Generalized Curvatures of Curves in $\mathbb{R}^n$
Lee-Peng Teo
TL;DR
This work develops an explicit, parameterization-robust method to compute the generalized curvatures $\\kappa_1,...,\\kappa_{n-1}$ of a curve $\\boldsymbol{\\gamma}:I\\to\\mathbb{R}^n$ of order $n-1$. It introduces the canonical matrix $\\mathbf{A}(t)$ and the Gram-Schmidt/QR perspective via $\\mathbf{A}(t)=\\mathbf{F}(t)\\mathbf{R}(t)$, then expresses the curvatures purely in terms of leading principal minors of $\\mathbf{B}(t)=\\mathbf{A}(t)^T\\mathbf{A}(t)$. The main result yields closed-form formulas: $\\kappa_1(t)=\\frac{\\sqrt{\\det\\mathbf{M}_2(t)}}{\\|\\boldsymbol{\\gamma}'(t)\\|^3}$, $\\kappa_i(t)=\\frac{\\sqrt{\\det\\mathbf{M}_{i+1}(t)\\det\\mathbf{M}_{i-1}(t)}}{\\|\\boldsymbol{\\gamma}'(t)\\|\\det\\mathbf{M}_i(t)}$ for $2\\le i\\le n-2$, and $\\kappa_{n-1}(t)=\\frac{\\det\\mathbf{A}(t)}{\\|\\boldsymbol{\\gamma}'(t)\\|\\det\\mathbf{M}_{n-1}(t)}\\sqrt{\\det\\mathbf{M}_{n-2}(t)}$, with sign information encoded by $\\det\\mathbf{A}(t)$. This provides an efficient, algebraic computation of curvatures from derivatives $\\boldsymbol{\\gamma}'(t),\\ldots,\\boldsymbol{\\gamma}^{(n)}(t)$ without performing Gram-Schmidt, and extends to piecewise-regular segments for general order. The results connect to classical 3D curvature and torsion as a special case and offer practical tools for high-dimensional curve analysis.
Abstract
For a curve $\boldsymbolγ:I\to\mathbb{R}^n$ of order $n-1$, we prove that the generalized curvatures $κ_1, \ldots, κ_{n-1}$ 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 $\boldsymbolγ^{(i)}(t)$. This gives an efficient algorithm to calculate the curvatures.