Geometry of Calugareanu's theorem
Geometry of Calugareanu's theorem
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DOI：
10.1098/rspa.2005.1527
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发表时间：
2005-10-08
影响因子：
3.5
通讯作者：
Hannay, JH
中科院分区：
文献类型：
--
作者：
Dennis, MR;Hannay, JH
封闭扭曲带的空间几何学的一个中心结果是卡卢卡努定理（也称为白色公式，或卡卢卡努-白色-富勒定理）。这使得带的两个边缘的整数连接数可以写成带扭曲（带绕其轴的旋转速率）和其扭曲的总和。我们表明，两倍的扭曲是平均值，在所有的投影方向上，丝带出现边缘上（签署适当）的地方的数量-“本地”交叉数的丝带边缘。这补充了通常将扭曲解释为带轴曲线的有符号自交叉的平均数量。使用我们开发的形式主义，我们还构建了一个几何上的任何封闭的空间曲线的自然丝带的“扭曲框架”丝带。根据定义，这条丝带的扭曲补偿了它的扭动，所以它的连接数总是零。
A central result in the space geometry of closed twisted ribbons is Calugareanu's theorem (also known as White's formula, or the Calugareanu-White-Fuller theorem). This enables the integer linking number of the two edges of the ribbon to be written as the sum of the ribbon twist (the rate of rotation of the ribbon about its axis) and its writhe. We show that twice the twist is the average, over all projection directions, of the number of places where the ribbon appears edge-on (signed appropriately)-the 'local' crossing number of the ribbon edges. This complements the common interpretation of writhe as the average number of signed self-crossings of the ribbon axis curve. Using the formalism we develop, we also construct a geometrically natural ribbon on any closed space curve the 'writhe framing' ribbon. By definition, the twist of this ribbon compensates its writhe, so its linking number is always zero.