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Beta Skeleton
In computational geometry and geometric graph theory, a ''β''-skeleton or beta skeleton is an undirected graph defined from a set of points in the Euclidean plane. Two points ''p'' and ''q'' are connected by an edge whenever all the angles ''prq'' are sharper than a threshold determined from the numerical parameter ''β''. Circle-based definition Let ''β'' be a positive real number, and calculate an angle ''θ'' using the formulas :\theta = \begin \sin^ \frac, & \text\beta \ge 1 \\ \pi - \sin^, & \text\beta\le 1\end For any two points ''p'' and ''q'' in the plane, let ''R''''pq'' be the set of points for which angle ''prq'' is greater than ''θ''. Then ''R''''pq'' takes the form of a union of two open disks with diameter ''βd''(''p'',''q'') for ''β'' ≥ 1 and ''θ'' ≤ π/2, and it takes the form of the intersection of two open disks with diameter ''d''(''p'',''q'')/''β'' for ''β'' ≤ 1 and ''θ'' ≥ π/2. When ''β''&n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta-skeleton
In computational geometry and geometric graph theory, a ''β''-skeleton or beta skeleton is an undirected graph defined from a set of points in the Euclidean plane. Two points ''p'' and ''q'' are connected by an edge whenever all the angles ''prq'' are sharper than a threshold determined from the numerical parameter ''β''. Circle-based definition Let ''β'' be a positive real number, and calculate an angle ''θ'' using the formulas :\theta = \begin \sin^ \frac, & \text\beta \ge 1 \\ \pi - \sin^, & \text\beta\le 1\end For any two points ''p'' and ''q'' in the plane, let ''R''''pq'' be the set of points for which angle ''prq'' is greater than ''θ''. Then ''R''''pq'' takes the form of a union of two open disks with diameter ''βd''(''p'',''q'') for ''β'' ≥ 1 and ''θ'' ≤ π/2, and it takes the form of the intersection of two open disks with diameter ''d''(''p'',''q'')/''β'' for ''β'' ≤ 1 and ''θ'' ≥ π/2. When ''β''&nb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gabriel Graph
In mathematics and computational geometry, the Gabriel graph of a set S of points in the Euclidean plane expresses one notion of proximity or nearness of those points. Formally, it is the graph G with vertex set S in which any two distinct points p \in S and q \in S are adjacent precisely when the closed disc having pq as a diameter contains no other points. Another way of expressing the same adjacency criterion is that p and q should be the two closest given points to their midpoint, with no other given point being as close. Gabriel graphs naturally generalize to higher dimensions, with the empty disks replaced by empty closed balls. Gabriel graphs are named after K. Ruben Gabriel, who introduced them in a paper with Robert R. Sokal in 1969. Percolation For Gabriel graphs of infinite random point sets, the finite site percolation threshold gives the fraction of points needed to support connectivity: if a random subset of fewer vertices than the threshold is given, the remaini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geographic Information System
A geographic information system (GIS) consists of integrated computer hardware and Geographic information system software, software that store, manage, Spatial analysis, analyze, edit, output, and Cartographic design, visualize Geographic data and information, geographic data. Much of this often happens within a spatial database; however, this is not essential to meet the definition of a GIS. In a broader sense, one may consider such a system also to include human users and support staff, procedures and workflows, the Geographic Information Science and Technology Body of Knowledge, body of knowledge of relevant concepts and methods, and institutional organizations. The uncounted plural, ''geographic information systems'', also abbreviated GIS, is the most common term for the industry and profession concerned with these systems. The academic discipline that studies these systems and their underlying geographic principles, may also be abbreviated as GIS, but the unambiguous GIScie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curvature
In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or surface is contained in a larger space, curvature can be defined ''extrinsically'' relative to the ambient space. Curvature of Riemannian manifolds of dimension at least two can be defined ''intrinsically'' without reference to a larger space. For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature ''at a point'' of a differentiable curve is the curvature of its osculating circle — that is, the circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connect The Dots
Connect the dots (also known as connect-the-dots, dot to dot, join the dots or follow the dots) is a form of puzzle containing a sequence of numbered dots. When a line is drawn connecting the dots the outline of an object is revealed. The puzzles frequently contain simple line art to enhance the image created or to assist in rendering a complex section of the image. Connect the dots puzzles are generally created for children. The use of numbers can be replaced with letters or other symbols. Versions for older solvers frequently have extra solving steps to discover the order, such as those used in puzzlehunts and the connect-the-dots crosswords invented by Liz Gorski. The roots of connecting dots to create pictures or help with calligraphy can be traced back to the 19th century. The Nine Dots Puzzle is the first known puzzle game where the player has to connect dots. But in this variant the goal is not to draw a picture, but to solve a logic puzzle. The emergence of connect the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Image Analysis
Image analysis or imagery analysis is the extraction of meaningful information from images; mainly from digital images by means of digital image processing techniques. Image analysis tasks can be as simple as reading barcode, bar coded tags or as sophisticated as facial recognition system, identifying a person from their face. Computers are indispensable for the analysis of large amounts of data, for tasks that require complex computation, or for the extraction of quantitative information. On the other hand, the human visual cortex is an excellent image analysis apparatus, especially for extracting higher-level information, and for many applications — including medicine, security, and remote sensing — human analysts still cannot be replaced by computers. For this reason, many important image analysis tools such as edge detection, edge detectors and Artificial neural network, neural networks are inspired by human visual perception models. Digital Digital Image Analy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dense Graph
In mathematics, a dense graph is a graph in which the number of edges is close to the maximal number of edges (where every pair of vertices is connected by one edge). The opposite, a graph with only a few edges, is a sparse graph. The distinction of what constitutes a dense or sparse graph is ill-defined, and is often represented by 'roughly equal to' statements. Due to this, the way that density is defined often depends on the context of the problem. The graph density of simple graphs is defined to be the ratio of the number of edges with respect to the maximum possible edges. For undirected simple graphs, the graph density is: :D = \frac = \frac For directed, simple graphs, the maximum possible edges is twice that of undirected graphs (as there are two directions to an edge) so the density is: :D = \frac = \frac where is the number of edges and is the number of vertices in the graph. The maximum number of edges for an undirected graph is = \frac2, so the maximal density is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex polygon, convex'' or ''star polygon, star''. In the limit (mathematics), limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon (effectively a Line (geometry), straight line), if the edge length is fixed. General properties These properties apply to all regular polygons, whether convex or star polygon, star: *A regular ''n''-sided polygon has rotational symmetry of order ''n''. *All vertices of a regular polygon lie on a common circle (the circumscribed circle); i.e., they are concyclic points. That is, a regular polygon is a cyclic polygon. *Together with the property of equal-length sides, this implies that every regular polygon also h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Naïve Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of mathematically rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use conditionals to divert the code execution through various routes (referred to as automated decision-making) and deduce valid inferences (referred to as automated reasoning). In contrast, a heuristic is an approach to solving problems without well-defined correct or optimal results.David A. Grossman, Ophir Frieder, ''Information Retrieval: Algorithms and Heuristics'', 2nd edition, 2004, For example, although social media recommender systems are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation. As an effective method, an algorithm can be expressed within a finite amount of space and time"Any classical mathe ... [...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] |
Stretch Factor
The stretch factor (i.e., Lipschitz continuity#Definition, bilipschitz constant) of an embedding measures the factor by which the embedding distorts distances. Suppose that one metric space is embedded into another metric space by a metric map, a continuous one-to-one function that preserves or reduces the distance between every pair of points. Then the embedding gives rise to two different notions of distance between pairs of points in . Any pair of points in has both an intrinsic metric, intrinsic distance, the distance from to in , and a smaller extrinsic distance, the distance from to in . The stretch factor of the pair is the ratio between these two distances, . The stretch factor of the whole mapping is the supremum of the stretch factors of all pairs of points. The stretch factor has also been called the distortion or dilation of the mapping. The stretch factor is important in the theory of geometric spanners, weighted graphs that approximate the Euclidean distanc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
