Given a pair of real functions (k, f), we study the conditions they must satisfy for \(k+\lambda f\) to be the curvature in the arc-length of a closed planar curve for all real \(\lambda \). Several equivalent conditions are pointed out, certain periodic behaviours are shown as essential and a family of such pairs is explicitely constructed. The discrete counterpart of the problem is also studied.

中文翻译：

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封闭平面曲线曲率函数的仿射子空间

给定一对实函数 ( k ,  f )，我们研究它们必须满足的条件，以使\(k+\lambda f\)成为所有实数\(\lambda \ ) . 指出了几个等效条件，某些周期性行为被证明是必不可少的，并且明确地构建了这样的对族。还研究了该问题的离散对应物。