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Laguerre and Minkowski planes produced by dilatations

Authors:	Rafael Artzy Hansjoachim Groh

Institution:	(1) Dept. of Mathematics, University of Haifa, 31999 Haifa, Israel;(2) FB 4 - AG 2 Geometrie und Algebra, Technische Hochschule Darmstadt, D-6100 Darmstadt, Germany

Abstract:	We show that each parabolic curve f in R² produces a Laguerre plane $\mathbb{L}(f)$ if f and all its images under dilatations are cycles. Likewise, two hyperbolic curves f₁,f₂ produce a Minkowski planeM(f₁,f₂). We determine for which curves $\mathbb{L}(f)$ is miquelian resp. ovoidal, and for which pairs f₁,f₂,M(f₁,f₂) is miquelian resp. satisfies the rectangle axiom, thus providing many examples of non-embeddable planes.

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