TY - JOUR

T1 - On the unfolding of simple closed curves

AU - Pardon, John

N1 - Copyright:
Copyright 2010 Elsevier B.V., All rights reserved.

PY - 2009/4

Y1 - 2009/4

N2 - We 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. We also formulate a generalization of their program, thereby making steps toward a fully continuous proof of the result. To facilitate this, we 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.

AB - We 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. We also formulate a generalization of their program, thereby making steps toward a fully continuous proof of the result. To facilitate this, we 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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U2 - 10.1090/S0002-9947-08-04781-8

DO - 10.1090/S0002-9947-08-04781-8

M3 - Article

AN - SCOPUS:77950681792

VL - 361

SP - 1749

EP - 1764

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 4

ER -