## Abstract

Given a rational algebraic surface in the rational parametric representation s(u,v) with unit normal vectors n(u,v) = (s_{u} × s_{v}) t ́|s_{u} × s_{v} t ́| the offset surface at distance d is s_{d}(u,v) = s(u,v) + dn(u,v). This is in general not a rational representation, since t ́|s_{u} × s_{v} is in general not rational. In this paper, we present an explicit representation of all rational surfaces with a continuous set of rational offsets s_{d}(u,v). The analogous question is solved for curves, which is an extension of Farouki's Pythagorean hodograph curves to the rationals. Additionally, we describe all rational curves c(t) whose arc length parameter s(t) is a rational function of t. Offsets arise in the mathematical description of milling processes and in the representation of thick plates, such that the presented curves and surfaces possess a very attractive property for practical use.

Original language | English (US) |
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Pages (from-to) | 175-192 |

Number of pages | 18 |

Journal | Computer Aided Geometric Design |

Volume | 12 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1995 |

Externally published | Yes |

## Keywords

- Dual Bézier curves and surfaces
- Isophote
- Offset curve
- Offset surface
- Rational Bézier representation
- Rational curve
- Rational surface
- Spherical Bézier patch

## ASJC Scopus subject areas

- Modeling and Simulation
- Automotive Engineering
- Aerospace Engineering
- Computer Graphics and Computer-Aided Design