FrenetSerretrammen

The Frenet-Serret frame, sometimes referred to as the Frenet frame, is a moving orthonormal trihedron attached to a smooth space curve in three-dimensional Euclidean space. For a regular curve r(s) parameterized by arc length s, the frame is defined by the tangent T(s) = r'(s), the principal normal N(s) = T'(s)/|T'(s)|, and the binormal B(s) = T(s) × N(s). The curvature κ(s) = |r''(s)| = |T'(s)| measures how sharply the curve bends, while the torsion τ(s) = -B'(s)·N(s) (equivalently τ = det(r', r'', r''')/|r''|^2) measures the rate at which the curve twists out of the osculating plane.

Frenet-Serret formulas describe how the frame evolves along the curve with respect to arc length: T'(s) = κ(s) N(s), N'(s) = -κ(s) T(s) + τ(s) B(s), and B'(s) = -τ(s) N(s). These equations encode the intrinsic geometric properties of the curve, with κ and τ providing complete local information about bending and twisting when the curve is sufficiently smooth. If the curve is not arc-length parameterized, the formulas can be adjusted to include the speed factor, expressing derivatives with respect to the chosen parameter.

Historically, the frame is named after Jean Frédéric Frenet and Joseph-Serret, who introduced the concepts in the 19th century. In addition to its role in differential geometry, the Frenet-Serret apparatus finds applications in kinematics, computer graphics, robotics, and the analysis of space curves, where curvature and torsion describe the local geometry and motion of a curve in space.