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 is calculated by the formula

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

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

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.