Star-shaped trajectories of certain billiards around a triangle

We explore the triangle outer billiards map in points at infinity in the hyperbolic plane, focusing on the rotation number. Building on Dogru and Tabachnikov’s work, which established the conditions for triangles where the rotation number of the billiard map is $1\mathrm{/}3$, we examine cases where the rotation number is $2\mathrm{/}5$. We provide a sufficient condition for this rotation number and show its necessity for large isosceles triangles. The results are framed within the context of the Beltrami-Klein model. We concludes with a conjecture based on the findings.