{
  "abstract": "We study polygonal projective billiards whose reflection law is constant on each side. We give a single explicit family with an open set of periodic trajectories for every period at least three, including every odd period, and identify the previously known centrally projective examples as a special parameter value. For period five we describe the full scalar-monodromy variety, prove its irreducibility, and show that every configuration preserves a nonzero quadratic form. Under the stated distinct-centre and non-collinearity hypotheses that form is nondegenerate; this yields an explicit unirational classification of the convex five-reflective locus. We also relate the definite members to spherical billiards and prove that, among analytic simply connected constant-curvature surfaces with geodesic walls, only the sphere can carry an open family of periodic trajectories. For general complete analytic ambients with global wall reflections we prove compactness and a common period for the geodesic flow. The arguments reduce the billiard return map to products of harmonic homologies and separate the algebraic closure condition from the strict first-impact inequalities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28817",
  "author_total": 1,
  "authors": [
    {
      "name": "Jorge Lucas González"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5etm07ar79g26.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28817",
  "fulltext_url": null,
  "id": "5etm07ar79g26",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28817",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7214355675",
  "pdf_url": "https://arxiv.org/pdf/2609.28817",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5etm07ar79g26.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28817",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28817"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Reflective projective billiards in the plane: odd periods, period five, and metric rigidity",
  "title_text": "Reflective projective billiards in the plane: odd periods, period five, and metric rigidity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/5etm07ar79g26",
  "venue": "arXiv",
  "versions": []
}