0. Overview

Polynomial parametric curves of high degree have a disadvantage: Requirements placed on one stretch of such a curve can have a very strong effect some distance away. In Figure , the jump in the height of the data points near the middle has a strong effect on the interpolating polynomial curve near the ends.

Figure: Lagrange interpolation of data points

book/09dir/lagrange.eps

In contrast, Figure shows an example of a ``cubic spline'' curve through the same data points. Notice how it follows them much more closely.

Figure: Spline interpolation of the same data points

book/09dir/spline.1.eps

The spline curve was constructed by using a different cubic polynomial curve between each two data points. In other words, it is a piecewise cubic curve, made of pieces of different cubic curves glued together. The pieces are so well matched where they are glued that the gluing is not obvious.

In fact, if the whole curve shown is described with a single function $P(t)$, then $P(t)$ is so smooth that it has a second derivative everywhere and this derivative is continuous.

Definition. A cubic spline curve is a piecewise cubic curve with continuous second derivative.

The word ``spline'' actually refers to a thin strip of wood or metal. At one time curves were designed for ships and planes by mounting actual strips of wood or metal so that they went through the desired data points but were otherwise free to move. For reasons of physics, such curves are approximately piecewise cubic with continuous second derivative, if they are suitably parameterized.

You may recall from calculus that the curvature of a curve at each point depends on the second derivative there. At the end points, an actual wood or metal strip has no reason to bend, and the second derivative of its curve is zero.

Definition. A cubic spline curve is relaxed if its second derivative is zero at each endpoint.

We shall concentrate on relaxed cubic spline curves. As you will see, they can be used either for controlled design (B-splines) or for interpolation. To describe the cubic pieces simply and conveniently, we shall use cubic Bézier curves.