Aproximação isotópica suave de curvas e superfícies implícitas.

Abstract

This dissertation contains a study about plane curves and surfaces. The two most common way to define this elements are reviewed: the parametric and the implicit definition, with emphasis on the latter. An analysis of some methods to represent plane curves and surfaces is made. One notices that this job is relatively simple when the parametric definition is used, however with the implicit definition this requires a larger number of operations.

This works also presents some methods to find approximations of curves and surfaces implicitly defined that preserves the topology and that generate objects smooth enough. This is achieved basically by a subdivision of the plane (respectivelly the space), which is used to find a piecewise linear approximation of the curve (respectivelly the surface), then this approximation is subdivided to make the result smooth. In the case of surfaces the output is a triangular mesh. Some treatments are also made to improve the quality of the mesh.