polyhedra_quickref.rst

Quick reference for polyhedra in Sage

Jean-Philippe Labbé <labbe@math.fu-berlin.de> Vincent Delecroix <vincent.delecroix@u-bordeaux.fr>

List of Polyhedron methods

H and V-representation

~sage.geometry.polyhedron.base.Polyhedron_base.base_ring	ring on which the polyhedron is defined
~sage.geometry.polyhedron.base.Polyhedron_base.ambient_space	ambient vector space or free module
~sage.geometry.polyhedron.base.Polyhedron_base.Hrepresentation_space	vector space or free module used for the vectors of the H-representation
~sage.geometry.polyhedron.base.Polyhedron_base.Vrepresentation_space	vector space or free module used for the vectors of the V-representation
~sage.geometry.polyhedron.base.Polyhedron_base.n_Hrepresentation	number of elements in the H-representation (sum of the number of equations and inequalities)
~sage.geometry.polyhedron.base.Polyhedron_base.n_Vrepresentation	number of elements in the V-representation (sum of vertices, rays and lines)
~sage.geometry.polyhedron.base.Polyhedron_base.n_equations	number of equations
~sage.geometry.polyhedron.base.Polyhedron_base.n_inequalities	number of inequalities
~sage.geometry.polyhedron.base.Polyhedron_base.n_vertices	number of vertices
~sage.geometry.polyhedron.base.Polyhedron_base.n_rays	number of rays
~sage.geometry.polyhedron.base.Polyhedron_base.n_lines	number of lines
~sage.geometry.polyhedron.base.Polyhedron_base.n_facets	number of facets

Polyhedron boolean properties:

~sage.geometry.polyhedron.base.Polyhedron_base.is_empty	tests emptyness
~sage.geometry.polyhedron.base.Polyhedron_base.is_universe	tests whether a polyhedra is the whole ambient space
~sage.geometry.polyhedron.base.Polyhedron_base.is_full_dimensional	tests if the polyhedron has the same dimension as the ambient space
~sage.geometry.polyhedron.base.Polyhedron_base.is_combinatorially_isomorphic	tests whether two polyhedra are combinatorially isomorphic
~sage.geometry.polyhedron.base.Polyhedron_base.is_compact	tests compactness, or boundedness of a polyhedron
~sage.geometry.polyhedron.base.Polyhedron_base.is_lattice_polytope	tests whether a polyhedron is a lattice polytope
~sage.geometry.polyhedron.base.Polyhedron_base.is_inscribed	tests whether the polyhedron is inscribed in a sphere
~sage.geometry.polyhedron.base.Polyhedron_base.is_minkowski_summand	tests if the polyhedron can be used to produce another given polyhedron using a Minkowski sum.
~sage.geometry.polyhedron.base.Polyhedron_base.is_neighborly	tests whether the polyhedron has full skeleton until half of the dimension (or up to a certain dimension)
~sage.geometry.polyhedron.base_ZZ.Polyhedron_ZZ.is_reflexive	tests if the polar of a lattice polytope is also a lattice polytope (only for Polyhedron over ZZ <sage.geometry.polyhedron.base_ZZ.Polyhedron_ZZ>)
~sage.geometry.polyhedron.base.Polyhedron_base.is_simple	checks whether the degree of all vertices is equal to the dimension of the polytope
~sage.geometry.polyhedron.base.Polyhedron_base.is_simplex	test whether a polytope is a simplex
~sage.geometry.polyhedron.base.Polyhedron_base.is_simplicial	checks whether all faces of the polyhedron are simplices
~sage.geometry.polyhedron.base.Polyhedron_base.is_lawrence_polytope	tests whether self is a Lawrence polytope

Enumerative properties

~sage.geometry.polyhedron.base.Polyhedron_base.ambient_dim	the dimension of the ambient vector space
~sage.geometry.polyhedron.base.Polyhedron_base.dim	the dimension of the polytope
~sage.geometry.polyhedron.base.Polyhedron_base.dimension	alias of dim
~sage.geometry.polyhedron.base.Polyhedron_base.f_vector	the f-vector (number of faces of each dimension)
~sage.geometry.polyhedron.base.Polyhedron_base.neighborliness	highest cardinality for which all k-subsets of the vertices are faces of the polyhedron

Implementation properties

~sage.geometry.polyhedron.base.Polyhedron_base.backend	gives the backend used
~sage.geometry.polyhedron.base.Polyhedron_base.base_ring	gives the base ring used
~sage.geometry.polyhedron.base.Polyhedron_base.change_ring	changes the base ring

Transforming polyhedra

~sage.geometry.polyhedron.base.Polyhedron_base.minkowski_sum	Minkowski sum of two polyhedra
~sage.geometry.polyhedron.base.Polyhedron_base.minkowski_difference	Minkowski difference of two polyhedra
~sage.geometry.polyhedron.base_ZZ.Polyhedron_ZZ.minkowski_decompositions	Minkowski decomposition (only for Polyhedron over ZZ <sage.geometry.polyhedron.base_ZZ.Polyhedron_ZZ>)
~sage.geometry.polyhedron.base.Polyhedron_base.product	cartesian product of two polyhedra
~sage.geometry.polyhedron.base.Polyhedron_base.intersection	intersection of two polyhedra
~sage.geometry.polyhedron.base.Polyhedron_base.join	join of two polyhedra
~sage.geometry.polyhedron.base.Polyhedron_base.convex_hull	convex hull of the union of two polyhedra
~sage.geometry.polyhedron.base.Polyhedron_base.affine_hull	constructs an affinely equivalent full dimensional polyhedra
~sage.geometry.polyhedron.base.Polyhedron_base.faces	the list of faces
~sage.geometry.polyhedron.base.Polyhedron_base.barycentric_subdivision	constructs a geometric realization of the barycentric subdivision
~sage.geometry.polyhedron.base.Polyhedron_base.dilation	scalar dilation
~sage.geometry.polyhedron.base.Polyhedron_base.face_truncation	truncates a specific face
~sage.geometry.polyhedron.base.Polyhedron_base.face_split	returns the face splitting of a face of self
~sage.geometry.polyhedron.base.Polyhedron_base.one_point_suspension	the one-point suspension over a vertex of self (face splitting of a vertex)
~sage.geometry.polyhedron.base.Polyhedron_base.stack	stack a face of the polyhedron
~sage.geometry.polyhedron.base.Polyhedron_base.lattice_polytope	returns an encompassing lattice polytope.
~sage.geometry.polyhedron.base.Polyhedron_base.polar	returns the polar of a polytope (needs to be compact)
~sage.geometry.polyhedron.base.Polyhedron_base.prism	prism over a polyhedron (increases both the dimension of the polyhedron and the dimension of the ambient space)
~sage.geometry.polyhedron.base.Polyhedron_base.pyramid	pyramid over a polyhedron (increases both the dimension of the polyhedron and the dimension of the ambient space)
~sage.geometry.polyhedron.base.Polyhedron_base.bipyramid	bipyramid over a polyhedron (increases both the dimension of the polyhedron and the dimension of the ambient)
~sage.geometry.polyhedron.base.Polyhedron_base.translation	translates by a given vector
~sage.geometry.polyhedron.base.Polyhedron_base.truncation	truncates all vertices simultaneously
~sage.geometry.polyhedron.base.Polyhedron_base.lawrence_extension	returns the Lawrence extension of self on a given point
~sage.geometry.polyhedron.base.Polyhedron_base.lawrence_polytope	returns the Lawrence polytope of self

Combinatorics

~sage.geometry.polyhedron.base.Polyhedron_base.face_lattice	the face lattice
~sage.geometry.polyhedron.base.Polyhedron_base.combinatorial_automorphism_group	the automorphism group of the underlying combinatorial polytope
~sage.geometry.polyhedron.base.Polyhedron_base.graph, ~sage.geometry.polyhedron.base.Polyhedron_base.vertex_graph	underlying graph
~sage.geometry.polyhedron.base.Polyhedron_base.vertex_digraph	digraph (orientation of edges determined by a linear form)
~sage.geometry.polyhedron.base.Polyhedron_base.vertex_facet_graph	bipartite digraph given vertex-facet adjacency
~sage.geometry.polyhedron.base.Polyhedron_base.adjacency_matrix	adjacency matrix
~sage.geometry.polyhedron.base.Polyhedron_base.incidence_matrix	incidence matrix
~sage.geometry.polyhedron.base.Polyhedron_base.facet_adjacency_matrix	adjacency matrix of the facets
~sage.geometry.polyhedron.base.Polyhedron_base.vertex_adjacency_matrix	adjacency matrix of the vertices

Integral points

~sage.geometry.polyhedron.base_ZZ.Polyhedron_ZZ.ehrhart_polynomial	the Ehrhart polynomial (only for Polyhedron over ZZ <sage.geometry.polyhedron.base_ZZ.Polyhedron_ZZ>)
~sage.geometry.polyhedron.base.Polyhedron_base.integral_points	list of integral points
~sage.geometry.polyhedron.base.Polyhedron_base.integral_points_count	number of integral points
~sage.geometry.polyhedron.base.Polyhedron_base.get_integral_point	get the i-th integral point without computing all interior lattice points
~sage.geometry.polyhedron.base_ZZ.Polyhedron_ZZ.has_IP_property	checks whether the origin is an interior lattice point and compactness (only for Polyhedron over ZZ <sage.geometry.polyhedron.base_ZZ.Polyhedron_ZZ>)
~sage.geometry.polyhedron.base.Polyhedron_base.random_integral_point	get a random integral point

Getting related geometric objects

~sage.geometry.polyhedron.base.Polyhedron_base.boundary_complex	returns the boundary complex of simplicial compact polyhedron
~sage.geometry.polyhedron.base.Polyhedron_base.center	returns the average of the vertices of the polyhedron
~sage.geometry.polyhedron.base.Polyhedron_base.representative_point	returns the sum of the center and the rays
~sage.geometry.polyhedron.base.Polyhedron_base.face_fan	returns the fan spanned by the faces of the polyhedron
~sage.geometry.polyhedron.base.Polyhedron_base.normal_fan	returns the fan spanned by the normals of the supporting hyperplanes of the polyhedron
~sage.geometry.polyhedron.base.Polyhedron_base.gale_transform	returns the (affine) Gale transform of the vertices of the polyhedron
~sage.geometry.polyhedron.base.Polyhedron_base.hyperplane_arrangement	returns the hyperplane arrangement given by the defining facets of the polyhedron
~sage.geometry.polyhedron.base.Polyhedron_base.to_linear_program	transform the polyhedra into a Linear Program
~sage.geometry.polyhedron.base.Polyhedron_base.triangulate	returns a triangulation of the polyhedron
~sage.geometry.polyhedron.base_ZZ.Polyhedron_ZZ.fibration_generator	returns an iterator of the fibrations of the lattice polytope (only for Polyhedron over ZZ <sage.geometry.polyhedron.base_ZZ.Polyhedron_ZZ>)

Other

~sage.geometry.polyhedron.base.Polyhedron_base.bounded_edges	generator for bounded edges
~sage.geometry.polyhedron.base.Polyhedron_base.bounding_box	returns the vertices of an encompassing cube
~sage.geometry.polyhedron.base.Polyhedron_base.contains	tests whether the polyhedron contains a vector
~sage.geometry.polyhedron.base.Polyhedron_base.interior_contains	tests whether the polyhedron contains a vector in its interior using the ambient topology
~sage.geometry.polyhedron.base.Polyhedron_base.relative_interior_contains	tests whether the polyhedron contains a vector in its relative interior
~sage.geometry.polyhedron.base_ZZ.Polyhedron_ZZ.find_translation	returns the translation vector between two translation of two polyhedron (only for Polyhedron over ZZ <sage.geometry.polyhedron.base_ZZ.Polyhedron_ZZ>)
~sage.geometry.polyhedron.base.Polyhedron_base.integrate	computes the integral of a polynomial over the polyhedron
~sage.geometry.polyhedron.base.Polyhedron_base.radius	returns the radius of the smallest sphere containing the polyhedron
~sage.geometry.polyhedron.base.Polyhedron_base.radius_square	returns the square of the radius of the smallest sphere containing the polyhedron
~sage.geometry.polyhedron.base.Polyhedron_base.volume	computes different volumes of the polyhedron
~sage.geometry.polyhedron.base.Polyhedron_base.restricted_automorphism_group	returns the restricted automorphism group
~sage.geometry.polyhedron.ppl_lattice_polytope.LatticePolytope_PPL_class.lattice_automorphism_group	returns the lattice automorphism group. Only for PPL Lattice Polytope <sage.geometry.polyhedron.ppl_lattice_polytope.LatticePolytope_PPL_class>