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Some Connections between Minkowski and Hyperbolic Planes

Jarosław KosiorekAndrzej Matraś — 2004

Bulletin of the Polish Academy of Sciences. Mathematics

The model of the Minkowski plane in the projective plane with a fixed conic sheds a new light on the connection between the Minkowski and hyperbolic geometries. The construction of the Minkowski plane in a hyperbolic plane over a Euclidean field is given. It is also proved that the geometry in an orthogonal bundle of circles is hyperbolic in a natural way.

The Double Tangency Symmetries in Laguerre Plane

Jarosław KosiorekAndrzej Matraś — 2007

Bulletin of the Polish Academy of Sciences. Mathematics

The group generated by double tangency symmetries in a Laguerre plane is investigated. The geometric classification of involutions of a symmetric Laguerre plane is given. We introduce the notion of projective automorphisms using the double tangency and parallel perspectivities. We give the description of the groups of projective automorphisms and automorphisms generated by double tangency symmetries as subgroups of the group M(𝔽,ℝ) of automorphisms of a chain geometry Σ(𝔽,ℝ) following Benz.

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