欧拉关于曲线曲率的研究

Euler’s Research on Curvature of Curves

曲率是描述几何对象弯曲程度的量,针对不同对象的曲率研究体现了微分几何的发展历程。之前的相关研究认为,欧拉在1786年发表的文章中引入辅助单位球面和密切平面,用纯数学的方式第一次给出空间曲线曲率的计算公式。事实上,在此之前,为了描述弹性细绳在受到外力时形成的曲线,欧拉已经在物理背景下给出了空间曲线曲率的求解方法。基于相关原始文献,围绕弹性细绳形成的曲线与曲率问题之间的关系展开研究,以新的视角解读欧拉对于空间曲线曲率的研究思想和具体过程,这有助于更好地理解微分几何的早期发展历史。

Curvature is applied to measure the bending degree of geometric objects.In the history of differential geometry,solving curvature of different objects was an important issue.The existing studies hold the idea that in the paper published in 1786,Euler solved curvature of spatial curves in a mathematical way for the first time,by introducing the unit sphere and the osculating plane.In fact,before that,with the physical background of"describing a curve formed by a flexible fiber acting by a force on some point",Euler had already obtained the solution to curvature of spatial curves.Based on the primary literature,this paper discusses the relationship between the"flexible fibers problem"and the"curvature problem",in order to reexamine Euler’s process of solving curvature of curves,so as to better understand the early history of differential geometry from a new perspective.