An application of the maximum principle to the geometry of plane curves
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- by Harold H. Johnson
- Proc. Amer. Math. Soc. 44 (1974), 432-435
- DOI: https://doi.org/10.1090/S0002-9939-1974-0348631-6
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Abstract:
The maximum principle of control theory is used to find necessary and sufficient conditions for a plane curve, which has bounded piecewise continuous curvature and prescribed initial and terminal points and directions, to have minimal length. This result is used to prove that such a closed curve having length $L$ and curvature $k$ satisfying $|k| \leqq K$ can be contained in a circle of radius $R$, where $R \leqq L/4 - (\pi - 2)/2K$.References
- V. G. Boltyanskiĭ, Matematicheskie metody optimal′nogo upravleniya, Second revised and supplemented edition, Physico-Mathematical Library for the Engineer, Izdat. “Nauka”, Moscow, 1969 (Russian). MR 0353082
- J. C. C. Nitsche, The smallest sphere containing a rectifiable curve, Amer. Math. Monthly 78 (1971), 881–882. MR 291387, DOI 10.2307/2316484
Bibliographic Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 44 (1974), 432-435
- MSC: Primary 52A40; Secondary 49B10
- DOI: https://doi.org/10.1090/S0002-9939-1974-0348631-6
- MathSciNet review: 0348631