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.
The logarithmic spiral or the pictured Archimedean spiral provide examples of curves whose curvature is monotonic for the entire curve. This monotonicity cannot happen for a simple closed curve (by the four-vertex theorem, there are at least four vertices where the curvature reaches an extreme point) but for such curves the theorem can be applied to the arcs of the curves between its vertices.

The theorem is named after Peter Tait, who published it in 1896, and Adolf Kneser, who rediscovered it and published it in 1912. 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.
This arc length must be greater than the straight-line distance between the same two centers, so the two circles have centers closer together than the difference of their radii, from which the theorem follows.

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.
Q188444
differential geometry
Concept
Q1060874
plane curve
Concept
Q2166240
osculating circle
UNK
Q724944
logarithmic spiral
Concept
Q28967
Archimedean spiral
Concept
Q260928
simple closed curve
Concept
Q994401
four-vertex theorem
UNK
Q362765
vertices
UNK
Q742536
Peter Tait
PER
Q68519
Adolf Kneser
PER
Q658654
evolute
Concept
Q131187
Taylor polynomial
Concept
Q7106440
osculating conics
Concept
Q85804930
Tait–Kneser theorem
Concept
1896-01-01T00:00:00Z^^http://www.w3.org/2001/XMLSchema#dateTime
1896
DATE
1912-01-01T00:00:00Z^^http://www.w3.org/2001/XMLSchema#dateTime
1912
DATE
named after
named after