Torsion of a curve

For example, they are coefficients in the system of differential equations for the Frenet frame given by the Frenet–Serret formulas.

The negative sign is simply a matter of convention: it is a byproduct of the historical development of the subject.

Geometric relevance: The torsion τ(s) measures the turnaround of the binormal vector.

In the animated figure the rotation of the binormal vector is clearly visible at the peaks of the torsion function.

Analytically, r(t) is a three times differentiable function of t with values in R3 and the vectors are linearly independent.