Conics and Osculating Circles in Hyperbolic Plane

The Cayley-Klein model is suitable for representing hyperbolic plane for creating geometric constructions, because the projective-geometric point of view in this model for Euclidean and hyperbolic plane are the same. Thus we show the classification of conics in Cayley-Klein model of hyperbolic plane, which can be constructed with perspective collineation as a collineary related image to the absolute conic. It is shown how to "translate" an Euclidean construction of an osculating circle in an arbitrary point of a conic which is given by a general data into Hyperbolic plane