Non-local isotopic approximation of nonsingular surfaces

ABSTRACT: We consider the problem of computing isotopic approximations of nonsingular surfaceswhich are implicitly represented by equations of the form f(x, y, z) = 0. This mesh generation problemhas seen much recent progress. We focus on methods based on domain subdivision using numericalprimitives because of their practical adaptive complexity. Previously, Snyder (1992) and Plantinga-Vegter(2004) have introduced techniques based on parametrizability and non-local isotopy, respectively. In ourprevious work (SoCG 2009), we synthesized these two techniques into a highly efficient and practicalalgorithm for curves. In this paper, we extend our approach to surfaces. The extension is by no meansroutine, as the correctness arguments and analysis are considerably more complex. Unlike the 2-D case, anew phenomenon arises in which local rules for constructing surfaces are no longer sufficient.We treat two important extensions, to exploit anisotropic subdivision and to allow arbitrary geometryfor the region-of-interest (ROI). Anisotropy means that we allow boxes to be split into 2, 4 or 8 childrenwhich are rectangular boxes with bounded aspect ratio. Using ROI allows our algorithms to be extremely”local”, and anisotropy increases their adaptivity.Our algorithms are relatively easy to implement, as the underlying primitives are based on intervalarithmetic and exact BigFloat numbers. We report on very encouraging preliminary experimental results.