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International Journal of Mathematics and Mathematical Sciences
Volume 2003, Issue 49, Pages 3115-3122
http://dx.doi.org/10.1155/S0161171203210371

An equality for the curvature function of a simple and closed curve on the plane

Department of Mathematics, University of Toledo, Toledo 43606, OH, USA

Received 22 October 2002

Copyright © 2003 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove an equality for the curvature function of a simple and closed curve on the plane. This equality leads to another proof of the four-vertex theorem in differential geometry. While examining the regularity assumption on the curve for the equality, we make comments on the relation between the boundary regularity of a Riemann mapping and two important subjects, the Schauder theory and the strong maximum principle, for elliptic partial differential equations of second order. We take a note on the curvature function itself in recognizing people's handwriting by a calculating device, as an afterthought on the equality.