The cut locus is a mathematical structure defined for a closed set
in a space
as the closure of the set of all points
that have two or more distinct shortest paths in
from
to
.
Definition in a special case
Let
be a metric space, equipped with the metric
, and let
be a point. The cut locus of
in
(
), is the locus of all the points in
for which there exists at least two distinct shortest paths to
in
. More formally,
for a point
in
if and only if there exists two paths
such that
,
,
, and the trajectories of the two paths are distinct.
Examples
For example, let ''S'' be the boundary of a simple
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two t ...
, and ''X'' the interior of the polygon.
Then the cut locus is the
medial axis
The medial axis of an object is the set of all points having more than one closest point on the object's boundary. Originally referred to as the topological skeleton, it was introduced in 1967 by Harry Blum as a tool for biological shape reco ...
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
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on ...
''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.
Example for the special case
Let
, that is the regular
2-sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
. Then the cut locus of every point on the sphere consists of exactly one point, namely the antipodal one.
References
Mathematical structures
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