Open Problems on Billiards and Geometric Optics

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

An oval is a smooth closed strictly convex plane mirror. Fix a point source in its interior. For each integer $n\ge1$, the $n$th caustic by reflection is the envelope of the one-parameter family of rays after $n$ specular reflections. Cusps are the ordinary cusp singularities of a generic envelope; the source restricts to generic caustics and source positions in general position.

Statement recorded in the supplied source.

For an oval $\gamma$ and a light source inside it, call the envelope of rays after $n$ reflections the $n$th caustic by reflection. Is every generic caustic by reflection in an ellipse a curve with exactly four cusps? Does this four-cusp property characterize ellipses? [S2]

Short English statement

Do all generic reflected caustics of every order in an ellipse have exactly four cusps, and does having this property for interior light sources characterize ellipses?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5200016 (AMR-051-0016). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 M. Bialy, C. Fierobe, A. Glutsyuk, M. Levi, A. Plakhov and S. Tabachnikov, Open problems on billiards and geometric optics (2021), arXiv:2110.10750v2, Serge Tabachnikov, Problem 4, p. 12. https://arxiv.org/pdf/2110.10750v2.