NURBS surface

1 Introduction

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

2 Definition

A NURBS surface is parametric surface defined by its , an array of $n+1$ rows and $m+1$ columns weighted control points and a knot vector in each direction. It is defined as

$$c(u,v)=\frac{{\sum }_{i=0}^n{\sum }_{j=0}^m{N}_{i,p}(u)N_{j,q}(v)w_{i,j}{P}_{i,j}}{{\sum }_{i=0}^n{\sum }_{j=0}^m{N}_{i,p}(u)N_{j,q}(v)w_{i,j}}\mathit{\hspace{1em}\hspace{1em}}0\le u\le 1,0\le v\le 1$$

where $u$ and $v$ are the parameters in each direction, $p$ is the in the $u$-direction, $q$ is the in the $v$-direction, $N_{i,p}$ and $N_{j,q}$ are the B-spline basis functions, $P_{i,j}$ are the control points and $w_{i,j}$ are the weights.