Discrete fairing and variational subdivision for freeform surface design

The representation of free-form surfaces by sufficiently refined polygonal meshes has become common in many geometric modeling applications where comp licated objects have to be handled. While working with triangle meshes is f lexible and efficient, prominent difficulties arise from the lack of infini tesimal smoothness and the prohibitive complexity of highly detailed 3D mod els. In this paper, we discuss the generation of fair triangle meshes that are optimal with respect to some discretized curvature energy functional. T he key issues are the proper definition of discrete curvature, the smoothin g of high-resolution meshes by filter operators, and the efficient generati on of optimal meshes by solving a sparse linear system that characterizes t he global minimum of an energy functional. Results and techniques from diff erential geometry, variational surface design (fairing), and numerical anal ysis are combined to find efficient and robust algorithms that generate smo oth meshes of arbitrary topology that interpolate or approximate a given se t of data points.