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Edge offset meshes in Laguerre geometry

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Abstract

A mesh with planar faces is called an edge offset (EO) mesh if there exists a combinatorially equivalent mesh such that corresponding edges of and lie on parallel lines of constant distance d. The edges emanating from a vertex of lie on a right circular cone. Viewing as set of these vertex cones, we show that the image of 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.

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Authors and Affiliations

  1. Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstr. 8–10/104, 1040, Wien, Austria

    Helmut Pottmann, Philipp Grohs & Bernhard Blaschitz

Authors
  1. Helmut Pottmann
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  2. Philipp Grohs
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  3. Bernhard Blaschitz
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Correspondence to Helmut Pottmann.

Additional information

Communicated by Rida Farouki.

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Pottmann, H., Grohs, P. & Blaschitz, B. Edge offset meshes in Laguerre geometry. Adv Comput Math 33, 45–73 (2010). https://doi.org/10.1007/s10444-009-9119-6

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  • DOI: https://doi.org/10.1007/s10444-009-9119-6

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