The Darboux Mate and the Higher Order Curvatures of Spherical Legendre Curves

We introduce and study a Darboux mate of a given spherical Legendre curve LC for the Euclidean \(S^2\) . Also, a triple sequence of curvatures is provided by the higher order derivatives of the last Frenet equation for the frontal of LC. These curvatures are expressed by a recurrence starting with the pair \((-k_1, -k_2, 0)\) , where \((k_1, k_2)\) is the classical curvature function of LC. Several examples are discussed, some of them in relationship with the usual theory of regular space curves. The case of Lorentz–Minkowski sphere \(S^2_1\) is sketched only from the point of view of the geodesic curvature.