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On elementary circle-geometry in Cayley-Klein planes. (English) Zbl 1236.51002
Summary: This contribution can be seen as an addendum to a recently paper published by H. Martini and M. Spirova on circle-geometries in Cayley-Klein planes [Publ. Math. 72, No. 3–4, 371–383 (2008; Zbl 1174.51005)], as it deals with further generalisations and extensions of the results therein to circle-geometries in all Cayley-Klein planes. The main methods in this paper are the interpretation of planar figures in space and dualizing according to the duality principle of projective spaces. There are, in principle, only three types of \(\mathbb R^{2}\)-ring structures and, thus, only three types of corresponding circle-geometries [W. Benz, Vorlesungen über Geometrie der Algebren. Geometrien von Möbius, Laguerre-Lie, Minkowski in einheitlicher und grundlagengeometrischer Behandlung. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0258.50024)]. Therefore, each generalisation to non-Euclidean planes must turn out to be just another representation of the classical Euclidean cases. This point of view gives more insight into why some elementary geometric theorems remain valid when changing the place of action from the Euclidean plane to non-Euclidean circle planes and makes explicit proofs of such elementary geometric theorems in non-Euclidean circle planes superfluous. We believe that even the Euclidean cases of circle-geometries comprise, in principle, already all non-Euclidean cases. Representations of such non-Euclidean circle-geometries might also be of interest in their own. For example, among the planar Cayley-Klein geometries the quasi-elliptic and quasi-hyperbolic geometry usually are neglected. They can be treated, similar to the isotropic Möbius geometry, by suitable projections of the Blaschke cylinder.

51A25 Algebraization in linear incidence geometry
51M09 Elementary problems in hyperbolic and elliptic geometries
51M15 Geometric constructions in real or complex geometry
Full Text: DOI
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