Representing curved shape is a fundamental problem in geometric modeling and design. A variety of approaches currently exist for modeling curved shape, by far the most popular of which is a boundary representation (B-rep) whose surfaces are trimmed NURBS (non-uniform rational B-splines). Unfortunately, building a modeler based on trimmed NURBS is exceedingly difficult for at least two reasons. First, trimming two NURBS patches to agree along a common edge usually involves complex surface/surface intersection algorithms and substantial approximation. Second, even in the absence of sharp edges, patching is often required since a single NURBS patch is capable of modeling only surfaces topologically equivalent to planes, cylinders, and tori. Modeling surfaces of other topological types (even those as simple as a sphere) require the surface be decomposed into a quiltwork of individual NURBS patches. Complicated continuity conditions must then be imposed to guarantee that adjacent patches join smoothly. Subdivision provides a promising alternative to NURBS. Using subdivision, a polyhedron is mapped to a sequence of denser polyhedra whose limit is a curved object. Each new polyhedron is derived from the previous polyhedron by splitting each of its faces into four new faces and positioning its new vertices using a fixed averaging mask. Different masks yield surface features such as corners, curved edges, or smooth faces in the limit object. The method is simple, requiring only polyhedral modeling. The method is general, producing objects with smooth faces and sharp curved edges without trimming or patching. Finally, subdivision provides a hierarchical representation that allows the design of efficient multi-resolution processing algorithms. This research addresses a combination of theoretical and applied problems whose solution is essential if subdivision is to be of use as a practical method for modeling curved shape. Specifically it develops: Curvature continuous subdivision methods; Subdivision methods for local refinement and introduction of shapes and edges; Functional subdivision methods over irregular triangulations (necessary for FEA) and algorithms for: Conversion to and from other representations such as NURBS, Editing and display of subdividion surfaces, Finite element analysis over subdivision surfaces, Computing the intersection, union, or difference of subdivision surfaces.
