d, n) → PolytopeProduce a combinatorial description of a multiplex with parameters d and n. This yields a self-dual d-dimensional polytope with n+1 vertices.
They are introduced by
T. Bisztriczky,On a class of generalized simplices, Mathematika 43:27-285, 1996,see also
M.M. Bayer, A.M. Bruening, and J.D. Stewart,A combinatorial study of multiplexes and ordinary polytopes,Discrete Comput. Geom. 27(1):49--63, 2002.neighborly_cubical (d, n) → PolytopeProduce the combinatorial description of a neighborly cubical polytope. The facets are labelled in oriented matroid notation as in the cubical Gale evenness criterion.
See Joswig and Ziegler, Discr. Comput. Geom. 24:315-344 (2000).newton (p) → LatticePolytopeProduce the Newton polytope of a polynomial p.
permutahedron (d) → PolytopeProduce a d-dimensional permutahedron. The vertices correspond to the elements of the symmetric group of degree d+1.
pile (sizes) → PolytopeProduce a (d+1)-dimensional polytope from a pile of cubes. Start with a d-dimensional pile of cubes. Take a generic convex function to lift this polytopal complex to the boundary of a (d+1)-polytope.
Parameters
Vector<Int> sizes a vector (s1,...,sd, where si specifies the number of boxes in the i-th dimension.Returns
Polytope rand_box (d, n, b) → PolytopeComputes the convex hull of n points sampled uniformly at random from the integer points in the cube [0,b]d.
rand_metric <Scalar> (n) → MatrixProduce an n-point metric with random distances. The values are uniformily distributed in [1,2].
rand_sphere (d, n) → PolytopeProduce a d-dimensional polytope with n random vertices uniformly distributed on the unit sphere.
rss_associahedron (l) → PolytopeProduce a polytope of constrained expansions in dimension l according to
Rote, Santos, and Streinu: Expansive motions and the polytope of pointed pseudo-triangulations.Discrete and computational geometry, 699--736, Algorithms Combin., 25, Springer, Berlin, 2003.signed_permutahedron (d) → PolytopeProduce a d-dimensional signed permutahedron.
transportation (r, c) → PolytopeProduce the transportation polytope from two vectors r of length m and c of length n, i.e. all positive m×n Matrizes with row sums equal to r and column sums equal to c.
- Subdivisionsbarycentric_subdivision (pc) → PointConfiguration
Create a point configuration as a barycentric subdivision of a given one.
Parameters
PointConfiguration pc input point configurationOptions
Bool no_labels do not write any labelsReturns
PointConfiguration common_refinement (points, sub1, sub2, dim) → Array<Set<Int>>Computes the common refinement of two subdivisions of points. It is assumed that there exists a common refinement of the two subdivisions.
Parameters
Matrix points Array<Set> sub1 first subdivisionArray<Set> sub2 second subdivisionInt dim dimension of the point configurationReturns
Array<Set<Int>> the common refinementcommon_refinement (p1, p2) → PolytopeComputes the common refinement of two subdivisions of the same polytope p1, p2. It is assumed that there exists a common refinement of the two subdivisions. It is not checked if p1 and p2 are indeed the same!
is_regular (points, subdiv) → Pair<bool,Cone>For a given subdivision subdiv of points tests if the subdivision is regular and if yes computes a weight vector inducing this subdivsion. The output is a pair of bool and the weight vector. Options can be used to ensure properties of the resulting vector. The default is having 0 on all vertices of the first face of subdiv.
Parameters
Matrix points Array<Set<Int> > subdiv Options
Matrix<Rational> equations system of linear equation the cone is cut with.Set<Int> lift_to_zero gives only lifting functions lifting the designated vertices to 0Int lift_face_to_zero gives only lifting functions lifting all vertices of the designated face to 0Returns
Pair<bool,Cone> is_subdivision (points, faces)Checks whether faces forms a valid subdivision of points, where points is a set of points, and faces is a collection of subsets of (indices of) points. If the set of interior points of points is known, this set can be passed by assigning it to the option interior_points. If points are in convex position (i.e., if they are vertices of a polytope), the option interior_points should be set to [ ] (the empty set).
placing_triangulation (Points, permutation) → Array<Set<Int>>Compute the placing triangulation of the given point set using the beneath-beyond algorithm.
regular_subdivision (points, weights) → Array<Set<Int>>Compute a regular subdivision of the polytope obtained by lifting points to weights and taking the lower complex of the resulting polytope. If the weight is generic the output is a triangulation.
secondary_cone (points, subdiv) → ConeFor a given subdivision subdiv of points tests computes the corresponding secondary cone. If the subdivision is not regular, the result will be the trivial cone. Options can be used to make the Cone POINTED.
Parameters
Matrix points Array<Set<Int> > subdiv Options
Matrix<Rational> equations system of linear equation the cone is cut with.Set<Int> lift_to_zero gives only lifting functions lifting the designated vertices to 0Int lift_face_to_zero gives only lifting functions lifting all vertices of the designated face to 0Returns
Cone splits_in_subdivision (vertices, subdivision, splits) → Set<Int>Tests which of the splits of a polyhedron are coarsenings of the given subdivision.
Parameters
Matrix vertices the vertices of the polyhedronArray<Set<Int>> subdivision a subdivision of the polyhedronMatrix splits the splits of the polyhedronReturns
Set<Int> split_compatibility_graph (splits, P) → GraphDOC_FIXME: Incomprehensible description! Computes the compatibility graph among the splits of a polytope P.
split_polyhedron (P) → PolytopeComputes the split polyhedron of a full-dimensional polyhdron P.
staircase_weight (k, l) → Vector<Rational>Gives a weight vector for the staircase triangulation of the product of a k- and an l-dimensional simplex.
Parameters
Int k the dimension of the first simplexInt l the dimension of the second simplexReturns
Vector<Rational> stellar_subdivision (pc, faces) → PointConfigurationComputes the complex obtained by stellar subdivision of all faces of the TRIANGULATION of the PointConfiguration.
Parameters
PointConfiguration pc input point configurationArray<Set<Int>> faces list of faces to subdivideOptions
Bool no_labels : do not write any labelsReturns
PointConfiguration tight_span (points, weight, full) → PolytopeCompute the tight span dual to the regular subdivision obtained by lifting points to weight and taking the lower complex of the resulting polytope.
tight_span (P) → PolytopeCompute the tight span dual to the regular subdivision of a polytope P obtained by the WEIGHTS and taking the lower complex of the resulting polytope.
- Symmetrylattice_automorphisms_smooth_polytope (P)
Returns a generating set for the lattice automorphism group of a smooth polytope by comparing lattice distances between vertices and facets.
Parameters
LatticePolytope P - Tight Spanmax_metric (n) → Matrix
Compute a metric such that the f-vector of its tight span is maximal among all metrics with n points.
S. Herrmann and M. Joswig: Bounds on the f-vectors of tight spans.Contrib. Discrete Math., Vol.2, 2007 161-184metric2hyp_triang (FMS) → PolytopeGiven a generic finite metric space FMS, construct the associated (i.e. dual) triangulation of the hypersimplex.
metric2splits (D) → Array<Pair<Set>>Computes the split decomposition of a metric space D (encoded as a symmetric distance matrix).
min_metric (n) → MatrixCompute a metric such that the f-vector of its tight span is minimal among all metrics with n points.
S. Herrmann and M. Joswig: Bounds on the f-vectors of tight spans.Contrib. Discrete Math., Vol.2, 2007 161-184points2metric (points) → MatrixDefine a metric by restricting the Euclidean distance function to a given set of points. Due to floating point computations (sqrt is used) the metric defined may not be exact. If the option max or l1 is set to true the max-norm or l1-nomr is used instead (with exact computation).
poly2metric (P) → MatrixDefine a metric by restricting the Euclidean distance function to the vertex set of a given polytope P. Due to floating point computations (sqrt is used) the metric defined may not be exact. If the option max or l1 is set to true the max-norm or l1-nomr is used instead (with exact computation).
thrackle_metric (n) → MatrixCompute a metric such that the f-vector of its tight span is maximal among all metrics with n points. This metric can be interpreted as a lifting function for the thrackle triangulation (see de Loera, Sturmfels and Thomas: Groebner Basis and triangultaions of the second hypersimplex)
ts_max_metric (n) → TightSpanComputes the tight span of a metric such that its f-vector is maximal among all metrics with n points.
S. Herrmann and M. Joswig: Bounds on the f-vectors of tight spans.Contrib. Discrete Math., Vol.2, 2007 161-184ts_min_metric (n) → TightSpanCompute the tight span of a metric such its f-vector is minimal among all metrics with n points.
S. Herrmann and M. Joswig: Bounds on the f-vectors of tight spans.Contrib. Discrete Math., Vol.2, 2007 161-184ts_thrackle_metric (n) → TightSpanCompute a tight span of a metric such that its f-vector is maximal among all metrics with n points. This metric can be interpreted as a lifting function for the thrackle triangulation (see de Loera, Sturmfels and Thomas: Groebner Basis and triangultaions of the second hypersimplex)
- Transforming a lattice polyhedronambient_lattice_normalization (p) → Polytope
Transform to a full-dimensional polytope while preserving the ambient lattice Z^n
vertex_lattice_normalization (p) → PolytopeTransform to a full-dimensional polytope while preserving the lattice spanned by vertices induced lattice of new vertices = Z^dim
- Transforming a polyhedronbound (P) → Polytope
Make a positive polyhedron bounded. Apply a projective linear transformation to a polyhedron mapping the far hyperplane to the hyperplane spanned by the points (1,0,...,0,1,0,...). The origin (1,0,...,0) is fixed.
The input polyhedron should be POSITIVE; i.e. no negative coordinates.
orthantify (P, v) → PolytopeMake a polyhedron POSITIVE. Apply an affine transformation to a polyhedron such that the vertex v is mapped to the origin (1,0,...,0) and as many facets through this vertex as possible are mapped to the bounding facets of the first orthant.
revert (P) → PolytopeApply a reverse transformation to a given polyhedron P. All transformation clients keep track of the polytope's history. They write or update the attachment REVERSE_TRANSFORMATION.
Applying revert to the transformed polytope reconstructs the original polyhedron.
transform (P, trans, store) → PolytopeTransform a polyhedron P according to the linear transformation trans.
translate (P, trans, store) → PolytopeTranslate a polyhedron P by a given translation vector trans.
- Utilitieslp2poly (file) → Polytope<Float>
Read a linear programming problem given in LP-Format (as used by cplex & Co.) and convert it to a Polytope<Float> object
poly2lp (P, LP, maximize, file)Convert a polymake description of a polyhedron to LP format (as used by CPLEX and other linear problem solvers) and write it to standard output or to a file. If LP comes with an attachment 'INTEGER_VARIABLES' (of type Array<Bool>), the output will contain an additional section 'GENERAL', allowing for IP computations in CPLEX.
Parameters
Polytope P LinearProgram LP default value: P->LPBool maximize produces a maximization problem; default value: 0 (minimize)String file default value: standard outputporta2poly (file) → Polytope<Rational>Read an .ieq or .poi file (porta input) or .poi.ieq or .ieq.poi (porta output) and convert it to a Polytope<Rational> object
print_face_lattice (VIF, dual)Write the face lattice of a vertex-facet incidence matrix VIF to stdout. If dual is set true the face lattice of the dual is printed.
Parameters
IncidenceMatrix VIF Bool dual - Visualizationbounding_box (V, surplus_k, voronoi) → Matrix
Introduce artificial boundary facets (which are always vertical, i.e., the last coordinate is zero) to allow for bounded images of unbounded polyhedra (e.g. Voronoi polyhedra). If the voronoi flag is set, the last direction is left unbounded.
clip_graph (p, V, G)Clip a graph with respect to a given bounding box. Used for the visualization of Voronoi diagrams.
splitstree (vis_obj ...)Call wiki:external_software#SplitsTree with the given visual objects.
Parameters
Visual::Object vis_obj ... objects to displayOptions
String File "filename" or "AUTO" Only create a NEXUS format file, don't start the GUI.The.nexsuffix is automatically added to the file name.Specify AUTO if you want the filename be automatically derived from the drawing title.You can also use any expression allowed for theopenfunction, including "-" for terminal output, "&HANDLE" for an already opened file handle, or "| program" for a pipe.
Common Option Lists
- Visualizationschlegel_init
Initial properties of the Schlegel diagram to be displayed.
Options
Int FACET index of the projection facet, see SchlegelDiagram::FACETRational ZOOM zoom factor, see SchlegelDiagram::ZOOMVector FACET_POINT Vector INNER_POINT