Fast Hierarchical Discretization of Parametric Boundary Representations

Discretization of parametric surfaces not only expedites their rendering, but also facilitates other applications such as motion planning, collision detection, and engineering analysis. Currently, discretization methods such as Delaunay Triangulation and Advancing Front techniques, which sample and mesh simultaneously and can guarantee well shaped triangles, are popular despite their high computational cost. In this paper we present a novel two-part sampling and meshing algorithm, which produces topologically correct meshes on arbitrary parametric boundary representations, without additional topological information such as an initial coarse mesh. The algorithm offers a performance increase of approximately two orders of magnitude over Delaunay based methods and at least one order of magnitude over advancing front methods. Based on spatial partitioning, our rule-based approach allows the user to employ arbitrary subdivision criteria to obtain meshes with differing geometric properties. In practice we afford very good control over the properties of the mesh, however the performance increase we offer comes at the expense of analytical guarantees concerning triangle shape. The hierarchical nature of our surface decomposition results in multi-resolution meshes well suited to rendering, collision detection, and path planning, where hierarchical pruning is heavily employed. Moreover, this hierarchy provides an intuitive control structure, which facilitates a future multithreaded implementation.