关于曲面上拐线的概念及其求法

The Concept of Inflexion-curve on Surface and Its Solution

曲面的凹凸是微分几何学中的经典问题之一,划分出曲面上不同的凹凸区域,对于研究曲面的凹凸以及其他性质是一件很有意义的工作。在三维欧氏空间中,根据不同情况曲面可分为上凹和下凹、左凹和右凹,或者前凹和后凹。曲面的拐线就是曲面上不同凹凸区域之间的公共边界。它为划分曲面上不同的凹凸区域提供了一种有效的工具和方法。利用传统微分学和经典微分几何学的理论和方法,提出三维欧氏空间中曲面拐线的概念并加以定义,通过研究曲面与其垂截线之间的关系和性质,得出拐线的求法和判定方法,可以用于对曲面不同上下凹区域的划分。同时类似的理论和方法也完全可以用于对曲面不同左右凹或前后凹区域的划分。对于曲面拐线本身所具有的性质还有待今后作更进一步的探索和证明。

The concave-convex property of a surface is one of the important and classical issues in differential geometry. Dividing a surface into several different concave-convex areas is of great value when studying the concave-convex property of the surface and its other properties. There are three different cases in the research of concave-convex areas of a surface in 3-dimensional Euclidean space, such as updown concave, left-right concave and fore-and-aft concave. The inflexion-curve of a surface is just a boundary shared by two contiguous （but not intersectant） areas on the surface the concave-convex directions of which are opposite to each other. It is a useful method to partition a surface into different concave-convex areas. The new concept of inflexion-curve of a surface is defined by using theories of traditional differential calculus and classical differential geometry. Through studying the characteristics of vertieal section line and its relations with the surface, the ways discovered can be used to find out a inflexion-curve and to determine whether it is a inflexion-curve or not on a surface. It is convenient to divide a surface into different up-down concave areas. Similarly, it can also be used to distinguish the parts of a surface, the left-right concave and fore-and-aft concave directions which differ from each other. Finally, it seems to be better to find out more about inflexion-curve itself in future.