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Delaunay Triangulation
In computational geometry, a Delaunay triangulation or Delone triangulation of a set of points in the plane subdivides their convex hull into triangles whose circumcircles do not contain any of the points; that is, each circumcircle has its generating points on its circumference, but all other points in the set are outside of it. This maximizes the size of the smallest angle in any of the triangles, and tends to avoid sliver triangles. The triangulation is named after Boris Delaunay for his work on it from 1934. If the points all lie on a straight line, the notion of triangulation becomes degenerate and there is no Delaunay triangulation. For four or more points on the same circle (e.g., the vertices of a rectangle) the Delaunay triangulation is not unique: each of the two possible triangulations that split the quadrangle into two triangles satisfies the "Delaunay condition", i.e., the requirement that the circumcircles of all triangles have empty interiors. By considering ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Delaunay Circumcircles Vectorial
Delaunay is a French surname. Notable people with the surname include: People Arts * Catherine Delaunay (born 1969), French jazz clarinet player and composer * Charles Delaunay (1911–1988), French author and jazz expert * Constance Delaunay (1922-2013), French translator, playwright, novelist and short stories writer * Jules-Élie Delaunay (1828–1891), French painter * Louis Arsene Delaunay (1826–1903), French actor * Robert Delaunay (1885–1941), French artist * Rose Delaunay (1857–after 1937), French opera ainger * Sonia Delaunay (1885–1979), Ukrainian-French artist * Vadim Delaunay (1947–1983), Russian poet and dissident Football * Henri Delaunay (1883–1955), French football administrator * Jean-Pierre Delaunay (born 1966), French footballer * Pierre Delaunay, football administrator Science * Boris Delaunay (1890–1980), Soviet/Russian mathematician, inventor of Delaunay triangulation * Charles-Eugène Delaunay (1816–1872), French astronomer and mathematician ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bowyer–Watson Algorithm
In computational geometry, the Bowyer–Watson algorithm is a method for computing the Delaunay triangulation of a finite set of points in any number of dimensions. The algorithm can be also used to obtain a Voronoi diagram of the points, which is the dual graph of the Delaunay triangulation. Description The Bowyer–Watson algorithm is an incremental algorithm. It works by adding points, one at a time, to a valid Delaunay triangulation of a subset of the desired points. After every insertion, any triangles whose circumcircles contain the new point are deleted, leaving a star-shaped polygonal hole which is then re-triangulated using the new point. By using the connectivity of the triangulation to efficiently locate triangles to remove, the algorithm can take ''O(N log N)'' operations to triangulate N points, although special degenerate cases exist where this goes up to ''O(N2)''. File:Bowyer-Watson 0.png, First step: insert a node in an enclosing "super"-triangle File:Bowyer-W ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Spanner
A geometric spanner or a -spanner graph or a -spanner was initially introduced as a weighted graph over a set of points as its vertices for which there is a -path between any pair of vertices for a fixed parameter . A -path is defined as a path through the graph with weight at most times the spatial distance between its endpoints. The parameter is called the stretch factor or dilation factor of the spanner. In computational geometry, the concept was first discussed by L.P. Chew in 1986, although the term "spanner" was not used in the original paper. The notion of graph spanners has been known in graph theory: -spanners are spanning subgraphs of graphs with similar dilation property, where distances between graph vertices are defined in graph-theoretical terms. Therefore geometric spanners are graph spanners of complete graphs embedded in the plane with edge weights equal to the distances between the embedded vertices in the corresponding metric. Spanners may be used in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nearest Neighbor Graph
The nearest neighbor graph (NNG) is a directed graph defined for a set of points in a metric space, such as the Euclidean distance in the plane. The NNG has a vertex for each point, and a directed edge from ''p'' to ''q'' whenever ''q'' is a nearest neighbor of ''p'', a point whose distance from ''p'' is minimum among all the given points other than ''p'' itself. In many uses of these graphs, the directions of the edges are ignored and the NNG is defined instead as an undirected graph. However, the nearest neighbor relation is not a symmetric one, i.e., ''p'' from the definition is not necessarily a nearest neighbor for ''q''. In theoretical discussions of algorithms a kind of general position is often assumed, namely, the nearest (k-nearest) neighbor is unique for each object. In implementations of the algorithms it is necessary to bear in mind that this is not always the case. For situations in which it is necessary to make the nearest neighbor for each object unique, the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paraboloid
In geometry, a paraboloid is a quadric surface that has exactly one axial symmetry, axis of symmetry and no central symmetry, center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every plane section of a paraboloid made by a plane Parallel (geometry)#A line and a plane, parallel to the axis of symmetry is a parabola. The paraboloid is hyperbolic if every other plane section is either a hyperbola, or two crossing lines (in the case of a section by a tangent plane). The paraboloid is elliptic if every other nonempty plane section is either an ellipse, or a single point (in the case of a section by a tangent plane). A paraboloid is either elliptic or hyperbolic. Equivalently, a paraboloid may be defined as a quadric surface that is not a cylinder, and has an implicit surface, implicit equation whose part of degree two may be factored over the complex numbers into two different linear factors. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Process
In probability theory, statistics and related fields, a Poisson point process (also known as: Poisson random measure, Poisson random point field and Poisson point field) is a type of mathematical object that consists of Point (geometry), points randomly located on a Space (mathematics), mathematical space with the essential feature that the points occur independently of one another. The process's name derives from the fact that the number of points in any given finite region follows a Poisson distribution. The process and the distribution are named after French mathematician Siméon Denis Poisson. The process itself was discovered independently and repeatedly in several settings, including experiments on radioactive decay, telephone call arrivals and actuarial science. This point process is used as a mathematical model for seemingly random processes in numerous disciplines including astronomy,G. J. Babu and E. D. Feigelson. Spatial point processes in astronomy. ''Journal of st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by \chi (Greek alphabet, Greek lower-case letter chi (letter), chi). The Euler characteristic was originally defined for polyhedron, polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico. Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant. In modern mathematics, the Euler characteristic arises from homology (mathematics), homology and, more abstractly, homological algebra. Polyhedra The Euler characteristic was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Delaunay Triangulation Does Not Minimize Edge Length
Delaunay is a French surname. Notable people with the surname include: People Arts * Catherine Delaunay (born 1969), French jazz clarinet player and composer * Charles Delaunay (1911–1988), French author and jazz expert * Constance Delaunay (1922-2013), French translator, playwright, novelist and short stories writer * Jules-Élie Delaunay (1828–1891), French painter * Louis Arsene Delaunay (1826–1903), French actor * Robert Delaunay (1885–1941), French artist * Rose Delaunay (1857–after 1937), French opera ainger * Sonia Delaunay (1885–1979), Ukrainian-French artist * Vadim Delaunay (1947–1983), Russian poet and dissident Football * Henri Delaunay (1883–1955), French football administrator * Jean-Pierre Delaunay (born 1966), French footballer * Pierre Delaunay, football administrator Science * Boris Delaunay (1890–1980), Soviet/Russian mathematician, inventor of Delaunay triangulation * Charles-Eugène Delaunay (1816–1872), French astronomer and m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Example Steps In Delauney Triangularization
Example may refer to: * ''exempli gratia'' (e.g.), usually read out in English as "for example" * .example, reserved as a domain name that may not be installed as a top-level domain of the Internet ** example.com, example.net, example.org, and example.edu: second-level domain names reserved for use in documentation as examples * HMS ''Example'' (P165), an Archer-class patrol and training vessel of the Royal Navy Arts * ''The Example'', a 1634 play by James Shirley * ''The Example'' (comics), a 2009 graphic novel by Tom Taylor and Colin Wilson * Example (musician), the British dance musician Elliot John Gleave (born 1982) * ''Example'' (album), a 1995 album by American rock band For Squirrels See also * Exemplar (other), a prototype or model which others can use to understand a topic better * Exemplum, medieval collections of short stories to be told in sermons * Eixample The Eixample (, ) is a district of Barcelona between the old city (Ciutat Vella) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypersphere
In mathematics, an -sphere or hypersphere is an - dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer . The circle is considered 1-dimensional and the sphere 2-dimensional because a point within them has one and two degrees of freedom respectively. However, the typical embedding of the 1-dimensional circle is in 2-dimensional space, the 2-dimensional sphere is usually depicted embedded in 3-dimensional space, and a general -sphere is embedded in an -dimensional space. The term ''hyper''sphere is commonly used to distinguish spheres of dimension which are thus embedded in a space of dimension , which means that they cannot be easily visualized. The -sphere is the setting for -dimensional spherical geometry. Considered extrinsically, as a hypersurface embedded in -dimensional Euclidean space, an -sphere is the locus of points at equal distance (the ''radius'') from a given '' center'' point. Its interior, consisting of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Position
In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that are possible, which is referred to as special position. Its precise meaning differs in different settings. For example, generically, two lines in the plane intersect in a single point (they are not parallel or coincident). One also says "two generic lines intersect in a point", which is formalized by the notion of a ''generic point''. Similarly, three generic points in the plane are not collinear; if three points are collinear (even stronger, if two coincide), this is a degenerate case. This notion is important in mathematics and its applications, because degenerate cases may require an exceptional treatment; for example, when stating general theorems or giving precise statements thereof, and when writing computer programs (see '' generic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, * a 0-dimensional simplex is a point, * a 1-dimensional simplex is a line segment, * a 2-dimensional simplex is a triangle, * a 3-dimensional simplex is a tetrahedron, and * a 4-dimensional simplex is a 5-cell. Specifically, a -simplex is a -dimensional polytope that is the convex hull of its vertices. More formally, suppose the points u_0, \dots, u_k are affinely independent, which means that the vectors u_1 - u_0,\dots, u_k-u_0 are linearly independent. Then, the simplex determined by them is the set of points C = \left\. A regular simplex is a simplex that is also a regular polytope. A regular -simplex may be constructed from a regular -simplex by connecting a new vertex to all original vertices by the common ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
