On cutting a torus

Methods are presented for cutting the surface of a polygonal shape that is homeomorphic to the torus. A number of issues are discussed as to how to find loops of a desired characteristic on the surface of a polyhedrally-defined torus and how to cut the torus into two parts by using the loops discovered. A method for finding a homeomorphism for toroidal surfaces is also described, which is important for shape transformation and texture mapping in graphics.