Chapter 15 - Algebraic Methods for Computer Aided Geometric Design

Abstract

This chapter presents a collection of gleanings from algebra or algebraic geometry that hold practical value for the field of computer aided geometric design (CAGD). It focuses on the insights, algorithm enhancements and practical capabilities that algebraic methods have contributed to CAGD. CAGD draws from several branches of mathematics and computer science, such as approximation theory, differential geometry, and numerical analysis. The chapter reviews some tools of algebra and algebraic geometry that have been brought to bear on problems in CAGD. Most of the free-form curves and surfaces used in CAGD are given by parametric equations. The chapter examines resultants and Gröbner basis, and discusses their applications in implicitization, inversion, parameterization, and intersection algorithms. The process of finding the implicit equation of a parametric curve or surface is called implicitization, whereas the process of finding the rational parametric equations of implicitly defined algebraic curves and surfaces is called parametrization. Other topics of CAGD research work using algebraic methods are also outlined in the chapter.