Chaos game in an extended hyperbolic plane

We obtain formulas for the midpoint and quasimidpoint of parabolic and nonparabolic segments in the canonical frame of the second type on the extended hyperbolic plane $H^2$ whose components in the projective Cayley–Klein model are the Lobachevsky plane $\Lambda^2$ and a positive-curvature hyperbolic plane $\widehat{H}$. We propose an algorithm for the Chaos game in the $H^2$ plane and present the results of this game played with the prepared software package pyv on triangles in the $\Lambda^2$ plane and trihedrals in the $\widehat{H}$ plane.