Cones, polytopes and polyhedra have similar combinatorial features: they are composed of faces of various dimensions that meet in faces of lower dimensions. Some functions apply to all of these geometrical objects, some do not.
A face of a cone C is the intersection of C with an affine hyperplane whose defining equation is non-negative on C. A facet of a cone C is a codimension 1 face of C. Defintions for polytopes and polyhedra are similar.
A sequence containing the vertices of the polyhedron P.
The number of vertices of the polyhedron P.
The sequence of generators of the rays of the cone C (returned as primitive lattice points). If C is strictly convex, this is the same as the minimal ( R)-generators.
The ith ray of the of the cone C (in the order returned by Rays(C)).
A sequence of equations defining the minimal linear subspace containing the cone C (or the sequence of toric lattice points Q).
A sequence of generators of the minimal linear subspace containing the cone C (or the sequence of toric lattice points Q).
A basis of the maximal linear subspace contained in the cone C.
If the cone C or polyhedron P lies in a toric lattice L, then return a finite sequence of vectors in the dual lattice check(L) which define supporting hyperplanes of C or P. The list is forced to be minimal for MinimalInequalities, but might not be otherwise. If the argument is a polyhedron, then an integer k is returned as a second return value, so that the first k inequalities correspond to the facets of P while the remaining cut out the subspace containing P.
> P := Polytope([[1,0],[0,1],[-1,-1]]); > Inequalities(P); [ <(2, -1), -1>, <(-1, -1), -1>, <(-1, 2), -1> ] 3For each inequality use HalfspaceToPolyhedron(m,h) to define the corresponding halfspace, and then intersect them all to recover P---and finally recover the vertices we started with.
> PP := &meet [HalfspaceToPolyhedron(H[1],H[2]) : H in Inequalities(P)]; > PP eq P; true > PP; 2-dimensional polytope PP with 3 vertices: ( 0, 1), (-1, -1), ( 1, 0)It can happen that a polytope does not span the ambient toric lattice in which it lies and that some of the inequalities are used to cut out the affine subspace that it does span.
> Q := Polytope([[1,0,2],[0,1,2],[-1,0,2],[0,-1,2]]); > Inequalities(Q); [ <(1, -1, 0), -1>, <(-1, -1, 0), -1>, <(-1, 1, 0), -1>, <(1, 1, 0), -1>, <(0, 0, 1), 2>, <(0, 0, -1), -2> ] 4In this case the final two inequalities are opposites of one another and cut out the affine hyperplane z=2. The second return value 4 indicates that the first 4 inequalities are cutting out facets of the polytope while the remaining inequalities are cutting out the affine hyperplane.
The f-vector of the polyhedron P or cone C.
The h-vector of the polyhedron P or cone C.
A sequence containing all facets of the toric cone C or polyhedron P.
A sequence of sets describing the facets of the polytope P. The jth set gives the indices of the vertices of P which define the jth facet of P.
The number of facets of the polyhedron P.
A sequence containing all face cones of the toric cone C or polyhedron P, or only those of dimension i if an integer i is also specified.
A sequence of sets describing the i-dimensional faces of the polyhedron P. The jth set gives the indices of the vertices of P which define the jth i-dimensional face.
A sequence containing all the edges of the polyhedron P.
A sequence of sets describing the edges of the polhedron P. The jth set gives the indices of the vertices of P which define the jth edge of P.
The graph of the face lattice of the polyhedron P. The vertices of the graph are labeled by the dimension of the corresponding face.
The face of the toric cone C supported by the toric lattice element H in the dual lattice to the one containing C (so H is a linear form on the ambient lattice of C).
Return true if and only if the hyperplane defined by v.u = h is a supporting hyperplane of the polyhedron P, where v is a lattice point of the dual ambient lattice of P and h is a rational number. If so, also gives the sign τsuch the hyperplane is a support of P (i.e. τin {-1, 0, + 1} such that Sign(v.u - h) is either 0 or τfor all u in P). If P is contained within the hyperplane, then τwill be 0.
The cone C such that C + v is a supporting cone of the polyhedron P, where v is a vertex of P.[Next][Prev] [Right] [Left] [Up] [Index] [Root]