# 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

 $c(u,v)=\frac{\sum_{i=0}^{n}\sum_{j=0}^{m}N_{i,p}(u)N_{j,q}(v)w_{i,j}P_{i,j}}{% \sum_{i=0}^{n}\sum_{j=0}^{m}N_{i,p}(u)N_{j,q}(v)w_{i,j}}\quad\quad 0\leq u\leq 1% ,\quad 0\leq v\leq 1$

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.

Title NURBS surface NURBSSurface 2013-03-22 17:23:51 2013-03-22 17:23:51 stitch (17269) stitch (17269) 6 stitch (17269) Definition msc 51N05 nonuniform rational B-spline surface NURBS