Frenet frame

Let $I\subset \reals$ be an interval and let $\gamma:I\to\reals^3$ be a parameterized space curve, assumed to be regular and free of points of inflection. We interpret $\gamma(t)$ as the trajectory of a particle moving through 3-dimensional space. The moving trihedron (also known as the Frenet frame, the Frenet trihedron, the repère mobile, and the moving frame) is an orthonormal basis of 3-vectors $T(t),N(t),B(t),$ defined and named as follows:

A straightforward application of the chain rule shows that these definitions are covariant with respect to reparameterizations. Hence, the above three vectors should be conceived as being attached to the point $\gamma(t)$ of the oriented space curve, rather than being functions of the parameter $t$ .

Corresponding to the above vectors are 3 planes, passing through each point $\gamma(t)$ of the space curve. The osculating plane at the point $\gamma(t)$ is the plane spanned by $T(t)$ and $N(t)$ ; the normal plane at $\gamma(t)$ is the plane spanned by $N(t)$ and $B(t)$ ; the rectifying plane at $\gamma(t)$ is the plane spanned by $T(t)$ and $B(t)$ .