Torsion

How fast a curve's osculating plane tilts as it twists out of that plane, per unit arc length.

Torsion $\tau$ is to $\mathbf{B}$ what curvature is to $\mathbf{T}$ : a turning rate per arc length, measuring how fast the osculating plane (spanned by $\mathbf{T}$ and $\mathbf{N}$ ) tilts as the curve twists through space. A curve confined to one flat plane has $\tau=0$ everywhere, since $\mathbf{B}$ stays one fixed direction the whole way along; a helix has nonzero constant torsion, curving and twisting at a steady rate at once.

Examples

A level circle has curvature but zero torsion; a rising spiral staircase pattern (a helix) has both, in constant proportion.