Affine curvature

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This article is about the curvature of affine plane curves, not to be confused with the curvature of an affine connection.

Affine curvature is a particular type of curvature that is defined on a plane curve that remains unchanged under an equi-affine transformation (an affine transformation that preserves area). Points where the affine curvature is zero retain this property under any affine transformation.

Consider an affine plane curve β(t). Choose co-ordinates for the affine plane such that the area of the parallelogram spanned by two vectors a = (a_1, \; a_2) and b = (b_1, \; b_2) is

\left \vert a\; b \right \vert = a_{1} b_{2} - a_{2} b_{1}.

Then the affine curvature is given by

k_a (t) = \left \vert \beta''(t) \; \beta'''(t) \right \vert .

Here β′ denotes the derivative of β with respect to t.

For a parametrically defined plane curve

t \mapsto (x(t), y(t)),

affine curvature is:

F[x,y]=\left| \frac{x''y'''-x'''y''}{(x'y''-x''y')^{5/2}}-\frac{1}{2}\left[\frac{1}{(x'y''-x''y')}\right]''\right|

[edit] See also

[edit] References

  • B. Opozda, Affine versions of Singer's theorem on locally homogeneous spaces, Ann. Global Anal. Geom. 15 (1997), 187-199.
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