Special Conics in a Hyperbolic Plane

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 hyperbolicelementary 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 ore more properties of the three Euclidean special conics. Place of action will be the “universal hyperbolic plane” π, i.e., the full projective plane endowed with a hyperbolic polarity ruling distance and angle measure.