Poleni curves on surfaces of constant curvature

{aeres : ACL}International audienceIn 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