Chapter 9 - NURBS

Abstract

This chapter focuses on nonuniform rational B-spline (NURBS), which is a sum of simpler polynomial splines. B-spline is shorthand for “basis spline.” Nonuniform refers to the fact that the parameterization of the curve isn't uniform. Because NURBSs use knots to define their parameterization, these can vary from one span to another. Rational refers to the fact that NURBSs are based on a ratio of sums of polynomials. This allows each control point to have a weight. These weights define the extent of closeness between the curve and its control vertex. A stronger weight will pull the curve closer to the control vertex. A NURBS is a sum of simpler polynomial splines. These simpler splines are referred to as basis functions. Mathematically, NURBSs are parametric polynomial curves. NURBSs are a superset of splines, conics (parabolics, arcs, circles, ellipses, and so on), and beziers. Curves form a basic part of NURBS and are of many different types—namely, Bezier, Hermite, Catmull–Rom, uniform B-splines, nonuniform B-splines, Kochanek–Bartels, and so on. All curves start out as a set of points (control vertices). Depending on its type, the curve will either go through these points (interpolate), get close to them (approximate), or a mix of both. Although each curve is basically a weighting of control vertices along its length, some curves are more sophisticated than others. NURBS curves are the most general of all. Although Maya supports rational curves, this functionality is rarely used and is currently inaccessible through Maya's user interface.