Geodesic
A Generalization of Ivory's Theorem
Journal for geometry and graphics, Tome 18 (2014) no. 1, pp. 7-21 Cet article a éte moissonné depuis la source Heldermann Verlag

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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.
Classification : 53A05, 53A07
Mots-clés : Ivory's theorem, energy of a curve, action of a curve, holomorphic Liouville curve 
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     author = {C.-S. Barbat },
     title = {A {Generalization} of {Ivory's} {Theorem}},
     journal = {Journal for geometry and graphics},
     pages = {7--21},
     year = {2014},
     volume = {18},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JGG_2014_18_1_JGG_2014_18_1_a1/}
}
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C.-S. Barbat . A Generalization of Ivory's Theorem. Journal for geometry and graphics, Tome 18 (2014) no. 1, pp. 7-21. http://geodesic.mathdoc.fr/item/JGG_2014_18_1_JGG_2014_18_1_a1/