Differential Geometry (1): Curves in Space — Curvature, Torsion, and the Frenet Frame

Mon, 01 Nov 2021 09:00:00 +0000

I am going to start this series the way every honest course on differential geometry starts: with a single moving particle in $\mathbb{R}^3$ . Not a manifold, not a fiber bundle, not even a surface — just a dot tracing a path. Everything later — Gauss maps, second fundamental forms, connections, Riemannian curvature tensors — is in some sense an effort to do for higher-dimensional objects what we are about to do, very thoroughly, for a one-dimensional one. So bear with the warm-up. The pay-off is that by the end of this article we will own a complete local theory: two scalar functions ( $\kappa$ , $\tau$ ) and one orthonormal frame ( $\mathbf{T}, \mathbf{N}, \mathbf{B}$ ) that together pin down a curve up to rigid motion. The slogan is “curve = two numbers per point”, and that slogan deserves a proof.

Curves on Chen Kai Blog

Differential Geometry (1): Curves in Space — Curvature, Torsion, and the Frenet Frame