{"paper":{"title":"Ruled surfaces asymptotically normalized","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Ioannis Kaffas, Stylianos Stamatakis","submitted_at":"2013-07-23T16:41:27Z","abstract_excerpt":"We consider a skew ruled surface $\\Phi$ in the Euclidean space $E^{3}$ and relative normalizations of it, so that the relative normals at each point lie in the corresponding asymptotic plane of $\\Phi$. We call such relative normalizations and the resulting relative images of $\\Phi$ \\emph{asymptotic}. We determine all ruled surfaces and the asymptotic normalizations of them, for which $\\Phi$ is a relative sphere (proper or inproper) or the asymptotic image degenerates into a curve. Moreover we study the sequence of the ruled surfaces ${\\Psi_{i}}_{i\\in \\mathbb{N}}$, where $\\Psi_{1}$ is an asympt"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.6146","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"}