## Abstract

Laguerre geometry provides a simple approach to the design of rational curves and surfaces with rational offsets. These so-called PH curves and PN surfaces can be constructed from arbitrary rational curves or surfaces with help of a geometric transformation which describes a change between two models of Laguerre geometry. Closely related to that is their optical interpretation as anticaustics of arbitrary rational curves/surfaces for parallel illumination. A theorem on rational parametrizations for envelopes of natural quadrics leads to algorithms for the computation of rational parametrizations of surfaces; those include canal surfaces with rational spine curve and rational radius function, offsets of rational ruled surfaces or quadrics, and surfaces generated by peripheral milling with a cylindrical or conical cutter. Laguerre geometry is also useful for the construction of PN surfaces with rational principal curvature lines. New families of such principal PN surfaces are determined.

Original language | English (US) |
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Pages (from-to) | 223-249 |

Number of pages | 27 |

Journal | Computer Aided Geometric Design |

Volume | 15 |

Issue number | 3 |

State | Published - Mar 1998 |

Externally published | Yes |

## Keywords

- Geometrical optics
- Laguerre geometry
- NC milling
- Offset
- Principal curvature line
- Principal patch
- Rational curve
- Rational offset: Pythagorean-hodograph curve
- Rational surface

## ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design
- Geometry and Topology
- Modeling and Simulation