Handbook of Computer Aided Geometric Design

Chapter 6

Spline Basics

Carl de Boor

This chapter promotes, details and exploits the fact that (univariate) splines, i.e., smooth piecewise polynomial functions, are weighted sums of B-splines.

6.1 PIECEWISE POLYNOMIALS

A piecewise polynomial of order k with break sequence ξ (necessarily strictly increasing) is, by definition, any function f that, on each of the half-open intervals [ξ_j ξ_j+1), agrees with some polynomial of degree < k. The term ‘order’ used here is not standard but handy.

Note that this definition makes a piecewise polynomial function right-continuous, meaning that, for any x, f(x) = f(x+):= lim_h↓0 f(x + h). This choice is ...