Affine curvature

rdf:langString Affine curvature

rdf:langString A.P.

rdf:langString A/a010980

rdf:langString a/a010990

rdf:langString Shirokov

rdf:langString Affine curvature

rdf:langString Affine differential geometry

rdf:langString Special affine curvature, also known as the equiaffine 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 equiaffine curvature k are precisely all non-singular plane conics. Those with k > 0 are ellipses, those with k = 0 are parabolae, and those with k < 0 are hyperbolae. 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.