**Beiträge zur Algebra und Geometrie
**

Contributions to Algebra and Geometry

Vol. 39, No. 2, pp. 291-305 (1998)

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Galilei Laguerre Geometry and Rational Circular Offset Surfaces

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Johannes Wallner, Horng-Yang Chen, Helmut Pottmann

Institut für Geometrie, Technische Universität Wien, Wiedner Hauptstrasse 8-10/113, A-1040 Wien e-mail: hannes@geometrie.tuwien.ac.at

**Abstract:** In this paper the Laguerre geometry of three-dimensional Galilei space, that is, the geometry of oriented planes and cycles of an affine Cayley-Klein space with absolute line and two conjugate complex absolute points on it, is studied. Several different models are presented: A cyclographic model and an isotropic model together with their duals and an interpretation in terms of geometrical optics of Galilei space. We also have a closer look on Laguerre transformations. Galilei geometry arises naturally in the context of rational surfaces which possess rational circular offset surfaces, and it can also be used for modeling rational circular offset surfaces, e. g., with Galilei cyclides.

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