{"paper":{"title":"Translation-like isoptic surfaces and angle sums of translation triangles in $\\NIL$ geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"G\\'eza Csima, Jen\\H{o} Szirmai","submitted_at":"2023-02-15T13:37:29Z","abstract_excerpt":"After having investigated the geodesic and translation triangles and their angle sums in $\\SOL$ and $\\SLR$ geometries we consider the analogous problem in $\\NIL$ space that is one of the eight 3-dimensional Thurston geometries.\n  We analyze the interior angle sums of translation triangles in $\\NIL$ geometry and we provide a new approach to prove that it can be larger than or equal to $\\pi$.\n  Moreover, for the first time in non-constant curvature Thurston geometries we have developed a procedure for determining the equations of $\\NIL$ isoptic surfaces of translation-like segments and as a spec"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2302.07653","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2302.07653/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}