The curvature of a curve describes how quickly the curve changes direction at a point. Formally it's defined as the magnitude of the rate of change of the unit tangent vector with respect to arc length. The curvature of a curve is defined as

If the curve is already parameterized by arc length the curvature can simply be calculated as

Alternative Forms

Without Arc Length

When is simple to compute we can skip the arc length calculations entirely and use this equation for curvature instead.

Derivation

Without Tangent Unit Vectors

When the derivative of is difficult to compute we can use the following version

Derivation