- The image of ``K`` under the isomorphism should have full dimension::
-
- sage: K = random_cone(max_dim=10)
- sage: (phi, phi_inv) = span_iso(K)
- sage: phi(K).dim() == phi(K).lattice_dim()
- True
-
- The isomorphism should be an inner product space isomorphism, and
- thus it should preserve dual cones (and commute with the "dual"
- operation). But beware the automatic renaming of the dual lattice.
- OH AND YOU HAVE TO SORT THE CONES::
-
- sage: K = random_cone(max_dim=10, strictly_convex=False, solid=True)
- sage: L = K.lattice()
- sage: rename_lattice(L, 'L')
- sage: (phi, phi_inv) = span_iso(K)
- sage: sorted(phi_inv( phi(K).dual() )) == sorted(K.dual())
- True
-
- We may need to isomorph twice to make sure we stop moving down to
- smaller spaces. (Once you've done this on a cone and its dual, the
- result should be proper.) OH AND YOU HAVE TO SORT THE CONES::
-
- sage: K = random_cone(max_dim=10, strictly_convex=False, solid=False)
- sage: L = K.lattice()
- sage: rename_lattice(L, 'L')
- sage: (phi, phi_inv) = span_iso(K)
- sage: K_S = phi(K)
- sage: (phi_dual, phi_dual_inv) = span_iso(K_S.dual())
- sage: J_T = phi_dual(K_S.dual()).dual()
- sage: phi_inv(phi_dual_inv(J_T)) == K
- True
-
- """
- phi_domain = K.sublattice().vector_space()
- phi_codo = VectorSpace(phi_domain.base_field(), phi_domain.dimension())
-
- # S goes from the new space to the cone space.
- S = linear_transformation(phi_codo, phi_domain, phi_domain.basis())
-
- # phi goes from the cone space to the new space.
- def phi(J_orig):
- r"""
- Takes a cone ``J`` and sends it into the new space.
- """
- newrays = map(S.inverse(), J_orig.rays())
- L = None
- if len(newrays) == 0:
- L = ToricLattice(0)
-
- return Cone(newrays, lattice=L)
-
- def phi_inverse(J_sub):
- r"""
- The inverse to phi which goes from the new space to the cone space.
- """
- newrays = map(S, J_sub.rays())
- return Cone(newrays, lattice=K.lattice())