Generalized Curvatures of Curves in $\mathbb{R}^n$

TL;DR

This paper presents a new method to compute generalized curvatures of curves in n-dimensional space using principal minors of a specific matrix, enabling more efficient calculations.

Contribution

It introduces a formula expressing generalized curvatures in terms of principal minors, providing an efficient computational algorithm for curves in higher dimensions.

Findings

Derived explicit formulas for generalized curvatures in terms of matrix minors

Developed an efficient algorithm for curvature calculation in $\,\mathbb{R}^n$

Validated the method with theoretical proofs and computational examples

Abstract

For a curve $\boldsymbol{\gamma}:I\to\mathbb{R}^n$ of order $n-1$, we prove that the generalized curvatures $\kappa_1, \ldots, \kappa_{n-1}$ can be expressed in terms of the leading principal minors of the matrix $\mathbf{A}(t)^T\mathbf{A}(t)$, where $\mathbf{A}(t)$ is the $n\times n$ matrix whose $i$-th column is $\boldsymbol{\gamma}^{(i)}(t)$. This gives an efficient algorithm to calculate the curvatures.