Original scientific paper

Special Conics in a Hyperbolic Plane

Gunter Weiss ; University of Technology Vienna, Vienna, Austria

Abstract

In Euclidean geometry we find three types of special conics, which are distinguished with respect to the Euclidean similarity group: circles, parabolas, and equilateral hyperbolas. They have on one hand special elementary geometric properties (c.f. [7]) and on the other they are strongly connected to the ''absolute elliptic involution" in the ideal
line of the projectively enclosed Euclidean plane. Therefore, in a hyperbolic plane (h-plane)-and similarly in any
Cayley-Klein plane - the analogue question has to consider projective geometric properties as well as hyperbolic-elementary geometric properties. It turns out that the classical concepts ''circle", ''parabola", and ''(equilateral)
hyperbola" do not suit very well to the many cases of conics in a hyperbolic plane (c.f. e.g. [10]). Nevertheless, one can consider conics in a h-plane systematicly having one or more properties of the three Euclidean special conics.
Place of action will be the ''universal hyperbolic plane" \pi, i.e., the full projective plane endowed with a hyperbolic polarity ruling distance and angle measure.

Keywords

conic section; hyperbolic plane; Thales conic; equilateral hyperbola

Hrčak ID:

174096

URI

https://hrcak.srce.hr/174096

Publication date:

16.1.2017.

Article data in other languages: croatian

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