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

Abstract

For a curve $\mathbf{\gamma }:I\to {\mathbb{R}}^n$ of order $n-1$, we prove that the generalized curvatures ${\kappa }_1,\dots ,{\kappa }_{n-1}$ 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 ${\mathbf{\gamma }}^{(i)}(t)$. This gives an efficient algorithm to calculate the curvatures.

arXiv comment17 pages