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

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.