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arXiv · 2104.02170

Variations on the Tait-Kneser theorem

Gil Bor·Connor Jackman·Serge Tabachnikov
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Abstract

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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Gil Bor, Connor Jackman, Serge Tabachnikov. 2021-04-05. Variations on the Tait-Kneser theorem. https://arxiv.org/abs/2104.02170

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