# On the derivatives of curvature of framed space curve and their time-updating scheme: Extended version with MATLAB code

## TL;DR

This paper introduces a new method for calculating higher-order derivatives of curvature in framed space curves, along with a MATLAB implementation for updating these derivatives efficiently during small configuration changes.

## Contribution

It provides a closed-form formula for derivatives of curvature tensors using Gibbs vector parametrization and develops a linearized updating scheme with MATLAB code.

## Key findings

- Closed-form formula for derivatives of curvature tensors.
- Efficient linearized updating algorithm for curvature derivatives.
- MATLAB code implementation for practical computation.

## Abstract

This paper deals with the concept of curvature of framed space curves, their higher-order derivatives, variations, and co-rotational derivatives. We realize that parametrizing rotation tensor using the Gibbs vector is effective in deriving a closed form formula to obtain any order derivative of the curvature tensor as the summation of functions of the parametrizing quantity and its derivatives. We use these results for formulating a linearized updating algorithm for curvature and its derivatives when the configuration of the curve acquires a small increment. Finally, the MATLAB code to obtain updated curvature (spatial and material) and its derivatives is presented.