Circle Geometries Modeled in Projective Lines over $$ {{\mathbb{R}}^2} $$ -rings

Journal of Mathematical Sciences - Tập 191 - Trang 764-767 - 2013
M. Hamann1, G. Weiss1
1Faculty of Mathematics and Natural Sciences, Dresden University of Technology, Dresden, Germany

Tóm tắt

In this paper, we consider classical circle geometries and connect them with places of planar Cayley–Klein geometries. There are, in principle, only three types of $$ {{\mathbb{R}}^2} $$ -ring structures and, thus, only three types of corresponding circle geometries. Thus, each generalization to non-Euclidean planes turns out to be just another representation of the classical Euclidean cases. 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 themselves. For example, among the planar Cayley–Klein geometries, the quasi-elliptic and quasi-hyperbolic geometry usually are neglected. They can be treated similarly to the isotropic Möbius geometry by suitable projections of the Blaschke cylinder.

Tài liệu tham khảo

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