Journal for Geometry and Graphics 18 (2014), No. 1, 007--021
Copyright Heldermann Verlag 2014
A Generalization of Ivory's Theorem
Calin-Serban Barbat
Dieselstr. 19, München, Germany
calin.barbat@web.de
We use the definition of the energy of a curve on a surface and show that in a Liouville net over the complex plane C the energy integrals along the two diagonals of any curvilinear quadrangle of net curves are equal. In particular cases also the lengths of the two diagonals are equal. For Liouville nets in C we prove a theorem about the energy of certain approximating polygons for the diagonals, in which the well-known planar version of Ivory's Theorem is included as a special case. This new theorem can therefore be seen as a generalization of Ivory's Theorem in the plane. In addition, we prove that this theorem is valid on the holomorphic Liouville curves.
Keywords: Ivory's theorem, energy of a curve, action of a curve, holomorphic Liouville curve.
MSC: 53A05; 53A07
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