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Cut Locus
The cut locus is a mathematical structure defined for a closed set S in a space X as the closure of the set of all points p\in X that have two or more distinct shortest paths in X from S to p. Definition in a special case Let X be a metric space, equipped with the metric \mathrm_X, and let x \in X be a point. The cut locus of x in X (\operatorname_X(x)), is the locus of all the points in X for which there exists at least two distinct shortest paths to x in X. More formally, y \in \operatorname_X(x) for a point y in X if and only if there exists two paths \gamma,\gamma':I\to X such that \gamma(0) = \gamma'(0) = x, \gamma(1)=\gamma'(1)=y, , \gamma, =, \gamma', = \mathrm_X(x,y), and the trajectories of the two paths are distinct. Examples 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 bounda ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 minimizing geodesics connecting them from p, but it may contain additional points where the minimizing geodesic is unique, under certain circumstances. The distance function from ''p'' is a smooth function except at the point ''p'' itself and the cut locus. Definition Fix a point p in a complete Riemannian manifold (M,g), and consider the tangent space T_pM. It is a standard result that for sufficiently small v in T_p M, the curve defined by the Riemannian exponential map, \gamma(t) = \exp_p(tv) for t belonging to the interval ,1/math> is a minimizing geodesic, and is the unique minimizing geodesic connecting the two endpoints. Here \exp_p denotes the exponential map from p. The cut locus of p in the tangent space is defined to be the set of all vectors v in T_pM such that \gamma(t)=\exp_p(tv) is a minimizing geodesic for t \in ,1/math> bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 together, may be called a polygon. The segments of a polygonal circuit are called its '' edges'' or ''sides''. The points where two edges meet are the polygon's '' vertices'' (singular: vertex) or ''corners''. The interior of a solid polygon is sometimes called its ''body''. An ''n''-gon is a polygon with ''n'' sides; for example, a triangle is a 3-gon. A simple polygon is one which does not intersect itself. Mathematicians are often concerned only with the bounding polygonal chains of simple polygons and they often define a polygon accordingly. A polygonal boundary may be allowed to cross over itself, creating star polygons and other self-intersecting polygons. A polygon is a 2-dimensional example of the more general polytope in any ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 recognition. In mathematics the closure of the medial axis is known as the cut locus. In 2D, the medial axis of a subset ''S'' which is bounded by planar curve ''C'' is the locus of the centers of circles that are tangent to curve ''C'' in two or more points, where all such circles are contained in ''S''. (It follows that the medial axis itself is contained in ''S''.) The medial axis of a simple polygon is a tree whose leaves are the vertices of the polygon, and whose edges are either straight segments or arcs of parabolas. The medial axis together with the associated radius function of the maximally inscribed discs is called the medial axis transform (MAT). The medial axis transform is a complete shape descriptor (see also shape analysis) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 the same plane. Cubes and pyramids are examples of convex polyhedra. A polyhedron is a 3-dimensional example of a polytope, a more general concept in any number of dimensions. Definition Convex polyhedra are well-defined, with several equivalent standard definitions. However, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of "polyhedron" have been given within particular contexts,. some more rigorous than others, and there is not universal agreement over which of these to choose. Some of these definitions exclude shapes that have often been counted as polyhedra (such as the self-crossing polyhedra) or include shapes that are often not considered as valid p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Net (polyhedron)
In geometry, a net of a polyhedron is an arrangement of non-overlapping edge-joined polygons in the plane which can be folded (along edges) to become the faces of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and solid geometry in general, as they allow for physical models of polyhedra to be constructed from material such as thin cardboard. An early instance of polyhedral nets appears in the works of Albrecht Dürer, whose 1525 book ''A Course in the Art of Measurement with Compass and Ruler'' (''Unterweysung der Messung mit dem Zyrkel und Rychtscheyd '') included nets for the Platonic solids and several of the Archimedean solids. These constructions were first called nets in 1543 by Augustin Hirschvogel. Existence and uniqueness Many different nets can exist for a given polyhedron, depending on the choices of which edges are joined and which are separated. The edges that are cut from a convex polyhedron to form a net must form a spanning tree of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings. Basic terminology As mentioned earlier is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Structures
In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additional meaning or significance. A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, events, equivalence relations, differential structures, and categories. Sometimes, a set is endowed with more than one feature simultaneously, which allows mathematicians to study the interaction between the different structures more richly. For example, an ordering imposes a rigid form, shape, or topology on the set, and if a set has both a topology feature and a group feature, such that these two features are related in a certain way, then the structure becomes a topological group. Mappings between sets which preserve structures (i.e., structures in the do ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
