Affine curvature

A differential invariant of a plane curve in the geometry of the general affine group or a subgroup of it. The affine curvature is usually understood to mean the differential invariant of the curve in the geometry of the unimodular affine (or equi-affine) group. In this geometry the affine (or, more exactly, the equi-affine) curvature of a plane curve $y=y(x)$ is calculated by the formula

$$k=-\frac12[(y'')^{-2/3}]'',$$

while the affine (or, more exactly, equi-affine) arc length of the curve is

$$s=\int(y'')^{1/3}dx.$$

There is a geometrical interpretation of the affine curvature at a point $M_0$ of the curve: Let $M$ be a point on the curve close to $M_0$, let $s$ be the affine length of the arc $M_0M$ and let $\sigma$ be the affine length of the arc of the parabola tangent to this curve at $M_0$ and $M$. The affine curvature at $M_0$ then is

$$k_0=\pm\lim_{M\to M_0}\sqrt{\frac{720(\sigma-s)}{s^5}}.$$

In the affine theory of space curves and surfaces there are also notions of affine curvature which resemble the respective notions of Euclidean differential geometry. For references, see Affine differential geometry.