Cut locus

The cut locus is a mathematical structure defined for a closed set S {displaystyle S} in a space X {displaystyle X} in which the length of every path is well defined. The cut locus of S {displaystyle S} is the closure of the set of all points p ∈ X {displaystyle pin X} that have two or more distinct shortest paths in X {displaystyle X} from S {displaystyle S} to p {displaystyle p} . The cut locus is a mathematical structure defined for a closed set S {displaystyle S} in a space X {displaystyle X} in which the length of every path is well defined. The cut locus of S {displaystyle S} is the closure of the set of all points p ∈ X {displaystyle pin X} that have two or more distinct shortest paths in X {displaystyle X} from S {displaystyle S} to p {displaystyle p} . Let X {displaystyle X} be a metric space, equipped with the metric d X {displaystyle mathrm {d} _{X}} , and let x ∈ X {displaystyle xin X} be a point. The cut locus of x {displaystyle x} in X {displaystyle X} ( CL X ⁡ ( x ) {displaystyle operatorname {CL} _{X}(x)} ), is the locus of all the points in X {displaystyle X} for which there exists at least two distinct shortest paths to x {displaystyle x} in X {displaystyle X} . More formally, y ∈ CL X ⁡ ( x ) {displaystyle yin operatorname {CL} _{X}(x)} for a point y {displaystyle y} in X {displaystyle X} if and only if there exists two paths γ , γ ′ : I → X {displaystyle gamma ,gamma ':I o X} such that γ ( 0 ) = γ ′ ( 0 ) = x {displaystyle gamma (0)=gamma '(0)=x} , γ ( 1 ) = γ ′ ( 1 ) = y {displaystyle gamma (1)=gamma '(1)=y} , | γ | = | γ ′ | = d X ( x , y ) {displaystyle |gamma |=|gamma '|=mathrm {d} _{X}(x,y)} , and the trajectories of the two paths are distinct. 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 ofmaximal disks that touch the polygon boundary at two or more points, corresponding to two or moreshortest 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. Let X = S 2 {displaystyle X=S^{2}} , that is the regular 2-sphere. Then the cut locus of every point on the sphere consists of exactly one point, namely the antipodal one.