Variations on the Tait-Kneser theorem (Q6169267)

The Tait-Kneser theorem states that the osculating circles of a vertex-free curve in the Euclidean plane \(\mathbb R^{2}\) are disjoint and nested. The authors present a proof of the above result which is based on the following mapping: A circle with equation \((x-a)^2+(y-b)^2= r^2\), \(r>0\), is mapped to the point \((a,b,r)\) in the upper half space \(\mathbb R^{2,1}_{+}\) of the pseudo-Euclidean space \(\mathbb R^{2,1}\). Then the following version of the Tait-Kneser theorem is established: Theorem. In the centro-affine plane \(\mathbb R^{2}\), the osculating central conics of a star-shaped curve with monotone centro-affine curvature are disjoint and nested. The proof relies on sending a central conic \(ax^2+2bxy+cy^2=1\) to the point \((a,b,c)\) in \(\mathbb R^{2,1}\), which is equipped with an appropriate inner product. The next section deals with the space of those conics in the Euclidean plane \(\mathbb{R}^2\) which have a focus at the origin. Such conics are called Kepler conics and admit a parametrisation by the half space \(\mathbb R^{2,1}_{+}\). This leads to another variant of the Tait-Kneser theorem. This nicely illustrated paper is rounded off by several remarks about other three-parameter families of curves for which a version of the Tait-Kneser theorem holds. Reviewer's remark: The mapping appearing in the first part of the paper is just a small variation of the cyclographic mapping, a named coined by [\textit{W. Fiedler}, Cyklographie oder Construction der Aufgaben über Kreise und Kugeln, und elementare Geometrie der Kreis- und Kugelsysteme. Leipzig: Teubner (1882; JFM 14.0500.02)]. An analogue of Lemma~3, where a relationship between null curves in \(\mathbb R^{1,2}\) and osculating circles of planar curves is given, can be found on page 253 of the book [\textit{E. Müller} and \textit{J. L. Krames} (ed.), Vorlesungen über darstellende Geometrie. Bd. II: Die Zyklographie. Wien: F. Deuticke (1929; JFM 55.0347.06)].