Osculating curves: around the Tait-Kneser Theorem

The notion of osculating circle (or circle of curvature) of a smooth plane curve is familiar to every student of calculus and elementary differential geometry: this is the circle that approximates the curve at a point better than all other circles. One may say that the osculating circle passes through three infinitesimally close points on the curve. More specifically, pick three points on the curve and draw a circle through these points. As the points tend to each other, there is a limiting position of the circle: this is the osculating circle. Its radius is the radius of curvature of the curve, and the reciprocal of the radius is the curvature of the curve. If both the curve and the osculating circle are represented locally as graphs of smooth functions then not only the values of these functions but also their first and second derivatives coincide at the point of contact. Ask your mathematical friend to sketch an arc of a curve and a few osculating circles. Chances are, you will see something like Figure 1. Figure 1: Osculating circles? This is wrong! The following theorem was discovered by Peter Guthrie Tait in the end of the 19th century [9] and rediscovered by Adolf Kneser early in the 20th century [4]. Theorem 1 The osculating circles of an arc with monotonic positive curvature are pairwise disjoint and nested. Tait’s paper is so short that we quote it almost verbatim (omitting some old-fashioned terms): 1 a