Frenet-Serret Formulas

Differential Geometry · Axiom Academy

Discover how curvature and torsion completely determine a curve's moving frame in 3D space 1. The Three Frenet-Serret Equations As we move along a curve parametrized by arc length s , the Frenet-Serret frame T , N , B evolves according to three coupled differential equations: Here, T is the unit tangent vector, N is the principal normal, B is the binormal, κ (kappa) is curvature, and τ (tau) is torsion. Watch how these vectors rotate and evolve together. 2. Tangent Vector Evolution: T' = κN The first equation tells us that the tangent vector T changes in the direction of the principal normal N , with rate proportional to the curvature κ. Geometric interpretation: As we move along the curve, T rotates toward N . The curvature κ measures how fast the curve is bending - larger κ means faster rotation of T . 3. Normal Vector Evolution: N' = -κT + τB The principal normal N has two components of change: it rotates back toward T (with rate -κ) and simultaneously rotates toward B (with rate τ). Geometric interpretation: N lies in the plane spanned by T and B , and its evolution reflects both the curve's bending (κ term) and twisting (τ term). 4. Binormal Vector Evolution: B' = -τN The binormal B changes only in the direction of the principal normal N , with rate proportional to the torsion τ.

This is the written version of the interactive lesson above. See the full Differential Geometry course.