Variations on the Tait-Kneser theorem

At every point, a smooth plane curve can be approximated, to second order, by a circle; this circle is called osculating. One may think of the osculating circle as passing through three infinitesimally close points of the curve. A vertex of the curve is a point at which the osculating circle hyper-osculates: it approximates the curve to third order. Equivalently, a vertex is a critical point of the curvature function. Consider a (necessarily non-closed) curve, free from vertices. The classical Tait-Kneser theorem [13, 5] (see also [3, 10]), states that the osculating circles of the curve are pairwise disjoint, see Figure 1. This theorem is closely related to the four vertex theorem of S. Mukhopadhyaya [8] that a plane oval has at least 4 vertices (see again [3, 10]). Figure 1 illustrates the Tait-Kneser theorem: it shows an annulus foliated by osculating circles of a curve.