Deformation of Framed Curves

TL;DR

This paper introduces a method for deforming framed curves in 3D space while preserving specific curvature constraints, using a two-stage process involving frame deformation via parallel transport and reparametrization of the base curve.

Contribution

It presents a novel two-step deformation technique for constrained framed curves that maintains curvature constraints through parallel transport and reparametrization.

Findings

The method effectively deforms framed curves without violating curvature constraints.

Applications demonstrate the practical utility of the deformation process.

The approach generalizes to various types of curvature-prescribed frames.

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 $r'r^T$) is prescribed. One example is the Fr\'enet 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.