"Symbolic-Numeric Techniques for Cubic Surfaces"
, Serie RISC-Linz, 7-2005
Symbolic-Numeric Techniques for Cubic Surfaces
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In geometric modelling and related areas algebraic curves/surfaces typically are described either as the zero set of an algebraic equation (implicit representation), or as the image of a map given by rational functions (parametric representation). The availability of both representations often result in more efficient computations. Computational theories and techniques of algebraic geometry in floating point environment are of high interest in geometric modelling related communities. Therefore, deriving approximate algorithms that can be applied to numeric data have become a very active research area. In this thesis we focus on the two conversion problems, called implicitization and parametrization, from the numeric point of view. A very important issue in the implicitization problem is the perturbation behavior of parametric objects. For a numerically given parametrization we cannot compute an exact implicit equation, just an approximate one. We introduce a condition number of the implicitization problem to measure the worst effect on the solution, when the input data is perturbed by a small amount. Using this condition number we study the algebraic and geometric robustness of the implicitization process. Several techniques for parameterizing a rational algebraic surface as a whole exist. However, in many applications, it suffices to parameterize a small portion of the surface. This motivates the analysis of local parametrizations, i.e. parametrizations of a small neighborhood of a given point $P$ of the surface $S$. We introduce several techniques for generating such parameterizations for nonsingular cubic surfaces. For this class of surfaces, it is shown that the local parametrization problem can be solved for all points, and any such surface can be covered completely.