Framed Curves, Ribbons, and Parallel Transport on the Sphere

Abstract

AbstractWe consider curves $$\gamma : [0, 1]\rightarrow {\mathbb {R}}^3$$ γ : [ 0 , 1 ] → R 3 endowed with an adapted orthonormal frame $$r: [0, 1]\rightarrow SO(3)$$ r : [ 0 , 1 ] → S O ( 3 ) . We wish to deform such framed curves $$(\gamma , r)$$ ( γ , r ) while preserving two contraints: a local constraint prescribing one of its ‘curvatures’ (i.e., off-diagonal elements of $$r'r^T$$ r ′ r T ), and a global constraint prescribing the initial and terminal values of $$\gamma $$ γ and r. We proceed in two stages. First we deform the frame r in a way that is naturally compatible with the constraints on r, by interpreting the local constraint in terms of parallel transport on the sphere. This provides a link to the differential geometry of surfaces. The deformation of the base curve $$\gamma $$ γ is achieved in a second step, by means of a suitable reparametrization of the frame. We illustrate this deformation procedure by providing some applications: first, we characterize the boundary conditions on $$(\gamma , r)$$ ( γ , r ) that are accessible without violating the local constraint; then, we provide a short proof of a smooth approximation result for framed curves satisfying both the differential and the global constraints. Finally, we also apply these ideas to elastic ribbons with nonzero width.