Deformation of Framed Curves

Abstract

We consider curves $\gamma :[0,1]\to {\mathbb{R}}^3$ endowed with an adapted orthonormal frame $r:[0,1]\to SO(3)$. We are interested in the cases where the frame is constrained, in the sense that one of its `curvatures' (i.e., off-diagonal elements of $r^{\prime }{r}^T$) is prescribed. One example is the Frénet frame. In order to deform such constrained framed curves without spoiling the constraint, we proceed in two stages. First we deform the frame $r$ in a way that is naturally compatible with the differential constraint, by interpreting it in terms of parallel transport on the sphere. The deformation of the base curve $\gamma$ is achieved in a second step, by means of a suitable reparametrization of the frame. We illustrate this deformation procedure by providing some applications.