Approximate implicitization using linear algebra

TL;DR

This paper reviews algorithms for approximate implicitization of rational curves and surfaces using linear algebra, introduces new methods with orthogonal polynomials, and discusses their theoretical and practical advantages.

Contribution

It unifies existing approaches under common polynomial bases and proposes new least squares methods with orthogonal polynomials for improved speed and stability.

Findings

Orthogonal polynomial-based methods are faster and more numerically stable.

Unified framework for various implicitization algorithms.

Proposed propositions relate polynomial basis properties to approximation quality.

Abstract

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.