Add to ORKGIn 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) = 2α cosh(αs), the curve is known under the name “la courbe des forçats”, introduced in 1729 by Giovanni Poleni in relation with the tractrix [9]. 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. 1 Elastic Poleni curves In [7] the authors show that on surfaces of Gaussian curvature G, elastic curves γ are solutions of the intrinsic equation k′′g + 1 2 k3g + kg (G − λ) = 0, where kg denotes the geodesic curvature ...
