## Abstract

A mesh M with planar faces is called an edge offset (EO) mesh if there exists a combinatorially equivalent mesh M^{d} such that corresponding edges of M and M^{d} lie on parallel lines of constant distance d. The edges emanating from a vertex of M lie on a right circular cone. Viewing M as set of these vertex cones, we show that the image of M under any Laguerre transformation is again an EO mesh. As a generalization of this result, it is proved that the cyclographic mapping transforms any EO mesh in a hyperplane of Minkowksi 4-space into a pair of Euclidean EO meshes. This result leads to a derivation of EO meshes which are discrete versions of Laguerre minimal surfaces. Laguerre minimal EO meshes can also be constructed directly from certain pairs of Koebe meshes with help of a discrete Laguerre geometric counterpart of the classical Christoffel duality.

Original language | English (US) |
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Pages (from-to) | 45-73 |

Number of pages | 29 |

Journal | Advances in Computational Mathematics |

Volume | 33 |

Issue number | 1 |

DOIs | |

State | Published - Jul 1 2010 |

## Keywords

- Discrete differential geometry
- Edge offset mesh
- Koebe polyhedron
- Laguerre geometry
- Laguerre minimal surface
- Minimal surface

## ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics