{
  "schema_version": 1,
  "problem_number": "AMR-050-0056",
  "title": "Polar, Dual and Inversive Polygon Areas of N-Periodics in the Elliptic Billiard",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Reznik, Garcia and Koiller listed many quantities found experimentally to be constant along the families of N-periodic billiard trajectories in an ellipse. We treat five rows of their table of inversive objects (records AMR-050-0056, -0057, -0058, -0059 and -0061 of the UnsolvedMath collection), built from a trajectory P and one focus f: the inversion P† of P in the unit circle at f, the polar polygon (the antipedal polygon of P†), its dual polygon, and the focal pedal and antipedal polygons. For least period N, any winding number, a confocal elliptic caustic and signed areas we prove that the ratios of the polar area to the inversive area (row k813) and to the dual area (row k814, credited by the source to A. Akopyan) are constant for every N, the dual being the polar reciprocal of the polar polygon about the centre of the circle touched by its sides, as in the source's figure, or its point-wise inversion; for N = 4 the constants are 2a⁴/b⁴ and 2a⁶/b¹⁰. Rows k815 (odd N) and k816 (even N) use a symbol that the source does not define. Read as the area of the pedal polygon of P†, as in the source's row k908,b, row k815 is false (exact certificate for N = 3) and k816 is false for N = 4 (exact certificate) but true for every even N ≥ 6; read as the area of the inversion of the focal pedal polygon, k815 is false and k816 true; read as the focal pedal area, both hold. Row k818 (N ≡ 2 mod 4) follows from two earlier deposits of the author, doi:10.5281/zenodo.23094118 and doi:10.5281/zenodo.23079799; we reprove it, and its quotient is undefined on the families where the focal antipedal area vanishes identically, which are recorded in earlier deposits (for N = 6: exactly b/a = 1/2). The method is Stachel's Jacobi parametrisation, the phases at which a side of P is an isotropic line through f, and Liouville's theorem. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.MG",
    "math.DS",
    "math.CV"
  ],
  "keywords": [
    "elliptic billiard",
    "Poncelet porism",
    "N-periodic trajectories",
    "inversive polygon",
    "polar polygon",
    "dual polygon",
    "bicentric polygons",
    "pedal polygon",
    "antipedal polygon",
    "signed area",
    "Jacobi elliptic functions",
    "Liouville theorem",
    "Reznik-Garcia-Koiller invariants",
    "UnsolvedMath",
    "AMR-050-0056",
    "AMR-050-0057",
    "AMR-050-0058",
    "AMR-050-0059",
    "AMR-050-0061",
    "math.MG",
    "math.DS",
    "math.CV",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-15",
  "publication_date": "2026-10-15",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-15",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-050-0056/",
  "pdf_url": "https://eulersolve.org/papers/amr-050-0056/paper.pdf?v=6f5828a3c4ab",
  "doi": "10.5281/zenodo.23404903",
  "zenodo_record_url": "https://zenodo.org/records/23404903",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Five rows of the Reznik-Garcia-Koiller list of invariants of N-periodic billiard trajectories in an ellipse (records AMR-050-0056, -0057, -0058, -0059 and -0061): k813 and k814 are proved for every N; k815 and k816, whose pedal symbol is not defined in the source, are decided under each of three readings, with exact counterexamples where they fail; k818 follows from two earlier deposits of the author and is proved again here.",
  "files": {
    "paper.pdf": {
      "sha256": "6f5828a3c4ab053ba651ca4025fee22df141d78c924e468558a29a66b22bbfd7"
    },
    "source.zip": {
      "sha256": "a9aeab6f9a7271f894437b8e7eb1542b82f737d6f938a0f27c7d895d6329df56"
    },
    "verification_report.md": {
      "sha256": "7ea65bc06bf8b3fc083eaa33de79392008b88481b0b369bb6aa4ee90c69c5d86"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}