Pestov–Ionin theorem

The Pestov–Ionin theorem in the differential geometry of plane curves states that every simple closed curve of curvature at most one encloses a unit disk.

A smooth simple closed curve of curvature at most one, and a unit disk enclosed by it

History and generalizations

Although a version of this was published for convex curves by Wilhelm Blaschke in 1916, it is named for German Gavrilovich Pestov and Vladimir Kuzmich Ionin, who published a version of this theorem in 1959 for non-convex doubly differentiable ($C^{2}$) curves, the curves for which the curvature is well-defined at every point. The theorem has been generalized further, to curves of bounded average curvature (singly differentiable, and satisfying a Lipschitz condition on the derivative), and to curves of bounded convex curvature (each point of the curve touches a unit disk that, within some small neighborhood of the point, remains interior to the curve).

Applications

The theorem has been applied in algorithms for motion planning. In particular it has been used for finding Dubins paths, shortest routes for vehicles that can move only in a forwards direction and that can turn left or right with a bounded turning radius. It has also been used for planning the motion of the cutter in a milling machine for pocket machining, and in reconstructing curves from scattered data points.