# 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.

Title | NURBS surface |
---|---|

Canonical name | NURBSSurface |

Date of creation | 2013-03-22 17:23:51 |

Last modified on | 2013-03-22 17:23:51 |

Owner | stitch (17269) |

Last modified by | stitch (17269) |

Numerical id | 6 |

Author | stitch (17269) |

Entry type | Definition |

Classification | msc 51N05 |

Synonym | nonuniform rational B-spline surface |

Related topic | NURBS |