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Affine curvature

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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.

How to Cite This Entry:
Affine curvature. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Affine_curvature&oldid=32603
This article was adapted from an original article by A.P. Shirokov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article