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Relative Convex Hull
In discrete geometry and computational geometry, the relative convex hull or geodesic convex hull is an analogue of the convex hull for the points inside a simple polygon or a rectifiable simple closed curve. Definition Let P be a simple polygon or a rectifiable simple closed curve, and let X be any set enclosed by P. A geodesic between two points in P is a shortest path connecting those two points that stays entirely within P. A subset K of the points inside P is said to be relatively convex, geodesically convex, or P-convex if, for every two points of K, the geodesic between them in P stays within K. Then the relative convex hull of X can be defined as the intersection of all relatively convex sets containing X. Equivalently, the relative convex hull is the minimum-perimeter weakly simple polygon in P that encloses X. This was the original formulation of relative convex hulls, by . However this definition is complicated by the need to use weakly simple polygons (intuitively, po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relative Convex Hull
In discrete geometry and computational geometry, the relative convex hull or geodesic convex hull is an analogue of the convex hull for the points inside a simple polygon or a rectifiable simple closed curve. Definition Let P be a simple polygon or a rectifiable simple closed curve, and let X be any set enclosed by P. A geodesic between two points in P is a shortest path connecting those two points that stays entirely within P. A subset K of the points inside P is said to be relatively convex, geodesically convex, or P-convex if, for every two points of K, the geodesic between them in P stays within K. Then the relative convex hull of X can be defined as the intersection of all relatively convex sets containing X. Equivalently, the relative convex hull is the minimum-perimeter weakly simple polygon in P that encloses X. This was the original formulation of relative convex hulls, by . However this definition is complicated by the need to use weakly simple polygons (intuitively, po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamic Convex Hull
The dynamic convex hull problem is a class of dynamic problems in computational geometry. The problem consists in the maintenance, i.e., keeping track, of the convex hull for input data undergoing a sequence of discrete changes, i.e., when input data elements may be inserted, deleted, or modified. It should be distinguished from the kinetic convex hull, which studies similar problems for continuously moving points. Dynamic convex hull problems may be distinguished by the types of the input data and the allowed types of modification of the input data. Planar point set It is easy to construct an example for which the convex hull contains all input points, but after the insertion of a single point the convex hull becomes a triangle. And conversely, the deletion of a single point may produce the opposite drastic change of the size of the output. Therefore, if the convex hull is required to be reported in traditional way as a polygon, the lower bound for the worst-case computational c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Computer And System Sciences
The ''Journal of Computer and System Sciences'' (JCSS) is a peer-reviewed scientific journal in the field of computer science. ''JCSS'' is published by Elsevier, and it was started in 1967. Many influential scientific articles have been published in ''JCSS''; these include five papers that have won the Gödel Prize. an 2014 Gödel Prize Its managing editor is |
Computational Geometry (journal)
''Computational Geometry'', also known as ''Computational Geometry: Theory and Applications'', is a peer-reviewed mathematics journal for research in theoretical and applied computational geometry, its applications, techniques, and design and analysis of geometric algorithms. All aspects of computational geometry are covered, including the numerical, graph theoretical and combinatorial aspects, as well as fundamental problems in various areas of application of computational geometry: in computer graphics, pattern recognition, image processing, robotics, electronic design automation, CAD/CAM, and geographical information systems. The journal was founded in 1991 by Jörg-Rüdiger Sack and Jorge Urrutia.. It is indexed by ''Mathematical Reviews'', Zentralblatt MATH, Science Citation Index, and Current Contents ''Current Contents'' is a rapid alerting service database from Clarivate Analytics, formerly the Institute for Scientific Information and Thomson Reuters. It is publis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Collision Detection
Collision detection is the computational problem of detecting the intersection (Euclidean geometry), intersection of two or more objects. Collision detection is a classic issue of computational geometry and has applications in various computing fields, primarily in computer graphics, computer games, computer simulations, robotics and computational physics. Collision detection algorithms can be divided into operating on 2D and 3D objects. Overview In physical simulation, experiments such as playing billiards, are conducted. The physics of bouncing billiard balls are well understood, under the umbrella of rigid body motion and elastic collisions. An initial description of the situation would be given, with a very precise physical description of the billiard table and balls, as well as initial positions of all the balls. Given a force applied to the cue ball (probably resulting from a player hitting the ball with their cue stick), we want to calculate the trajectories, precise ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path Planning
Motion planning, also path planning (also known as the navigation problem or the piano mover's problem) is a computational problem to find a sequence of valid configurations that moves the object from the source to destination. The term is used in computational geometry, computer animation, robotics and computer games. For example, consider navigating a mobile robot inside a building to a distant waypoint. It should execute this task while avoiding walls and not falling down stairs. A motion planning algorithm would take a description of these tasks as input, and produce the speed and turning commands sent to the robot's wheels. Motion planning algorithms might address robots with a larger number of joints (e.g., industrial manipulators), more complex tasks (e.g. manipulation of objects), different constraints (e.g., a car that can only drive forward), and uncertainty (e.g. imperfect models of the environment or robot). Motion planning has several robotics applications, such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Hull Of A Simple Polygon
In discrete geometry and computational geometry, the convex hull of a simple polygon is the polygon of minimum perimeter that contains a given simple polygon. It is a special case of the more general concept of a convex hull. It can be computed in linear time, faster than algorithms for convex hulls of point sets. The convex hull of a simple polygon can be subdivided into the given polygon itself and into polygonal ''pockets'' bounded by a polygonal chain of the polygon together with a single convex hull edge. Repeatedly reflecting an arbitrarily chosen pocket across this convex hull edge produces a sequence of larger simple polygons; according to the Erdős–Nagy theorem, this process eventually terminates with a convex polygon. Structure The convex hull of a simple polygon is itself a convex polygon. Overlaying the original simple polygon onto its convex hull partitions this convex polygon into regions, one of which is the original polygon. The remaining regions are called ''p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is generally expresse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygon Triangulation
In computational geometry, polygon triangulation is the partition of a polygonal area (simple polygon) into a set of triangles, i.e., finding a set of triangles with pairwise non-intersecting interiors whose union is . Triangulations may be viewed as special cases of planar straight-line graphs. When there are no holes or added points, triangulations form maximal outerplanar graphs. Polygon triangulation without extra vertices Over time, a number of algorithms have been proposed to triangulate a polygon. Special cases It is trivial to triangulate any convex polygon in linear time into a fan triangulation, by adding diagonals from one vertex to all other non-nearest neighbor vertices. The total number of ways to triangulate a convex ''n''-gon by non-intersecting diagonals is the (''n''−2)nd Catalan number, which equals :\frac, a formula found by Leonhard Euler. A monotone polygon can be triangulated in linear time with either the algorithm of A. Fournier and D.Y. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Geometry
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object. Discrete geometry has a large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory, toric geometry, and combinatorial topology. History Although polyhedra and tessellations had been studied for many years by people such as Kepler and Cauchy, modern discrete geometry has its origins in the late 19th century. Early topics studie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weakly Simple Polygon
In geometry, a simple polygon is a polygon that does not intersect itself and has no holes. That is, it is a flat shape consisting of straight, non-intersecting line segments or "sides" that are joined pairwise to form a single closed path. If the sides intersect then the polygon is not simple. The qualifier "simple" is frequently omitted, with the above definition then being understood to define a polygon in general. The definition given above ensures the following properties: * A polygon encloses a region (called its interior) which always has a measurable area. * The line segments that make up a polygon (called sides or edges) meet only at their endpoints, called vertices (singular: vertex) or less formally "corners". * Exactly two edges meet at each vertex. * The number of edges always equals the number of vertices. Two edges meeting at a corner are usually required to form an angle that is not straight (180°); otherwise, the collinear line segments will be considered part ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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