Loewner's ``forgotten" theorem
Loewner's ``forgotten" theorem
复制标题
勒纳“被遗忘”定理
DOI：
10.1007/s00283-021-10144-z
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Albers, P.
中科院分区：
文献类型：
--
作者：
Albers, P.
有方向的光滑闭合曲线将平面划分为若干区域。对于不在曲线上的每一点，曲线围绕该点的完整旋转次数称为曲线围绕该点的旋转次数。很明显，一个区域内任意点的旋转数与该区域内其他点的旋转数相同。旋转数为整数；它的符号表示总的旋转是逆时针还是顺时针。作为一个整数，如果点和曲线连续变化，旋转数不会改变，除非点穿过曲线，或者（视观点而定）曲线经过该点，在这种情况下，旋转数一般会根据图1中的规则改变1。
An oriented smooth closed curve partitions the plane into a number of regions. For every point not on the curve, the number of complete turns that the curve makes about this point is called the rotation number of the curve about the point. Clearly, the rotation number about any point in a region is the same as that for every other point in the region. The rotation number is an integer; its sign indicates whether the total rotation is counterclockwise or clockwise. Being an integer, the rotation number does not change if the point and the curve change continuously, unless the point crosses the curve or—depending on the point of view—the curve passes through the point, in which case the rotation number changes, generically by one, according to the rule in Figure 1.