{"paper":{"title":"On Cayley conditions for billiards inside ellipsoids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["nlin.SI"],"primary_cat":"math.DS","authors_text":"Rafael Ramirez-Ros","submitted_at":"2012-11-28T09:54:26Z","abstract_excerpt":"All the segments (or their continuations) of a billiard trajectory inside an ellipsoid of $\\Rset^n$ are tangent to n-1 quadrics of the pencil of confocal quadrics determined by the ellipsoid. The quadrics associated to periodic billiard trajectories verify certain algebraic conditions. Cayley found them in the planar case. Dragovi\\'{c} and Radnovi\\'{c} generalized them to any dimension. We rewrite the original matrix formulation of these generalized Cayley conditions as a simpler polynomial one. We find several remarkable algebraic relations between caustic parameters and ellipsoidal parameter"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.6557","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}