Czechoslovak Mathematical Journal, first online, pp. 1-38


General-affine invariants of plane curves and space curves

Shimpei Kobayashi, Takeshi Sasaki

Received April 3, 2018.   Published online September 16, 2019.

Abstract:  We present a fundamental theory of curves in the affine plane and the affine space, equipped with the general-affine groups ${\rm GA}(2)={\rm GL}(2,{\bb R})łtimes{\bb R}^2$ and ${\rm GA}(3)={\rm GL}(3,{\bb R})\ltimes{\bb R}^3$, 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.
Keywords:  plane curve; space curve; general-affine group; general-affine curvature; variational problem
Classification MSC:  53A15, 53A55, 53A20
DOI:  10.21136/CMJ.2019.0165-18

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Affiliations:   Shimpei Kobayashi, Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan, e-mail: shimpei@math.sci.hokudai.ac.jp; Takeshi Sasaki, Department of Mathematics, Kobe University, 1-1, Rokkodai, Nada-ku, Kobe, 657-8501, Japan, e-mail: sasaki@math.kobe-u.ac.jp


 
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