Affine curvature

Special affine curvature, also known as the equi-affine curvature or affine curvature, is a particular type of curvature that is defined on a plane curve that remains unchanged under a special affine transformation (an affine transformation that preserves area). The curves of constant equi-affine curvature k are precisely all non-singular plane conics. Those with k > 0 are ellipses, those with k = 0 are parabolas, and those with k < 0 are hyperbolas.

The usual Euclidean curvature of a curve at a point is the curvature of its osculating circle, the unique circle making second order contact (having three point contact) with the curve at the point. In the same way, the special affine curvature of a curve at a point P is the special affine curvature of its hyperosculating conic, which is the unique conic making fourth order contact (having five point contact) with the curve at P. In other words it is the limiting position of the (unique) conic through P and four points P₁, P₂, P₃, P₄ on the curve, as each of the points approaches P:

In some contexts, the affine curvature refers to a differential invariant κ of the general affine group, which may readily obtained from the special affine curvature k by κ = k^−3/2dk/ds, where s is the special affine arc length. Where the general affine group is not used, the special affine curvature k is sometimes also called the affine curvature (Shirokov 2001b).

To define the special affine curvature, it is necessary first to define the special affine arclength (also called the equi-affine arclength). Consider an affine plane curve ${\displaystyle \beta (t)}$. Choose co-ordinates for the affine plane such that the area of the parallelogram spanned by two vectors ${\displaystyle a=(a_{1},\;a_{2})}$ and ${\displaystyle b=(b_{1},\;b_{2})}$ is given by the determinant

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