A Focal Inversion Area Ratio for Four-Periodic Elliptic Billiards
Manuscript 30 September 2026 · Online 30 September 2026
Abstract
The experimental invariant list of Reznik, Garcia and Koiller prints the value 2 for the focal-inverse area of the outer tangent polygon divided by that of a four-periodic elliptic billiard orbit. We give a rational counterexample to this value and a self-contained area calculation showing that the correct constant is 1/2. The result concerns vertexwise inversion of both polygons about the same focus and signed shoelace area. It applies to primitive four-periodic orbits with a confocal elliptic caustic. This is a reciprocal correction to a tabulated constant, not a refutation of the underlying constancy or a claim about the corresponding general-period conjecture.
Record
- Affiliation
- Mercury Software GmbH
- Contact
- [email protected] · GitHub
- Result
- Counterexample and corrected formula
- Categories
- math.DS · math.MG
- Manuscript
- 30 September 2026
- Online release
- 30 September 2026
- Version
- 1.0
- License
- Creative Commons Attribution 4.0 International
Files and verification
The PDF is the canonical reading copy. The source archive contains the LaTeX manuscript, bibliography, reproducibility material, and audit documents without build artefacts.
Citation
Alper Ferudun, “A Focal Inversion Area Ratio for Four-Periodic Elliptic Billiards,” EulerSolve Research Papers, AMR-050-0048, 2026. https://doi.org/10.5281/zenodo.23065068.
BibTeX
@misc{Ferudun2026AMR0500048,
author = {Ferudun, Alper},
title = {A Focal Inversion Area Ratio for Four-Periodic Elliptic Billiards},
year = {2026},
howpublished = {EulerSolve Research Papers},
url = {https://eulersolve.org/papers/amr-050-0048/},
doi = {10.5281/zenodo.23065068},
note = {AMR-050-0048; unrefereed preprint}
}