volesti (version 1.1.2-9)

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, rotation = list())

Value

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

Arguments

P

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

rotation

A list that contains (a) the rotation matrix T and (b) the 'seed' to set a spesific seed for the number generator.

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

# 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)