Generalized Chaos game in an extended hyperbolic plane

We propose and theoretically substantiate an algorithm for conducting a generalized Chaos game with an arbitrary jump on finite convex polygons of the extended hyperbolic plane $H^2$ whose components in the Cayley–Klein projective model are the Lobachevsky plane and its ideal domain. In particular, the defining identities for a point dividing an elliptic, hyperbolic, or parabolic segment in a given ratio are proved, and formulas for calculating the coordinates of such a point at a canonical frame of the first type are obtained. The results of a generalized Chaos game conducted using the advanced software package pyv are presented.