Poleni curves on surfaces of constant curvature

Théodor Hangan (1) , Cornel Murea (1) , Tewfik Sari (1, 2)

1 LMIA - Laboratoire de Mathématiques Informatique et Applications [UHA]
2 MERE - Water Resource Modeling

In the euclidean plane, a regular curve can be defined through its intrinsic equation which relates its curvature $k$ to the arc length $s$. Elastic plane curves were determined this way. If $k(s)=\frac {2\alpha}{\cosh \left(\alpha s\right)}$, the curve is known under the name ''la courbe des for\c cats'', introduced in 1729 by Giovanni Poleni in relation with the tractrix \cite{Palais1976}. The above equation is yet meaningful on a surface if one interprets $k$ as the geodesic curvature of the curve. In this paper we solve the above equation on a surface of constant curvature.

Mots clés: elastic curve; geodesical curvature; non-euclidean geometry

Domaines: Géométrie différentielle [math.DG]

Théodor Hangan, Cornel Murea, Tewfik Sari. Poleni curves on surfaces of constant curvature. Rendiconti del Seminario Matematico, 2009, 67, pp.59-76. ⟨hal-00204551⟩