For the cut locus of a point in a Riemannian manifold, see cut locus (Riemannian manifold). In Riemannian geometry, the cut locus of a point p in a Manifold is roughly the set of all other points for which there are multiple geodesics
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. In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit Then the cut locus is the medial axis of the polygon. The medial axis is a method for representing the Shape of objects by finding the Topological skeleton, a set of curves which roughly run along the middle of an object 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. What is a polyhedron? We can at least say that a polyhedron is built up from different kinds of element or entity each associated with a different number of dimensions 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. In Geometry the net of a Polyhedron is an arrangement of edge-joined Polygons in the plane which can be folded (along edges to become the faces of the polyhedron
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