Approximate implicitization using linear algebra
For researchers in computer aided geometric design (CAGD), this work offers incremental improvements in speed and numerical stability for approximate implicitization algorithms.
This paper unifies and improves algorithms for approximate implicitization of rational parametric curves and surfaces using linear algebra, particularly singular value decomposition. It introduces faster and more numerically stable least squares methods with orthogonal polynomials, and proposes propositions linking polynomial basis properties to implicit approximation quality.
In this paper we consider a family of algorithms for approximate implicitization of rational parametric curves and surfaces. The main approximation tool in all of the approaches is the singular value decomposition, and they are therefore well suited to floating point implementation in computer aided geometric design (CAGD) systems. We unify the approaches under the names of commonly known polynomial basis functions, and consider various theoretical and practical aspects of the algorithms. We offer new methods for a least squares approach to approximate implicitization using orthogonal polynomials, which tend to be faster and more numerically stable than some existing algorithms. We propose several simple propositions relating the properties of the polynomial bases to their implicit approximation properties.