Parameterization Using Manifolds

There are a variety of surface types (such as meshes and implicit surfaces) that lack a natural parameterization. We believe that manifolds are a natural method for representing parameterizations because of their ability to handle arbitrary topology and represent smooth surfaces. Manifolds provide a formal mechanism for creating local, overlapping parameterization and defining the functions that map between them. In this paper we present specific manifolds for several genus types (sphere, plane, n-holed tori, and cylinder) and an algorithm for establishing a bijective function between an input mesh and the manifold of the appropriate genus. This bijective function is used to define a smooth embedding of the manifold that approximates or interpolates the mesh. The smooth embedding is used to calculate analytical quantities, such as curvature and area, which can then be mapped back to the mesh. Possible applications include texture mapping, surface fitting, arbitrary topology surface modeling, feature detection, and surface comparison.