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, ℝ) ⋉ ℝ 2 and GA(3) = GL(3, ℝ) ⋉ 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.

中文翻译：

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平面曲线和空间曲线的一般仿射不变量

我们提出了仿射平面和仿射空间中曲线的基本理论，配备了通用仿射群 GA(2) = GL(2, ℝ) ⋉ ℝ 2 和 GA(3) = GL(3, ℝ) ⋉ ℝ 3 。我们定义了一般仿射长度参数和曲率，并展示了这些不变量如何确定一般仿射运动的曲线。然后我们研究了一般仿射长度泛函的极值问题并推导出一个变分公式。我们给出了几个曲线的例子，并讨论了曲线的等仿射处理和投影处理的一些关系。