Illinois Journal of Mathematics

The total absolute curvature of open curves in $E^{3}$

Kazuyuki Enomoto, Jin-ichi Itoh, and Robert Sinclair

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The total absolute curvature of open curves in $E^3$ is discussed. We study the curves which attain the infimum of the total absolute curvature in the set of curves with fixed endpoints, end-directions, and length. We show that if the total absolute curvature of a sequence of curves in this set tends to the infimum, the limit curve must lie in a plane. Moreover, it is shown that the limit curve is either a subarc of a closed plane convex curve or a piecewise linear curve with at most three edges. The uniqueness of the curves minimizing the total absolute curvature is also discussed. This extends the results in [Yokohama Math. J. 48 (2000), 83–96], which deals with a similar problem for curves in $E^2$.

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Illinois J. Math., Volume 52, Number 1 (2008), 47-76.

First available in Project Euclid: 15 May 2009

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Zentralblatt MATH identifier

Primary: 53A04: Curves in Euclidean space


Enomoto, Kazuyuki; Itoh, Jin-ichi; Sinclair, Robert. The total absolute curvature of open curves in $E^{3}$. Illinois J. Math. 52 (2008), no. 1, 47--76. doi:10.1215/ijm/1242414121.

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