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Convex Hull Algorithms
Algorithms that construct convex hulls of various objects have a broad range of applications in mathematics and computer science. In computational geometry, numerous algorithms are proposed for computing the convex hull of a finite set of points, with various computational complexities. Computing the convex hull means that a non-ambiguous and efficient representation of the required convex shape is constructed. The complexity of the corresponding algorithms is usually estimated in terms of ''n'', the number of input points, and sometimes also in terms of ''h'', the number of points on the convex hull. Planar case Consider the general case when the input to the algorithm is a finite unordered set of points on a Cartesian plane. An important special case, in which the points are given in the order of traversal of a simple polygon's boundary, is described later in a separate subsection. If not all points are on the same line, then their convex hull is a convex polygon whose ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset. Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points. The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its dual problem of intersecting half-spaces, are fundamental problems of com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graham Scan
Graham's scan is a method of finding the convex hull of a finite set of points in the plane with time complexity O(''n'' log ''n''). It is named after Ronald Graham, who published the original algorithm in 1972. The algorithm finds all vertices of the convex hull ordered along its boundary. It uses a stack to detect and remove concavities in the boundary efficiently. Algorithm The first step in this algorithm is to find the point with the lowest y-coordinate. If the lowest y-coordinate exists in more than one point in the set, the point with the lowest x-coordinate out of the candidates should be chosen. Call this point ''P''. This step takes O(''n''), where ''n'' is the number of points in question. Next, the set of points must be sorted in increasing order of the angle they and the point ''P'' make with the x-axis. Any general-purpose sorting algorithm is appropriate for this, for example heapsort (which is O(''n'' log ''n'')). Sorting in order of angle does not require co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Franco P
Franco may refer to: Name * Franco (name) * Francisco Franco (1892–1975), Spanish general and dictator of Spain from 1939 to 1975 * Franco Luambo (1938–1989), Congolese musician, the "Grand Maître" Prefix * Franco, a prefix used when referring to France, a country * Franco, a prefix used when referring to French people and their diaspora, e.g. Franco-Americans, Franco-Mauritians * Franco, a prefix used when referring to Franks, a West Germanic tribe Places * El Franco, a municipality of Asturias in Spain * Presidente Franco District, in Paraguay * Franco, Virginia, an unincorporated community, in the United States Other uses * Franco (band), Filipino band * Franco (''General Hospital''), a fictional character on the American soap opera ''General Hospital'' * Franco, the Luccan franc, a 19th-century currency of Lucca, Italy * ''Franco, Ciccio e il pirata Barbanera'', a 1969 Italian comedy film directed by Mario Amendola * ''Franco, ese hombre'', a 1964 documentary fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divide And Conquer (Convex Hull)
Divide and rule policy ( la, divide et impera), or divide and conquer, in politics and sociology is gaining and maintaining power divisively. Historically, this strategy was used in many different ways by empires seeking to expand their territories. However, it has been hard to distinguish between the exploitation of pre-existing divisions by opponents, and the deliberate creation or strengthening of these divisions implied by "divide and rule". The strategy, but not the phrase, applies in many ancient cases: the example of Aulus Gabinius exists, parting the Jewish nation into five conventions, reported by Flavius Josephus in Book I, 169–170 of ''The Jewish War'' (''De bello Judaico''). Strabo also reports in '' Geographica'', 8.7.3 that the Achaean League was gradually dissolved when it became part of the Roman province of Macedonia, as the Romans treated the various states differently, wishing to preserve some and to destroy others. Elements of this technique involve: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quicksort
Quicksort is an efficient, general-purpose sorting algorithm. Quicksort was developed by British computer scientist Tony Hoare in 1959 and published in 1961, it is still a commonly used algorithm for sorting. Overall, it is slightly faster than merge sort and heapsort for randomized data, particularly on larger distributions. Quicksort is a divide-and-conquer algorithm. It works by selecting a 'pivot' element from the array and partitioning the other elements into two sub-arrays, according to whether they are less than or greater than the pivot. For this reason, it is sometimes called partition-exchange sort. The sub-arrays are then sorted recursively. This can be done in-place, requiring small additional amounts of memory to perform the sorting. Quicksort is a comparison sort, meaning that it can sort items of any type for which a "less-than" relation (formally, a total order) is defined. Most implementations of quicksort are not stable, meaning that the relative order of equal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quickhull
Quickhull is a method of computing the convex hull of a finite set of points in ''n''-dimensional space. It uses a divide and conquer approach similar to that of quicksort, from which its name derives. Its worst case time complexity for 2-dimensional and 3-dimensional space is O(n^2), but when the input precision is restricted to O(\log n) bits, its worst case time complexity is conjectured to be O(n \log r), where n is the number of input points and r is the number of processed points (up to n). N-dimensional Quickhull was invented in 1996 by C. Bradford Barber, David P. Dobkin, and Hannu Huhdanpaa. It was an extension of Jonathan Scott Greenfield's 1990 planar Quickhull algorithm, although the 1996 authors did not know of his methods. Instead, Barber et al. describe it as a deterministic variant of Clarkson and Shor's 1989 algorithm. Algorithm The 2-dimensional algorithm can be broken down into the following steps: # Find the points with minimum and maximum x coordinates, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ronald Graham
Ronald Lewis Graham (October 31, 1935July 6, 2020) was an American mathematician credited by the American Mathematical Society as "one of the principal architects of the rapid development worldwide of discrete mathematics in recent years". He was president of both the American Mathematical Society and the Mathematical Association of America, and his honors included the Leroy P. Steele Prize for lifetime achievement and election to the National Academy of Sciences. After graduate study at the University of California, Berkeley, Graham worked for many years at Bell Labs and later at the University of California, San Diego. He did important work in scheduling theory, computational geometry, Ramsey theory, and quasi-randomness, and many topics in mathematics are named after him. He published six books and about 400 papers, and had nearly 200 co-authors, including many collaborative works with his wife Fan Chung and with Paul Erdős. Graham has been featured in ''Ripley's Believe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big O Notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for ''Ordnung'', meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rates: d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gift Wrapping Algorithm
In computational geometry, the gift wrapping algorithm is an algorithm for computing the convex hull of a given set of points. Planar case In the two-dimensional case the algorithm is also known as Jarvis march, after R. A. Jarvis, who published it in 1973; it has O(''nh'') time complexity, where ''n'' is the number of points and ''h'' is the number of points on the convex hull. Its real-life performance compared with other convex hull algorithms is favorable when n is small or h is expected to be very small with respect to n. In general cases, the algorithm is outperformed by many others ( See Convex hull algorithms). Algorithm For the sake of simplicity, the description below assumes that the points are in general position, i.e., no three points are collinear. The algorithm may be easily modified to deal with collinearity, including the choice whether it should report only extreme points (vertices of the convex hull) or all points that lie on the convex hull. Also, the comple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chan's Algorithm
In computational geometry, Chan's algorithm, named after Timothy M. Chan, is an optimal output-sensitive algorithm to compute the convex hull of a set P of n points, in 2- or 3-dimensional space. The algorithm takes O(n \log h) time, where h is the number of vertices of the output (the convex hull). In the planar case, the algorithm combines an O(n \log n) algorithm (Graham scan, for example) with Jarvis march (O(nh)), in order to obtain an optimal O(n \log h) time. Chan's algorithm is notable because it is much simpler than the Kirkpatrick–Seidel algorithm, and it naturally extends to 3-dimensional space. This paradigm has been independently developed by Frank Nielsen in his Ph.D. thesis. Algorithm Overview A single pass of the algorithm requires a parameter m which is between 0 and n (number of points of our set P). Ideally, m = h but h, the number of vertices in the output convex hull, is not known at the start. Multiple passes with increasing values of m are done which then ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Timothy M
Timothy is a masculine name. It comes from the Greek language, Greek name (Timotheus (other), Timόtheos) meaning "honouring God", "in God's honour", or "honoured by God". Timothy (and its variations) is a common name in several countries. People Given name * Timothy (given name), including a list of people with the name * Tim (given name) * Timmy * Timo * Timotheus * Timothée Surname * Christopher Timothy (born 1940), Welsh actor. * Miriam Timothy (1879–1950), British harpist. * Nick Timothy (born 1980), British political adviser. Mononym * Saint Timothy, a companion and co-worker of Paul the Apostle * Timothy I (Nestorian patriarch) Education * Timothy Christian School (Illinois), a school system in Elmhurst, Illinois * Timothy Christian School (New Jersey), a school in Piscataway, New Jersey Arts and entertainment * Timothy (song), "Timothy" (song), a 1970 song by The Buoys * ''Timothy Goes to School'', a Canadian-Chinese children's animated series * Timo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultimate Convex Hull Algorithm
Ultimate or Ultimates may refer to: Arts, entertainment, and media Music Albums * ''Ultimate'' (Jolin Tsai album) * ''Ultimate'' (Pet Shop Boys album) *''Ultimate!'', an album by The Yardbirds *'' The Ultimate (Bryan Adams Album)'', a compilation album by Bryan Adams *''Ultimate Prince'' or just Ultimate, an album by Prince Songs * "Ultimate" (Denzel Curry song), a song by Denzel Curry from the double EP 32 Zel/Planet Shrooms *"Ultimate", a song by Lindsay Lohan from the ''Freaky Friday'' soundtrack Video games *''Super Smash Bros. Ultimate'', often referred to as simply ''Ultimate''. *''Ultimate General'', a series of computer games recreating the American Civil War *Ultimate Play the Game or just Ultimate, a video game developer, now known as Rare Other uses in arts, entertainment, and media *Ultimate (roller coaster), at Lightwater Valley amusement park near Ripon, North Yorkshire, England *Ultimates, a fictional superhero group in the Marvel Comics universe Philosophy *T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
