Interpolation

Abstract

Interpolation is not a branch of mathematics but rather a collection of techniques the reader will find useful when solving computer graphics problems. Basically, an interpolant is a way of changing one number into another. For example, to change 2 into 4 we simply add 2, which is not very useful. The real function of an interpolant is to change one number into another in, perhaps, 10 equal steps. Thus if we start with 2 and repeatedly added 0.2, it would generate the sequence 2.0, 2.2, 2.4, 2.6, 2.8, 3.0, 3.2, 3.4, 3.6, 3.8, and 4. These numbers could then be used to translate, scale, rotate an object, move a virtual camera, or change the position, color or brightness of a virtual light source.

In order to repeat the above interpolant for different numbers we require a formula, which is one of the first exercises of this chapter. We also need to explore ways of controlling the spacing between the interpolated values. In animation, for example, we often need to move an object very slowly and gradually increase its speed. Conversely, we may want to bring an object to a halt, making its speed less and less.

We start with the simplest of all interpolants: the linear interpolant.