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We study the geometry of reflection of a massive point-like particle at conicsection boundaries. Thereby the particle is subjected to a central forceassociated with either a Kepler or Hooke potential. The conic section isassumed to have a focus at the Kepler center, or have its center at the Hookiancenter respectively. When the particle hits the boundary it is ideallyreflected according to the law of reflection. These systems are known to beintegrable.  We describe the consecutive billiard orbits in terms of their foci. We showthat the second foci of these orbits always lie on a circle in the Kepler case.In the Hooke case, we show that the foci of the orbits lie on a Cassini oval.For both systems we analyze the envelope of the directrices of the orbits aswell.

Geometric properties of integrable Kepler and Hooke billiards with conic section boundaries