Improved algorithm for algebraic curve and surface fitting

Abstract

In the past we have used a simple first order approximation of the Euclidean distance from a point to an implicit curve or surface which yielded good results in the case of unconstrained algebraic curves or surfaces, and reasonable results in the case of bounded algebraic curves and surfaces. However, experiments with the exact Euclidean distance have shown the limitations of this simple approximation. In this paper we introduce a more complex, and better, approximation to the Euclidean distance from a point to an algebraic curve or surface. Evaluating this new approximate distance does not require iterative procedures either, and the fitting algorithm based on it produces results of the same quality as those based on the exact Euclidean distance.