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Planar Straight-line Graph
In computational geometry and geometric graph theory, a planar straight-line graph (or ''straight-line plane graph'', or ''plane straight-line graph''), in short ''PSLG'', is an embedding of a planar graph in the plane such that its edges are mapped into straight-line segments. Fáry's theorem (1948) states that every planar graph has this kind of embedding. In computational geometry, PSLGs have often been called planar subdivisions, with an assumption or assertion that subdivisions are polygonal rather than having curved boundaries. PSLGs may serve as representations of various maps, e.g., geographical maps in geographical information systems. Special cases of PSLGs are triangulations (polygon triangulation, point-set triangulation). Point-set triangulations are maximal PSLGs in the sense that it is impossible to add straight edges to them while keeping the graph planar. Triangulations have numerous applications in various areas. PSLGs may be seen as a special kind of Eucl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Point Location1
A point is a small dot or the sharp tip of something. Point or points may refer to: Mathematics * Point (geometry), an entity that has a location in space or on a plane, but has no extent; more generally, an element of some abstract topological space * Point, or Element (category theory), generalizes the set-theoretic concept of an element of a set to an object of any category * Critical point (mathematics), a stationary point of a function of an arbitrary number of variables * Decimal point * Point-free geometry * Stationary point, a point in the domain of a single-valued function where the value of the function ceases to change Places * Point, Cornwall, England, a settlement in Feock parish * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Points, West Virginia, an unincorporated community in the United States Business and fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Point-set Triangulation
A triangulation of a set of points \mathcal in the Euclidean space \mathbb^d is a simplicial complex that covers the convex hull of \mathcal, and whose vertices belong to \mathcal. In the plane (when \mathcal is a set of points in \mathbb^2), triangulations are made up of triangles, together with their edges and vertices. Some authors require that all the points of \mathcal are vertices of its triangulations. In this case, a triangulation of a set of points \mathcal in the plane can alternatively be defined as a maximal set of non-crossing edges between points of \mathcal. In the plane, triangulations are special cases of planar straight-line graphs. A particularly interesting kind of triangulations are the Delaunay triangulations. They are the geometric duals of Voronoi diagrams. The Delaunay triangulation of a set of points \mathcal in the plane contains the Gabriel graph, the nearest neighbor graph and the minimal spanning tree of \mathcal. Triangulations have a number of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Geometric Algorithms
An algorithm is fundamentally a set of rules or defined procedures that is typically designed and used to solve a specific problem or a broad set of problems. Broadly, algorithms define process(es), sets of rules, or methodologies that are to be followed in calculations, data processing, data mining, pattern recognition, automated reasoning or other problem-solving operations. With the increasing automation of services, more and more decisions are being made by algorithms. Some general examples are; risk assessments, anticipatory policing, and pattern recognition technology. The following is a list of well-known algorithms. Automated planning Combinatorial algorithms General combinatorial algorithms * Brent's algorithm: finds a cycle in function value iterations using only two iterators * Floyd's cycle-finding algorithm: finds a cycle in function value iterations * Gale–Shapley algorithm: solves the stable matching problem * Pseudorandom number generators (uniformly dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Local Feature Size
Local feature size refers to several related concepts in computer graphics and computational geometry for measuring the size of a geometric object near a particular point. *Given a smooth manifold M, the local feature size at any point x \in M is the distance between x and the medial axis of M. *Given a planar straight-line graph In computational geometry and geometric graph theory, a planar straight-line graph (or ''straight-line plane graph'', or ''plane straight-line graph''), in short ''PSLG'', is an embedding of a planar graph in the plane such that its edges are ma ..., the local feature size at any point x is the radius of the smallest closed ball centered at x which intersects any two disjoint features (vertices or edges) of the graph. See also * Nearest neighbour function References {{Reflist Geometric algorithms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Thematic Map
A thematic map is a type of map that portrays the geographic pattern of a particular subject matter (theme) in a geographic area. This usually involves the use of map symbols to Geovisualization, visualize selected properties of geographic features that are not naturally visible, such as temperature, language, or population. In this, they contrast with general reference maps, which focus on the location (more than the properties) of a diverse set of physical features, such as rivers, roads, and buildings. Alternative names have been suggested for this class, such as ''special-subject'' or ''special-purpose maps'', ''statistical maps'', or ''distribution maps'', but these have generally fallen out of common usage. Thematic mapping is closely allied with the field of Geovisualization. Several types of thematic maps have been invented, starting in the 18th and 19th centuries, as large amounts of statistical data began to be collected and published, such as Census, national census ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Map Overlay
A map is a symbolic depiction of interrelationships, commonly spatial, between things within a space. A map may be annotated with text and graphics. Like any graphic, a map may be fixed to paper or other durable media, or may be displayed on a transitory medium such as a computer screen. Some maps change interactively. Although maps are commonly used to depict geographic elements, they may represent any space, real or fictional. The subject being mapped may be two-dimensional such as Earth's surface, three-dimensional such as Earth's interior, or from an abstract space of any dimension. Maps of geographic territory have a very long tradition and have existed from ancient times. The word "map" comes from the , wherein ''mappa'' meant 'napkin' or 'cloth' and ''mundi'' 'of the world'. Thus, "map" became a shortened term referring to a flat representation of Earth's surface. History Maps have been one of the most important human inventions for millennia, allowing humans t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Point Location
The point location problem is a fundamental topic of computational geometry. It finds applications in areas that deal with processing geometrical data: computer graphics, geographic information systems (GIS), motion planning, and computer aided design (CAD). In its most general form, the problem is, given a partition of the space into disjoint regions, to determine the region where a query point lies. For example, the problem of determining which window of a graphical user interface contains a given mouse click can be formulated as an instance of point location, with a subdivision formed by the visible parts of each window, although specialized data structures may be more appropriate than general-purpose point location data structures in this application. Another special case is the point in polygon problem, in which one needs to determine whether a point is inside, outside, or on the boundary of a single polygon. In many applications, one needs to determine the location of sev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Quad-edge
A quad-edge data structure is a computer representation of the topology of a two-dimensional or three-dimensional map, that is, a graph drawn on a (closed) surface. It was first described by Jorge Stolfi and Leonidas J. Guibas. It is a variant of the earlier winged edge data structure. Overview The fundamental idea behind the quad-edge structure is the recognition that a single edge, in a closed polygonal mesh topology, sits between exactly two faces and exactly two vertices. The quad-edge data structure represents an edge, along with the edges it is connected to around the adjacent vertices and faces to encode the topology of the graph. An example implementation of the quad-edge data-type is as follows typedef struct quadedge; typedef struct quadedge_ref; Each quad-edge contains four references to adjacent quad-edges. Each of the four references points to the next edge counter-clockwise around either a vertex or a face. Each of these references represent either the ori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Doubly Connected Edge List
The doubly connected edge list (DCEL), also known as half-edge data structure, is a data structure to represent an embedding of a planar graph in the plane, and polytopes in 3D. This data structure provides efficient manipulation of the topological information associated with the objects in question (vertices, edges, faces). It is used in many algorithms of computational geometry to handle polygonal subdivisions of the plane, commonly called planar straight-line graphs (PSLG). For example, a Voronoi diagram is commonly represented by a DCEL inside a bounding box. This data structure was originally suggested by Muller and Preparata for representations of 3D convex polyhedra. Simplified versions of the data structure, as described here, only consider connected graphs, but the DCEL structure may be extended to handle disconnected graphs as well by introducing dummy edges between disconnected components. Data structure DCEL is more than just a doubly linked list of edges. In th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Winged Edge
In computer graphics, the winged edge data structure is a way to represent polygon meshes in computer memory. It is a type of boundary representation and describes both the geometry and topology of a model. Three types of records are used: vertex records, edge records, and face records. Given a reference to an edge record, one can answer several types of adjacency queries (queries about neighboring edges, vertices and faces) in constant time. This kind of adjacency information is useful for algorithms such as subdivision surface. Features The winged edge data structure explicitly describes the geometry and topology of faces, edges, and vertices when three or more surfaces come together and meet at a common edge. The ordering is such that the surfaces are ordered counter-clockwise with respect to the innate orientation of the intersection edge. Moreover the representation allows numerically unstable situations like that depicted below. The winged edge data structure allows ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
Euclidean Graph
Geometric graph theory in the broader sense is a large and amorphous subfield of graph theory, concerned with graphs defined by geometric means. In a stricter sense, geometric graph theory studies combinatorial and geometric properties of geometric graphs, meaning graphs drawn in the Euclidean plane with possibly intersecting straight-line edges, and topological graphs, where the edges are allowed to be arbitrary continuous curves connecting the vertices; thus, it can be described as "the theory of geometric and topological graphs" (Pach 2013). Geometric graphs are also known as spatial networks. Different types of geometric graphs A ''planar straight-line graph'' is a graph in which the vertices are embedded as points in the Euclidean plane, and the edges are embedded as non-crossing line segments. Fáry's theorem states that any planar graph may be represented as a planar straight line graph. A triangulation is a planar straight line graph to which no more edges may be added, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
