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arXiv · 1902.10926

General-affine invariants of plane curves and space curves

Shimpei Kobayashi·Takeshi Sasaki
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Abstract

We present a fundamental theory of curves in the affine plane and the affine space, equipped with the general-affine groups ${\rm GA}(2)={\rm GL}(2,{\bf R})\ltimes {\bf R}^2$ and ${\rm GA}(3)={\rm GL}(3,{\bf R})\ltimes {\bf R}^3$, 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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Shimpei Kobayashi, Takeshi Sasaki. 2019-02-28. General-affine invariants of plane curves and space curves. https://arxiv.org/abs/1902.10926

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