Definition:Frenet-Serret Frame

Definition

Let $\alpha : I \to \R^3$ be a (smooth) curve parameterized by arc length.

Let $s \in I$ be such that the curvature $\map \kappa s \ne 0$.

The Frenet-Serret frame of $\alpha$ at $s$ is the triple:

$\struct {\map t s, \map n s, \map b s}$

where:

$\map t s$ is the unit tangent vector

$\map n s$ is the normal vector

$\map b s$ is the binormal vector

Also known as

Also called:

Frenet trihedron

TNB frame

moving trihedron

Source of Name

This entry was named for Jean Frédéric Frenet and Joseph Alfred Serret.

Sources

• 2016: Manfredo P. do Carmo: Differential Geometry of Curves and Surfaces (2nd ed.): $1$-$5$: The Local Theory of Curves Parametrized by Arc Length