Frenet–Serret formulas

In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space $\mathbb {R} ^{3},$ or the geometric properties of the curve itself irrespective of any motion. More specifically, the formulas describe the derivatives of the so-called tangent, normal, and binormal unit vectors in terms of each other. The formulas are named after the two French mathematicians who independently discovered them: Jean Frédéric Frenet, in his thesis of 1847, and Joseph Alfred Serret, in 1851. Vector notation and linear algebra currently used to write these formulas were not yet available at the time of their discovery.

The tangent, normal, and binormal unit vectors, often called $T$ , $N$ , and $B$ , or collectively the Frenet–Serret basis (or TNB basis), together form an orthonormal basis that spans $\mathbb {R} ^{3},$ and are defined as follows:

$T$ is the unit vector tangent to the curve, pointing in the direction of motion.
$N$ is the normal unit vector, the derivative of $T$ with respect to the arclength parameter of the curve, divided by its length.
$B$ is the binormal unit vector, the cross product of $T$ and $N$ .

The above basis in conjunction with an origin at the point of evaluation on the curve define a moving frame, the Frenet–Serret frame (or TNB frame).

The Frenet–Serret formulas are: ${\begin{aligned}{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}}&=\kappa \mathbf {N} ,\\[4pt]{\frac {\mathrm {d} \mathbf {N} }{\mathrm {d} s}}&=-\kappa \mathbf {T} +\tau \mathbf {B} ,\\[4pt]{\frac {\mathrm {d} \mathbf {B} }{\mathrm {d} s}}&=-\tau \mathbf {N} ,\end{aligned}}$ where ${\tfrac {d}{ds}}$ is the derivative with respect to arclength, $κ$ is the curvature, and $τ$ is the torsion of the space curve. (Intuitively, curvature measures the failure of a curve to be a straight line, while torsion measures the failure of a curve to be planar.) The $TNB$ basis combined with the two scalars, $κ$ and $τ$ , is called collectively the Frenet–Serret apparatus.