rotate_polytope {volesti}

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())

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.

Value

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

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)