On the unfolding of simple closed curves

Details On the unfolding of simple closed curves On the unfolding of simple closed curves

2008-09-08

arxiv.org/abs/0809.1404v1

Trans. Amer. Math. Soc. 361 (2009), no. 4, 1749--1764

Mathematics

Differential Geometry

to appear in Transactions of the AMS

Scientific paper

10.1090/S0002-9947-08-04781-8

I show that every rectifiable simple closed curve in the plane can be continuously deformed into a convex curve in a motion which preserves arc length and does not decrease the Euclidean distance between any pair of points on the curve. This result is obtained by approximating the curve with polygons and invoking the result of Connelly, Demaine, and Rote that such a motion exists for polygons. I also formulate a generalization of their program, thereby making steps toward a fully continuous proof of the result. To facilitate this, I generalize two of the primary tools used in their program: the Farkas Lemma of linear programming to Banach spaces and the Maxwell-Cremona Theorem of rigidity theory to apply to stresses represented by measures on the plane.

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