Smooth interpolating curves of prescribed length and minimum curvature
Author:
Joseph W. Jerome
Journal:
Proc. Amer. Math. Soc. 51 (1975), 62-66
MSC:
Primary 49A05; Secondary 41A05
DOI:
https://doi.org/10.1090/S0002-9939-1975-0380551-4
MathSciNet review:
0380551
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Abstract | References | Similar Articles | Additional Information
Abstract: It is shown that, among all smooth curves of length not exceeding a prescribed upper bound which interpolate a finite set of planar points, there is at least one which minimizes the curvature in the ${L^2}$ sense. Thus, we show to be sufficient for the solution of the problem of minimum curvature a condition, viz., prescribed length, which has been known to be necessary for at least a decade. The proof extends immediately to curves in ${{\mathbf {R}}^n},n > 2$.
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- E. H. Lee and G. E. Forsythe, Variational study of nonlinear spline curves, SIAM Rev. 15 (1973), 120–133. MR 331716, DOI https://doi.org/10.1137/1015004
- Joseph W. Jerome, Minimization problems and linear and nonlinear spline functions. I. Existence, SIAM J. Numer. Anal. 10 (1973), 808–819. MR 410205, DOI https://doi.org/10.1137/0710066
- S. D. Fisher and J. W. Jerome, Stable and unstable elastica equilibrium and the problem of minimum curvature, J. Math. Anal. Appl. 53 (1976), no. 2, 367–376. MR 403384, DOI https://doi.org/10.1016/0022-247X%2876%2990116-5
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Additional Information
Keywords:
Minimum curvature,
mean square curvature,
interpolating,
prescribed length,
nonlinear open spline curve
Article copyright:
© Copyright 1975
American Mathematical Society
