{"paper":{"title":"A necessary condition for cylindrical curves in terms of curvature and torsion","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A curve lies on a circular cylinder only if its curvature and torsion satisfy a derived differential equation.","cross_cats":[],"primary_cat":"math.DG","authors_text":"Rafael L\\'opez","submitted_at":"2026-05-13T05:25:16Z","abstract_excerpt":"We establish necessary conditions for a regular curve to lie on a circular cylinder in terms of its curvature $\\kappa$ and torsion $\\tau$. By identifying a fundamental function $\\psi = \\sin^2 \\alpha$, representing the squared sine of the angle between the tangent vector and the axis of the cylinder, we reduce the geometric inclusion problem to a compatibility condition between an explicit eighth-degree polynomial equation and a differential equation for $\\psi$. This approach yields a single ODE involving only $\\kappa$ and $\\tau$ that governs the inclusion of the curve in the cylinder. The robu"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"This approach yields a single ODE involving only κ and τ that governs the inclusion of the curve in the cylinder.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The curve is regular (non-vanishing speed) and the cylinder is circular with a well-defined fixed axis so that the angle α between the tangent and the axis is globally consistent along the curve.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A regular curve lies on a circular cylinder only if its curvature κ and torsion τ satisfy a specific compatibility ODE obtained from an eighth-degree polynomial condition on the angle function ψ.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A curve lies on a circular cylinder only if its curvature and torsion satisfy a derived differential equation.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"d53a78814b6534d8ff420d5aecdd9b86b4d39dabcbeb2fd98ccaac1dc55ddad2"},"source":{"id":"2605.13022","kind":"arxiv","version":1},"verdict":{"id":"8f1d69b3-8e55-4d83-b074-a6766985b516","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T02:20:17.147387Z","strongest_claim":"This approach yields a single ODE involving only κ and τ that governs the inclusion of the curve in the cylinder.","one_line_summary":"A regular curve lies on a circular cylinder only if its curvature κ and torsion τ satisfy a specific compatibility ODE obtained from an eighth-degree polynomial condition on the angle function ψ.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The curve is regular (non-vanishing speed) and the cylinder is circular with a well-defined fixed axis so that the angle α between the tangent and the axis is globally consistent along the curve.","pith_extraction_headline":"A curve lies on a circular cylinder only if its curvature and torsion satisfy a derived differential equation."},"references":{"count":21,"sample":[{"doi":"","year":1975,"title":"R. L. Bishop, There is more than one way to frame a curve. Am. Math. Mon. 82 (1975), 246–251","work_id":"c6d14ed4-d915-4150-8b8a-5be45e4251f3","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"title":"L. C. B. Da Silva, Moving frames and the characterization of curves that lie on a surface. J. Geom. 108 (2017), 1091","work_id":"dd2557d5-444f-40e2-a8f5-18980b375a5a","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"L. C. B. Da Silva, J. D. Da Silva, Characterization of curves that lie on a geodesic sphere or on a totally geodesic hypersurface in a hyperbolic space or in a sphere. Mediterr. J. Math. 15 (2018), 70","work_id":"c584a4eb-d3b2-4469-bff0-4c6223ac7ed3","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1976,"title":"M. P. Do Carmo, Differential Geometry of Curves and Surfaces. Prentice Hall, Englewood Cliffs, NJ, 1976","work_id":"55b25608-3e88-45fd-b5cd-f337978e63dc","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1993,"title":"D. A. Forsyth, Recognizing algebraic surfaces from their outlines. In: International Conference on Computer Vision, Berlin, pp. 476–480, 1993","work_id":"03d39cbc-d89a-4ada-b942-9d050f593db9","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":21,"snapshot_sha256":"581140c3b2511d754a8aa2dfee1c80db5711d289f2f7392009a2a79e6c926f39","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"6aa044a6727f458e78018081dc277d6786375e2f89485561644e76f1325928a9"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}