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Keywords:
plane curve; space curve; general-affine group; general-affine curvature; variational problem
plane curve; space curve; general-affine group; general-affine curvature; variational problem
Summary:
We present a fundamental theory of curves in the affine plane and the affine space, equipped with the general-affine groups ${\rm GA}(2)={\rm GL}(2,{\mathbb R})\ltimes {\mathbb R}^2$ and ${\rm GA}(3)={\rm GL}(3,{\mathbb R})\ltimes {\mathbb R}^3$, respectively. We define general-affine length parameter and curvatures and show how such invariants determine the curve up to general-affine motions. We then study the extremal problem of the general-affine length functional and derive a variational formula. We give several examples of curves and also discuss some relations with equiaffine treatment and projective treatment of curves.
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