Integral characterizations and the theory of curves
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- by Victor Dannon PDF
- Proc. Amer. Math. Soc. 81 (1981), 600-602 Request permission
Abstract:
Spherical curves in ${E^4}$ are shown to be given by Frenet-like equations. Thus, finding an integral characterization for a spherical ${E^4}$ curve is identical to finding it for an ${E^3}$ Frenet curve. For an ${E^3}$ Frenet curve we obtain: Let $\alpha (s)$ be a unit speed ${C^4}$ curve in ${E^3}$ so that $\alpha ’(s) = T$. Then $\alpha$ is a Frenet curve with curvature $\kappa (s)$ and torsion $\tau (s)$ if and only if there are constant vectors ${\mathbf {a}}$ and ${\mathbf {b}}$ so that \[ {\mathbf {T’}}(s) = \kappa (s)\{ {{\mathbf {a}}\cos \xi (s) + {\mathbf {b}}\sin \xi (s) - \int _0^s {\cos [\xi (s) - \xi (\delta )]{\mathbf {T}}(\delta )\kappa (\delta )\;d\delta } } \},\] where $\xi (s) = \int _0^s {\tau (\delta )\;d\delta }$.References
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Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 81 (1981), 600-602
- MSC: Primary 53A07
- DOI: https://doi.org/10.1090/S0002-9939-1981-0601738-1
- MathSciNet review: 601738