FrenetSerret

Frenet-Serret refers to the Frenet-Serret apparatus, a formalism in differential geometry that describes the geometric properties of a smooth space curve in three-dimensional Euclidean space. It consists of an orthonormal moving frame, the tangent T, the normal N, and the binormal B, together with the curvature κ and torsion τ, which quantify bending and twisting along the curve.

For a smooth regular curve r(s) parameterized by arc length s, the unit tangent is T = r′(s). The curvature κ(s) is the magnitude of T′(s), and the principal normal is N = T′(s)/κ(s). The binormal B is defined as B = T × N, completing the orthonormal frame {T, N, B}.

The frame satisfies the Frenet-Serret formulas:

dT/ds = κ N

dN/ds = −κ T + τ B

dB/ds = −τ N

where τ(s) is the torsion, measuring how the curve twists out of its osculating plane.

The construction requires the curve to be sufficiently smooth (typically C^3) with nonzero curvature on the interval considered; points where κ = 0 require limiting interpretation. In higher dimensions, the concept generalizes to a moving orthonormal frame with a sequence of curvatures κ1, κ2, ..., and corresponding torsions, forming a generalized Frenet frame.

Historically, the frame is named after the French mathematicians Frenet and Serret, who introduced it in the 19th century. In practice, the Frenet-Serret apparatus is used to characterize local geometry of curves, analyze motion in physics and engineering, and support computations in computer graphics and robotics. Common examples include the circle (constant κ, τ = 0) and the helix (constant κ and τ).