Pestov–Ionin theorem

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

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.

History and generalizations

Although a version of this was published for convex curves by Wilhelm Blaschke in 1916,Template:R it is named for Template:Ill and Template:Ill, 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.Template:R The theorem has been generalized further, to curves of bounded average curvature (singly differentiable, and satisfying a Lipschitz condition on the derivative),Template:R 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).Template:R

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.Template:R It has also been used for planning the motion of the cutter in a milling machine for pocket machining,Template:R and in reconstructing curves from scattered data points.Template:R