Poleni Curves on Surfaces of Constant Curvature

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) = 2α cosh(αs) , the curve is known by 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 k3 g + kg (G−λ) = 0, where kg denotes the geodesic curvature of γ, λ is a constant and the primes indicate derivation with respect to the arc-length parameter s on γ. As the function f (s) = 2 cosh(s) satisfies the equation f ′′ + 1 2 f 3 − f = 0, there exist on a surface S of constant curvature elastic curves with intrinsic equation (1) kg(s) = 2 cosh(s) . If S is the euclidean plane, the solution of (1) is Giovanni Poleni’s curve introduced in 1729 in relation with the tractrix [9]. Its parametric equations are (2) x(s) = s−2tanh(s) , y(s) = 2 cosh(s) . It is plotted in Figure 1. Actually, for every α ∈R, the function f (s) = 2α cosh(αs) satisfies the differential equation f ′′ + 1 2 f 3 −α2 f = 0. Therefore there exist on surface S elastic curves with intrinsic equation (3) kg(s) = 2α cosh(αs) .