KoG, Vol. 20 No. 20, 2016.
Izvorni znanstveni članak
Special Conics in a Hyperbolic Plane
Gunter Weiss
; University of Technology Vienna, Vienna, Austria
Sažetak
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.
Ključne riječi
conic section; hyperbolic plane; Thales conic; equilateral hyperbola
Hrčak ID:
174096
URI
Datum izdavanja:
16.1.2017.
Posjeta: 1.353 *