Osculating Curves: Around the Tait-Kneser Theorem
Osculating Curves: Around the
Tait-Kneser Theorem
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密切曲线：围绕 Tait-Kneser 定理
DOI：
10.1007/s00283-012-9336-6
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
V. Timorin
中科院分区：
文献类型：
--
作者：
É. Ghys;S. Tabachnikov;V. Timorin
平滑平面曲线的密切圆（或曲率圆）的概念对于每个微积分和初等微分几何的学生来说都是熟悉的：这是比所有其他圆更好地在一点处逼近曲线的圆。人们可能会说密切圆穿过曲线上三个无限接近的点。更具体地说，在曲线上选取三个点并通过这些点画一个圆。由于这些点彼此趋向，所以圆有一个极限位置：这就是密切圆。它的半径就是曲线的曲率半径，半径的倒数就是曲线的曲率。如果曲线和密切圆都局部表示为平滑函数图，那么不仅这些函数的值而且它们的一阶和二阶导数在接触点处重合。请你的数学朋友画出一条曲线的弧线和一些密切圆。您很可能会看到如图 1 所示的内容。
The notion of osculating circle (or circle of curvature) of a smooth plane curve is familiar to every student of calculus and elementary differential geometry: this is the circle that approximates the curve at a point better than all other circles. One may say that the osculating circle passes through three infinitesimally close points on the curve. More specifically, pick three points on the curve and draw a circle through these points. As the points tend to each other, there is a limiting position of the circle: this is the osculating circle. Its radius is the radius of curvature of the curve, and the reciprocal of the radius is the curvature of the curve. If both the curve and the osculating circle are represented locally as graphs of smooth functions then not only the values of these functions but also their first and second derivatives coincide at the point of contact. Ask your mathematical friend to sketch an arc of a curve and a few osculating circles. Chances are, you will see something like Figure 1.