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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] |
Quickhull Example6
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, as t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quickhull Example3
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, as t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quickhull Example7
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, as t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Divide And Conquer Algorithm
In computer science, divide and conquer is an algorithm design paradigm. A divide-and-conquer algorithm recursively breaks down a problem into two or more sub-problems of the same or related type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem. The divide-and-conquer technique is the basis of efficient algorithms for many problems, such as sorting (e.g., quicksort, merge sort), multiplying large numbers (e.g., the Karatsuba algorithm), finding the closest pair of points, syntactic analysis (e.g., top-down parsers), and computing the discrete Fourier transform (FFT). Designing efficient divide-and-conquer algorithms can be difficult. As in mathematical induction, it is often necessary to generalize the problem to make it amenable to a recursive solution. The correctness of a divide-and-conquer algorithm is usually proved by mathematical induction, and its computational cost is ... [...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] |
David P
David (; , "beloved one") (traditional spelling), , ''Dāwūd''; grc-koi, Δαυΐδ, Dauíd; la, Davidus, David; gez , ዳዊት, ''Dawit''; xcl, Դաւիթ, ''Dawitʿ''; cu, Давíдъ, ''Davidŭ''; possibly meaning "beloved one". was, according to the Hebrew Bible, the third king of the United Kingdom of Israel. In the Books of Samuel, he is described as a young shepherd and harpist who gains fame by slaying Goliath, a champion of the Philistines, in southern Canaan. David becomes a favourite of Saul, the first king of Israel; he also forges a notably close friendship with Jonathan, a son of Saul. However, under the paranoia that David is seeking to usurp the throne, Saul attempts to kill David, forcing the latter to go into hiding and effectively operate as a fugitive for several years. After Saul and Jonathan are both killed in battle against the Philistines, a 30-year-old David is anointed king over all of Israel and Judah. Following his rise to power, David ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Animation Depicting The Quickhull Algorithm
Animation is a method by which still figures are manipulated to appear as moving images. In traditional animation, images are drawn or painted by hand on transparent celluloid sheets to be photographed and exhibited on film. Today, most animations are made with computer-generated imagery (CGI). Computer animation can be very detailed 3D animation, while 2D computer animation (which may have the look of traditional animation) can be used for stylistic reasons, low bandwidth, or faster real-time renderings. Other common animation methods apply a stop motion technique to two- and three-dimensional objects like paper cutouts, puppets, or clay figures. A cartoon is an animated film, usually a short film, featuring an exaggerated visual style. The style takes inspiration from comic strips, often featuring anthropomorphic animals, superheroes, or the adventures of human protagonists. Especially with animals that form a natural predator/prey relationship (e.g. cats and mice, coyot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
