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Cut Locus
In differential geometry, the cut locus of a point on a manifold is the closure of the set of all other points on the manifold that are connected to by two or more distinct shortest geodesics. More generally, the cut locus of a closed set on the manifold is the closure of the set of all other points on the manifold connected to by two or more distinct shortest geodesics. Examples In the Euclidean plane, a point ''p'' has an empty cut locus, because every other point is connected to ''p'' by a unique geodesic (the line segment between the points). On the sphere, the cut locus of a point consists of the single antipodal point diametrically opposite to it. On an infinitely long cylinder, the cut locus of a point consists of the line opposite the point. Let ''X'' be the boundary of a simple polygon in the Euclidean plane. Then the cut locus of ''X'' in the interior of the polygon is the polygon's medial axis. Points on the medial axis are centers of disks that touch the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geodesics And Geodesic Circles On An Oblate Ellipsoid
In geometry, a geodesic () is a curve representing in some sense the locally shortest path (arc (geometry), arc) between two points in a differential geometry of surfaces, surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection (mathematics), connection. It is a generalization of the notion of a "Line (geometry), straight line". The noun ''wikt:geodesic, geodesic'' and the adjective ''wikt:geodetic, geodetic'' come from ''geodesy'', the science of measuring the size and shape of Earth, though many of the underlying principles can be applied to any Ellipsoidal geodesic, ellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's Planetary surface, surface. For a spherical Earth, it is a line segment, segment of a great circle (see also great-circle distance). The term has since been generalized to more abstract mathematical spaces; for example, in graph the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Map (Riemannian Geometry)
In Riemannian geometry, an exponential map is a map from a subset of a tangent space T''p''''M'' of a Riemannian manifold (or pseudo-Riemannian manifold) ''M'' to ''M'' itself. The (pseudo) Riemannian metric determines a canonical affine connection, and the exponential map of the (pseudo) Riemannian manifold is given by the exponential map of this connection. Definition Let be a differentiable manifold and a point of . An affine connection on allows one to define the notion of a straight line through the point .A source for this section is , which uses the term "linear connection" where we use "affine connection" instead. Let be a tangent vector to the manifold at . Then there is a unique geodesic :[0,1] → satisfying with initial tangent vector . The corresponding exponential map is defined by . In general, the exponential map is only ''locally defined'', that is, it only takes a small neighborhood of the origin at , to a neighborhood of in the manifold. This is b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Structures
In mathematics, a structure on a set (or on some sets) refers to providing or endowing it (or them) with certain additional features (e.g. an operation, relation, metric, or topology). Τhe additional features are attached or related to the set (or to the sets), so as to provide it (or them) with some additional meaning or significance. A partial list of possible structures are measures, algebraic structures ( groups, fields, etc.), topologies, metric structures (geometries), orders, graphs, 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 grou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete & Computational Geometry
'' Discrete & Computational Geometry'' is a peer-reviewed mathematics journal published quarterly by Springer. Founded in 1986 by Jacob E. Goodman and Richard M. Pollack, the journal publishes articles on discrete geometry and computational geometry. Abstracting and indexing The journal is indexed in: * ''Mathematical Reviews'' * '' Zentralblatt MATH'' * ''Science Citation Index'' * ''Current Contents ''Current Contents'' is a rapid alerting service database from Clarivate, formerly the Institute for Scientific Information and Thomson Reuters. It is published online and in several different printed subject sections. History ''Current Contents ...'' Notable articles Two articles published in ''Discrete & Computational Geometry'', one by Gil Kalai in 1992 with a proof of a subexponential upper bound on the diameter of a polytope and another by Samuel Ferguson in 2006 on the Kepler conjecture on optimal three-dimensional sphere packing, earned their authors the Fulk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graphs And Combinatorics
''Graphs and Combinatorics'' (ISSN 0911-0119, abbreviated ''Graphs Combin.'') is a peer-reviewed academic journal in graph theory, combinatorics, and discrete geometry published by Springer Japan. Its editor-in-chief is Katsuhiro Ota of Keio University. The journal was first published in 1985. Its founding editor in chief was Hoon Heng Teh of Singapore, the president of the Southeast Asian Mathematics Society, and its managing editor was Jin Akiyama. Originally, it was subtitled "An Asian Journal". In most years since 1999, it has been ranked as a second-quartile journal in discrete mathematics and theoretical computer science by SCImago Journal Rank The SCImago Journal Rank (SJR) indicator is a measure of the prestige of scholarly journals that accounts for both the number of citations received by a journal and the prestige of the journals where the citations come from. Etymology SCImago ..... References {{reflist Academic journals established in 1985 Combinatorics jo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blooming (geometry)
In the geometry of convex polyhedra, blooming or continuous blooming is a continuous three-dimensional motion of the surface of the polyhedron, cut to form a polyhedral net, from the polyhedron into a flat and non-self-overlapping placement of the net in a plane. As in rigid origami, the polygons of the net must remain individually flat throughout the motion, and are not allowed to intersect or cross through each other. A blooming, reversed to go from the flat net to a polyhedron, can be thought of intuitively as a way to fold the polyhedron from a paper net without bending the paper except at its designated creases. An early work on blooming by Biedl, Lubiw, and Sun from 1999 showed that some nets for non-convex but topologically spherical polyhedra have no blooming. The question of whether every convex polyhedron admits a net with a blooming was posed by Robert Connelly, and came to be known as Connelly’s blooming conjecture. More specifically, Miller and Pak suggested in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Source Unfolding
In computational geometry, the source unfolding of a convex polyhedron is a net obtained by cutting the polyhedron along the cut locus of a point on the surface of the polyhedron. The cut locus of a point p consists of all points on the surface that have two or more shortest geodesics to p. For every convex polyhedron, and every choice of the point p on its surface, cutting the polyhedron on the cut locus will produce a result that can be unfolded into a flat plane, producing the source unfolding. The resulting net may, however, cut across some of the faces of the polyhedron rather than only cutting along its edges. The source unfolding can also be continuously transformed from the polyhedron to its flat net, keeping flat the parts of the net that do not lie along edges of the polyhedron, as a blooming of the polyhedron. The unfolded shape of the source unfolding is always a star-shaped polygon, with all of its points visible by straight line segments from the image of p; this is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexandrov's Uniqueness Theorem
Alexandrov's theorem on polyhedra is a rigidity theorem in mathematics, describing three-dimensional convex polyhedra in terms of the distances between points on their surfaces. It implies that convex polyhedra with distinct shapes from each other also have distinct metric spaces of surface distances, and it characterizes the metric spaces that come from the surface distances on polyhedra. It is named after Soviet mathematician Aleksandr Danilovich Aleksandrov, who published it in the 1940s. Statement of the theorem The surface of any convex polyhedron in Euclidean space forms a metric space, in which the distance between two points is measured by the length of the shortest path from one point to the other along the surface. Within a single shortest path, distances between pairs of points equal the distances between corresponding points of a line segment of the same length; a path with this property is known as a geodesic. This property of polyhedral surfaces, that every pair of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toponogov Theorem
In the mathematical field of Riemannian geometry, Toponogov's theorem (named after Victor Andreevich Toponogov) is a triangle comparison theorem. It is one of a family of comparison theorems that quantify the assertion that a pair of geodesics emanating from a point ''p'' spread apart more slowly in a region of high curvature than they would in a region of low curvature. Let ''M'' be an ''m''-dimensional Riemannian manifold with sectional curvature ''K'' satisfying K\ge \delta\,. Let ''pqr'' be a geodesic triangle, i.e. a triangle whose sides are geodesics, in ''M'', such that the geodesic ''pq'' is minimal and if δ > ''0'', the length of the side ''pr'' is less than \pi / \sqrt \delta. Let ''p''′''q''′''r''′ be a geodesic triangle in the model space ''M''δ, i.e. the simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Hessian Comparison Theorem
Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States Arts, entertainment, and media * ''Local'' (comics), a limited series comic book by Brian Wood and Ryan Kelly * ''Local'' (novel), a 2001 novel by Jaideep Varma * ''The Local'' (film), a 2008 action-drama film * ''The Local'', English-language news websites in several European countries Computing * .local, a network address component Mathematics * Local property, a property which occurs on ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points * Local ring, type of ring in commutative algebra Other uses * Pub, a drinking establishment, known as a "local" to its regulars See also * * * Local group (other) * Locale (other) * Localism (other) * Locality (other) * Localization (other) * Locus (other) * Lokal (other) Lokal may refer t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hessian Matrix
In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued Function (mathematics), function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Otto Hesse, Ludwig Otto Hesse and later named after him. Hesse originally used the term "functional determinants". The Hessian is sometimes denoted by H or \nabla\nabla or \nabla^2 or \nabla\otimes\nabla or D^2. Definitions and properties Suppose f : \R^n \to \R is a function taking as input a vector \mathbf \in \R^n and outputting a scalar f(\mathbf) \in \R. If all second-order partial derivatives of f exist, then the Hessian matrix \mathbf of f is a square n \times n matrix, usually defined and arranged as \mathbf H_f= \begin \dfrac & \dfrac & \cdots & \dfrac \\[2.2ex] \dfrac & \dfrac & \cdots & \dfrac \\[2.2ex] \vdots & \vdot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
