Chapter 7 - B-Spline Approximation and the de Boor Algorithm

Abstract

The purpose of this chapter is to illustrate the B-Spline Approximation. It also elucidates the de Boor Algorithm. B-splines are generalizations of Bernstein polynomials and share various analyticand geometric properties. A spline is a piecewise polynomial whose pieces fittogether smoothly at the joins. B-spline curves and surfaces have two advantagesover polynomial curves and surfaces. For a huge collection of control points, a Beziercurve or surface approximates the control polygon or polyhedron with a singlepolynomial of high degree. But high-degree polynomials take a long time to computeand are numerically unstable. Splines provide low-degree approximations,which are faster to compute and numerically more tractable. Splines can be manufactured by forming piecewise Bezier curves and surfaces. B-splines provide an approximation scheme where constraints on the location of the control points are not mandatory; B-spline curves and surfaces meet smoothly at their joins for completely arbitrary collections of control points. Thus B-splines provide a simpler, numerically more stable approach to approximating large amounts of data. For these reasons B-splines have become very popular in large-scale industrial applications. This chapter gives a foundation in the fundamentals of polynomial and spline interpolation and approximation.