{"paper":{"title":"Parametric CR-umbilical Locus of Ellipsoids in $\\mathbb{C}^2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Joel Merker (LM-Orsay & UMS 1786), The-Anh Ta (LM-Orsay), Wei-Guo Foo (LM-Orsay)","submitted_at":"2017-07-21T07:49:07Z","abstract_excerpt":"For every real numbers $a \\geqslant 1$, $b \\geqslant 1$ with $(a,b) \\neq (1,1)$, the curve parametrized by $\\theta \\in \\mathbb{R}$ valued in $\\mathbb{C}^2 \\cong \\mathbb{R}^4$ \\[ \\gamma\\, \\colon \\ \\ \\ \\theta \\,\\,\\,\\longmapsto\\,\\,\\, \\big( x(\\theta)+{\\scriptstyle{\\sqrt{-1}}}\\,y(\\theta),\\,\\, u(\\theta)+{\\scriptstyle{\\sqrt{-1}}}\\,v(\\theta) \\big) \\] with components: \\[ x(\\theta) \\,:=\\, {\\textstyle{\\sqrt{\\frac{a-1}{a\\,(ab-1)}}}}\\, \\cos\\,\\theta, \\ \\ \\ \\ \\ y(\\theta) \\,:=\\, {\\textstyle{\\sqrt{\\frac{b\\,(a-1)}{ab-1}}}}\\, \\sin\\,\\theta, \\ \\ \\ \\ \\ u(\\theta) \\,:=\\, {\\textstyle{\\sqrt{\\frac{b-1}{b\\,(ab-1)}}}}\\, \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.06787","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"}