NURBS curve

1 Introduction

A NURBS curve, which is an acronym for Non-Uniform Rational B-Spline curve, is a generalization of both Bézier (http://planetmath.org/BezierCurve) and BSpline (http://planetmath.org/B-splines) curves. NURBS are commonly used in computer graphics, computer-aided design (CAD), engineering (CAE), and manufacturing (CAM).

2 Definition

A NURBS curve is a parametric curve defined by its , a set of weighted control points, and a knot vector. It is defined as

$$c(u)=\frac{{\sum }_{i=0}^n{N}_{i,p}(u)w_i{P}_i}{{\sum }_{i=0}^n{N}_{i,p}(u)w_i}\mathit{\hspace{1em}\hspace{1em}}0\le u\le 1$$

where $u$ is the parameter, $p$ is the , $N_{i,p}$ are the B-spline basis functions, $P_i$ are the control points and $w_i$ are the weights.