IV.9 - Linear Form Curves

Abstract

This chapter focuses on linear form curves. Straight lines are simple curves—easy to understand, and easy to manipulate. A generalization, the multilinear form, underlies the common curves of computer graphics. Through every pair of distinct points there is a unique line; this is an axiom of Euclidean geometry. But use of points and lines in computer graphics is impossible without a numerical representation. Points are commonly represented using (x, y, z) coordinates. Lines, however, may be represented in a variety of ways—Pliicker coordinates, implicit equations, parametric equations—depending on the task. This chapter uses the concept of multilinearity and symmetry to explain Lagrange polynomials, Bézier and B-spline curves, the de Casteljau and de Boor algorithms, and Catmull–Rom splines; and it unites all these in a single routine. The code presented in the chapter generates curves of many different flavors. Specific possibilities include Lagrange interpolants, Bézier curves, B-spline curves, and Cn Catmull–Rom splines.