Variations on the Tait-Kneser theorem
Differential Geometry
2021-06-04 v2 Classical Analysis and ODEs
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.
Cite
@article{arxiv.2104.02170,
title = {Variations on the Tait-Kneser theorem},
author = {Gil Bor and Connor Jackman and Serge Tabachnikov},
journal= {arXiv preprint arXiv:2104.02170},
year = {2021}
}
Comments
11 pages, 7 figures. One coauthor added. A new result added, a Keplerian version of the 4 vertex theorem