General-Affine Invariants of Plane Curves and Space Curves

We present a fundamental theory of curves in the affine plane and the affine space, equipped with the general-affine groups GA(2) = GL(2, Double-struck capital R); Double-struck capital R-2 and GA(3) = GL(3, Double-struck capital R); Double-struck capital 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