volesti (version 1.1.0)

rotate_polytope: Apply a random rotation to a convex polytope (H-polytope, V-polytope, zonotope or intersection of two V-polytopes)

Description

Given a convex H- or V- polytope or a zonotope or an intersection of two V-polytopes as input, this function applies (a) a random rotation or (b) a given rotation by an input matrix \(T\).

Usage

rotate_polytope(P, T = NULL, seed = NULL)

Arguments

P

A convex polytope. It is an object from class (a) Hpolytope, (b) Vpolytope, (c) Zonotope, (d) intersection of two V-polytopes.

T

Optional. A \(d\times d\) rotation matrix.

seed

Optional. A fixed seed for the random linear map generator.

Value

A list that contains the rotated polytope and the matrix \(T\) of the linear transformation.

Details

Let \(P\) be the given polytope and \(Q\) the rotated one and \(T\) be the matrix of the linear transformation.

• If \(P\) is in H-representation and \(A\) is the matrix that contains the normal vectors of the facets of \(Q\) then \(AT\) contains the normal vactors of the facets of \(P\).

• If \(P\) is in V-representation and \(V\) is the matrix that contains column-wise the vertices of \(Q\) then \(T^TV\) contains the vertices of \(P\).

• If \(P\) is a zonotope and \(G\) is the matrix that contains column-wise the generators of \(Q\) then \(T^TG\) contains the generators of \(P\).

• If \(M\) is a matrix that contains column-wise points in \(Q\) then \(T^TM\) contains points in \(P\).

Examples

# NOT RUN {
# rotate a H-polytope (2d unit simplex)
P = gen_simplex(2,'H')
poly_matrix_list = rotate_polytope(P)

# rotate a V-polytope (3d cube)
P = gen_cube(3, 'V')
poly_matrix_list = rotate_polytope(P)

# rotate a 5-dimensional zonotope defined by the Minkowski sum of 15 segments
Z = gen_rand_zonotope(3,6)
poly_matrix_list = rotate_polytope(Z)
# }