Cut locus

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For the cut locus of a point in a Riemannian manifold, see cut locus (Riemannian manifold).

The cut locus is a mathematical structure defined for a closed set S in a Euclidean space X, that is, a space in which the length of a path is defined. The cut locus of S is the closure of the set of all points p of X that have two or more distinct shortest paths in X from S to p.

For example, let S be the boundary of a simple polygon, and X the interior of the polygon. Then the cut locus is the medial axis of the polygon. The points on the medial axis are centers of maximal disks that touch the polygon boundary at two or more points, corresponding to two or more shortest paths to the disk center. As a second example, let S be a point x on the surface of a convex polyhedron P, and X the surface itself. Then the cut locus of x is what is known as the ridge tree of P with respect to x. This ridge tree has the property that cutting the surface along its edges unfolds P to a simple planar polygon. This polygon can be viewed as a net for the polyhedron.