International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 49, Pages 3115-3122
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 Biao Ou. 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.
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.