Tait-Kneser定理的变形
Variations on the Tait-Kneser theorem
摘要：
Tait-Kneser定理首先由Peter G. Tait在1896年指出，沿着具有单调非零曲率的平面曲线的密切圆是两两不相交且嵌套的。这篇笔记包含了一个证明这个定理使用洛伦兹几何的空间的圆。我们展示了如何一个类似的证明适用于两个变化的定理，关于密切胡克和开普勒二次曲线沿平面曲线。我们还证明了一个版本的开普勒二次曲线的四顶点定理。
英文摘要：
The Tait-Kneser theorem, first demonstrated by Peter G. Tait in 1896, states that the osculating circles along a plane curve with monotone non-vanishing curvature are pairwise disjoint and nested. This note contains a proof of this theorem using the Lorentzian geometry of the space of circles. We show how a similar proof applies to two variations on the theorem, concerning the osculating Hooke and Kepler conics along a plane curve. We also prove a version of the 4-vertex theorem for Kepler conics.
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