Pith Number

pith:WD2AU3HQ

pith:2026:WD2AU3HQTGSR2UPIIFDSWMKPAN

not attested not anchored not stored refs resolved

A necessary condition for cylindrical curves in terms of curvature and torsion

Rafael L\'opez

A curve lies on a circular cylinder only if its curvature and torsion satisfy a derived differential equation.

arxiv:2605.13022 v1 · 2026-05-13 · math.DG

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\usepackage{pith}
\pithnumber{WD2AU3HQTGSR2UPIIFDSWMKPAN}

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Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

This approach yields a single ODE involving only κ and τ that governs the inclusion of the curve in the cylinder.

C2weakest 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.

C3one 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.

References

21 extracted · 21 resolved · 0 Pith anchors

[1] R. L. Bishop, There is more than one way to frame a curve. Am. Math. Mon. 82 (1975), 246–251 1975

[2] L. C. B. Da Silva, Moving frames and the characterization of curves that lie on a surface. J. Geom. 108 (2017), 1091 2017

[3] 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 2018

[4] M. P. Do Carmo, Differential Geometry of Curves and Surfaces. Prentice Hall, Englewood Cliffs, NJ, 1976 1976

[5] D. A. Forsyth, Recognizing algebraic surfaces from their outlines. In: International Conference on Computer Vision, Berlin, pp. 476–480, 1993 1993

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-18T03:08:59.995997Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

b0f40a6cf099a51d51e841472b314f03458522830971a2ccd9a730620e6fc352

Aliases

arxiv: 2605.13022 · arxiv_version: 2605.13022v1 · doi: 10.48550/arxiv.2605.13022 · pith_short_12: WD2AU3HQTGSR · pith_short_16: WD2AU3HQTGSR2UPI · pith_short_8: WD2AU3HQ

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/WD2AU3HQTGSR2UPIIFDSWMKPAN \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: b0f40a6cf099a51d51e841472b314f03458522830971a2ccd9a730620e6fc352

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "6190bf0d8240c05a6eb7c4e3484ec185848c12a3fbf138c6f834b84d2f9275ff",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.DG",
    "submitted_at": "2026-05-13T05:25:16Z",
    "title_canon_sha256": "ee7f4359fc31a21e263252bfc9df06b892e69baba5b653aa704f2ecc02711789"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13022",
    "kind": "arxiv",
    "version": 1
  }
}