The torsion of a curve describes how quickly the curve twists out of the osculating plane at a point. Formally it is defined as the magnitude of the rate of change of the unit binormal vector with respect to arc length. The torsion of a curve is defined as

A curve with everywhere is a planar curve. Positive or negative torsion indicates the direction in which the curve spirals out of its osculating plane.

Alternative Forms

Without Arc Length

When and its derivatives are straightforward to compute, torsion can be calculated directly as

Derivation

Starting from the Frenet–Serret relations, we know:

Differentiating twice more and using the Frenet–Serret formulas and :

Since , , are mutually orthogonal unit vectors, the cross product eliminates the component:

Taking the dot product with isolates the component, since :

We also have:

Dividing gives:

For a Plane Curve

For a curve given explicitly as , the torsion is identically zero since the curve never leaves the -plane: