Modeling surfaces of arbitrary topology using manifolds

Abstract

Manifolds describe complicated objects that are locally $\Re\sp{n}$ by defining a set of overlapping maps from the object to $\Re\sp{n}$. In this thesis we present a general technique for inverting that process: we define a complicated object from a set of overlapping subsets of $\Re\sp{n}$. We first present a constructive definition that describes how to perform such a construction in general. We then apply this construction to the particular problem of defining surfaces of arbitrary topology. The surface is built in two steps: we build a manifold with the correct topology then embed the manifold into $\Re\sp3$ using traditional spline techniques. The surface inherits many of the properties of B-splines: local control, a compact representation, and guaranteed continuity of arbitrary degree. The surface is specified using a polyhedral control mesh instead of a rectangular one; the resulting surface approximates the polyhedral mesh much as a B-spline approximates its rectangular control mesh. Like a B-spline, the surface is a single, continuous object.