Quick reference for polyhedra in Sage¶

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

List of Polyhedron methods¶

H and V-representation

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

Polyhedron boolean properties:

`is_empty()`	tests emptyness
`is_universe()`	tests whether a polyhedra is the whole ambient space
`is_full_dimensional()`	tests if the polyhedron has the same dimension as the ambient space
`is_combinatorially_isomorphic()`	tests whether two polyhedra are combinatorially isomorphic
`is_compact()`	tests compactness, or boundedness of a polyhedron
`is_lattice_polytope()`	tests whether a polyhedron is a lattice polytope
`is_inscribed()`	tests whether the polyhedron is inscribed in a sphere
`is_minkowski_summand()`	tests if the polyhedron can be used to produce another given polyhedron using a Minkowski sum.
`is_neighborly()`	tests whether the polyhedron has full skeleton until half of the dimension (or up to a certain dimension)
`is_reflexive()`	tests if the polar of a lattice polytope is also a lattice polytope (only for `Polyhedron over ZZ`)
`is_simple()`	checks whether the degree of all vertices is equal to the dimension of the polytope
`is_simplex()`	test whether a polytope is a simplex
`is_simplicial()`	checks whether all faces of the polyhedron are simplices
`is_lawrence_polytope()`	tests whether self is a Lawrence polytope
`is_self_dual()`	tests whether the polytope is self-dual
`is_pyramid()`	test whether the polytope is a pyramid over one of its facets
`is_bipyramid()`	test whether the polytope is combinatorially equivalent to a bipyramid over some polytope
`is_prism()`	test whether the polytope is combinatorially equivalent to a prism of some polytope

Enumerative properties

`ambient_dim()`	the dimension of the ambient vector space
`dim()`	the dimension of the polytope
`dimension()`	alias of dim
`f_vector()`	the \(f\)-vector (number of faces of each dimension)
`flag_f_vector()`	the flag-\(f\)-vector (number of chains of faces)
`neighborliness()`	highest cardinality for which all \(k\)-subsets of the vertices are faces of the polyhedron
`simpliciality()`	highest cardinality for which all \(k\)-faces are simplices
`simplicity()`	highest cardinality for which the polar is \(k\)-simplicial

Implementation properties

`backend()`	gives the backend used
`base_ring()`	gives the base ring used
`change_ring()`	changes the base ring

Transforming polyhedra

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

Combinatorics

`combinatorial_polyhedron()`	the combinatorial polyhedron
`face_lattice()`	the face lattice
`hasse_diagram()`	the hasse diagram
`combinatorial_automorphism_group()`	the automorphism group of the underlying combinatorial polytope
`graph()`, `vertex_graph()`	underlying graph
`vertex_digraph()`	digraph (orientation of edges determined by a linear form)
`vertex_facet_graph()`	bipartite digraph given vertex-facet adjacency
`adjacency_matrix()`	adjacency matrix
`incidence_matrix()`	incidence matrix
`slack_matrix()`	slack matrix
`facet_adjacency_matrix()`	adjacency matrix of the facets
`vertex_adjacency_matrix()`	adjacency matrix of the vertices

Integral points

`ehrhart_polynomial()`	the Ehrhart polynomial for `Polyhedron over ZZ`
`ehrhart_polynomial()`	the Ehrhart polynomial for `Polyhedron over QQ`
`ehrhart_quasipolynomial()`	the Ehrhart quasipolynomial for `Polyhedron over QQ`
`h_star_vector()`	the \(h^*\)-vector for polytopes with integral vertices
`integral_points()`	list of integral points
`integral_points_count()`	number of integral points
`get_integral_point()`	get the i-th integral point without computing all interior lattice points
`has_IP_property()`	checks whether the origin is an interior lattice point and compactness (only for `Polyhedron over ZZ`)
`random_integral_point()`	get a random integral point

Getting related geometric objects

`affine_hull()`	returns the smallest affine subspace containing the polyhedron
`boundary_complex()`	returns the boundary complex of simplicial compact polyhedron
`center()`	returns the average of the vertices of the polyhedron
`centroid()`	returns the center of the mass
`representative_point()`	returns the sum of the center and the rays
`a_maximal_chain()`	returns a maximal chain of faces
`face_fan()`	returns the fan spanned by the faces of the polyhedron
`face_generator()`	a generator over the faces
`faces()`	the list of faces
`facets()`	the list of facets
`join_of_Vrep()`, `least_common_superface_of_Vrep()`	smallest face containing specified Vrepresentatives
`meet_of_Hrep()`, `greatest_common_subface_of_Hrep()`	largest face contained in specified Hrepresentatives
`normal_fan()`	returns the fan spanned by the normals of the supporting hyperplanes of the polyhedron
`gale_transform()`	returns the (affine) Gale transform of the vertices of the polyhedron
`hyperplane_arrangement()`	returns the hyperplane arrangement given by the defining facets of the polyhedron
`to_linear_program()`	transform the polyhedra into a Linear Program
`triangulate()`	returns a triangulation of the polyhedron
`fibration_generator()`	returns an iterator of the fibrations of the lattice polytope (only for `Polyhedron over ZZ`)

Other

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