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If $\\rho$ denotes the distance from the axis of revolution and $\\Psi=\\sin\\psi$, where $\\psi$ is the tangent angle of the generating curve, then the profile satisfies \\begin{equation*} \\frac{\\left[\\Psi(\\rho\\Psi'-\\Psi)^2+2(\\rho\\Psi'-\\Psi)+2C_1\\rho\\right]^2}{1-\\Psi^2} +\\left[(\\rho\\Psi'-\\Psi)^2-2\\right]^2=C_2, \\end{equation*} where $C_1$ and $C_2$ are constants of integration and the prime denotes differentiation with respect to $\\rho$. This equation reduces the axisym"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2606.01001","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math-ph","submitted_at":"2026-05-31T04:32:29Z","cross_cats_sorted":["cond-mat.soft","math.AP","math.MP"],"title_canon_sha256":"5f73ef0456d21eef78328b3a42669dae33ed9c80f5d8a76cfa2f01904b64dbfe","abstract_canon_sha256":"264c2282621d4ebd8a04d25556ebc07bb0fec22620b5b5e16e9206c526d4246f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T01:04:17.878943Z","signature_b64":"hi2ZCp0HAqQfeP1zbvawfyEz243r/BKpH54Im1awT7vD12l4Wm3Bp8QKUFJtIVREtWx83ciYq0JvB0piya7IAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62f1b96c2a0fa330b7bd28db65fb00ad46f5b2d002d11b6c7f31f2c825730f31","last_reissued_at":"2026-06-02T01:04:17.878527Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T01:04:17.878527Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A first-order formulation for axisymmetric Willmore surfaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cond-mat.soft","math.AP","math.MP"],"primary_cat":"math-ph","authors_text":"Z. C. Tu","submitted_at":"2026-05-31T04:32:29Z","abstract_excerpt":"We show that axisymmetric Willmore surfaces admit a first-order formulation obtained by combining two independent first integrals. If $\\rho$ denotes the distance from the axis of revolution and $\\Psi=\\sin\\psi$, where $\\psi$ is the tangent angle of the generating curve, then the profile satisfies \\begin{equation*} \\frac{\\left[\\Psi(\\rho\\Psi'-\\Psi)^2+2(\\rho\\Psi'-\\Psi)+2C_1\\rho\\right]^2}{1-\\Psi^2} +\\left[(\\rho\\Psi'-\\Psi)^2-2\\right]^2=C_2, \\end{equation*} where $C_1$ and $C_2$ are constants of integration and the prime denotes differentiation with respect to $\\rho$. 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