General-affine invariants of plane curves and space curves

Kobayashi, Shimpei   ; Sasaki, Takeshi

Czechoslovak Mathematical Journal, Tome 70 (2020) no. 1, pp. 67-104

Abstract (VO)

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.

Détail

DOI : 10.21136/CMJ.2019.0165-18

Classification : 53A15, 53A20, 53A55
Keywords: plane curve; space curve; general-affine group; general-affine curvature; variational problem

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