Curvature

Given a parametrized curve , the curvature is the angular rate of change of the tangent vector with respect to arc length.

where is the arclength and is the tangent vector

Intuitively, the curvature measures how much the curve “bends”. A.k.a, a small curvature means the curve is flat, and a large curvature means the curve is tighter.

Curvature in 3D

The general definition of curvature is not that easy to use in practice. But in 3D, the formula can be simplified:

where is the normal vector at .

Reference