Chapter 5 - Rational Techniques

Abstract

This chapter focuses on rational parametric curves and surfaces and the way they are used in geometric modeling applications. It provides an overview of the state-of-the-art, and emphasizes the role of rational techniques in commercial computer-aided design (CAD) packages. It generalizes non-rational Bézier curves to the rational case. It gives a brief outline of standard algorithms and then places special emphasis on algorithms, which make use of the additional flexibility offered by rational curves. The computation of the derivative of a rational Bézier curve can be performed by using the quotient rule. The chapter generalizes non-rational B-Splines to the rational case. It introduces rational Bézier patches. It also introduces the rational B-Spline surface or non-uniform rational B-Spline (NURBS) surface. This representation comprises all the surface representations: rational and non-rational Bézier patches as well as non-rational B-Spline patches.