Unrolling and rolling of curves in non-convex surfaces

The notion of unrolling of a spherical curve is proved to coincide with its development into the tangent plane. The development of a curve in an arbit rary surface in the Euclidean 3-space is then studied from the point of vie w of unrolling. The inverse operation, called the rolling of a curve onto a surface, is also analysed and the relationship of such notions with the fu nctional defined by the square of curvature is stated. An application to th e construction of nonlinear splines on Riemannian surfaces is suggested.