Simple manifolds for surface modeling and parameterization

We present a surface modeling technique using manifolds. Our approach uses a single, simple parameterization for all surfaces of a given genus. This differs from previous approaches which build a parameterization based on the elements of a mesh. The simple parameterization is more appropriate for applications that perform complex operations in parameter space or on the mesh surface. We define a manifold and a corresponding embedding function for three genera (plane, sphere, and torus). The manifold can be used simply as a parameterization tool or as a smooth surface approximating the original mesh. We demonstrate how to build a correspondence between the mesh and the manifold, then how to build an embedding that approximates the mesh.