curvature

The curvature of a curve is the inverse of the curvature radius. It is null for a straight line (infinite curvature radius). For a parametric curve, it can be computed from the first and second derivatives of the defining equation:

$$C = \frac{\vert x'y''-y'x''\vert}{(x'^2+y'^2)^{3/2}}$$

In fact, we will be more interested in the signed curvature, which we define as being positive when the curve turns left and negative when it turns right when following the curve in the direction of parameter increase:

$$\widetilde{C} = \frac{x'y''-y'x''}{(x'^2+y'^2)^{3/2}}$$ (1)

The signed curvature is also an intrinsic geometrical property of the curve at the point considered, it is independant of the parametric equations used as long as they define the same orientation. If two different orientations are used, they will define opposite signed curvatures.