Tait–Kneser theorem

In differential geometry, the Tait–Kneser theorem states that, if a smooth plane curve has monotonic curvature, then the osculating circles of the curve are disjoint and nested within each other.

[1] The logarithmic spiral or the pictured Archimedean spiral provide examples of curves whose curvature is monotonic for the entire curve.

[1][2][3] Tait's proof follows simply from the properties of the evolute, the curve traced out by the centers of osculating circles.

For curves with monotone curvature, the arc length along the evolute between two centers equals the difference in radii of the corresponding circles.

[1][2] Analogous disjointness theorems can be proved for the family of Taylor polynomials of a given smooth function, and for the osculating conics to a given smooth curve.

The nested osculating circles of an Archimedean spiral . The spiral itself is not shown, but is visible where the circles are more dense.
A logarithmic spiral with some osculating circles. These do not intersect, hence satisfying the Tait–Kneser theorem.