Spline Interpolation

Abstract

Given a set of points, it is easy to compute a polynomial that passes through the points. The LP of Section 3.2 is an example of such a polynomial. However, as the discussion in Section 1.5 (especially exercise 1.20) illustrates, a curve based on a high-degree poly- nomial may wiggle wildly and its shape may be far from what the user has in mind. In practical work we are normally interested in a smooth, tight curve that proceeds from point to point such that each segment between two points is a smooth arc. The spline approach to curve design, discussed in this chapter, constructs such a curve from indi- vidual segments, each a simple curve, generally a parametric cubic (PC). This chapter illustrates spline interpolation with three examples, cubic splines (Section 5.1), cardinal splines (Section 5.4), and Kochanek-Bartels splines (Section 5.6). Another important type, the B-spline, is the topic of Chapter 7. Other types of splines are known and are discussed in the scientific literature. A short history of splines can be found in [Schumaker 81] and [Farin 04].