Elementary Constructions for Conics in Hyperbolic and Elliptic Planes

In the Euclidean plane there are well-known constructions of points and tangents of e.g. an ellipse c based on the given axes of c, which make use of the Apollonius definition of c via its focal points or via two perspective affinities (de la Hire’s construction). Even there is no parallel relation neither in a hyperbolic plane nor in an elliptic plane it is still possible to modify many of the elementary geometric constructions for conics, such that they also hold for those (regular) non-Euclidean planes. Some of these modifications just replace Euclidean straight lines by nonEuclidean circles. Furthermore we also study properties of Thales conics, which are generated by two pencils of (non-Euclidean) orthogonal lines.