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Convex Polygon
In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a simple polygon (not self-intersecting). Equivalently, a polygon is convex if every line that does not contain any edge intersects the polygon in at most two points. Strictly convex polygon A convex polygon is ''strictly'' convex if no line contains more than two vertices of the polygon. In a convex polygon, all interior angles are less than ''or equal'' to 180 degrees, while in a strictly convex polygon all interior angles are strictly less than 180 degrees. Properties The following properties of a simple polygon are all equivalent to convexity: *Every internal angle is less than or equal to 180 degrees. *Every point on every line segment between two points inside or on the boundary of the polygon remains inside or on the bou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagon
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ''regular pentagon'' (or ''star polygon, star pentagon'') is called a pentagram. Regular pentagons A ''regular polygon, regular pentagon'' has Schläfli symbol and interior angles of 108°. A ''regular polygon, regular pentagon'' has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex polygon, convex regular pentagon are in the golden ratio to its sides. Given its side length t, its height H (distance from one side to the opposite vertex), width W (distance between two farthest separated points, which equals the diagonal length D) and circumradius R are given by: :\begin H &= \frac~t \approx 1.539~t, \\ W= D &= \frac~t\approx 1.618~t, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Time
In theoretical 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 gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morgan Kaufmann
Morgan Kaufmann Publishers is a Burlington, Massachusetts (San Francisco, California until 2008) based publisher specializing in computer science and engineering content. Since 1984, Morgan Kaufmann has been publishing contents on information technology, computer architecture, data management, computer networking, computer systems, human computer interaction, computer graphics, multimedia information and systems, artificial intelligence, computer security, and software engineering. Morgan Kaufmann's audience includes the research and development communities, information technology (IS/IT) managers, and students in professional degree programs. The company was founded in 1984 by publishers Michael B. Morgan and William Kaufmann and computer scientist Nils Nilsson. It was held privately until 1998, when it was acquired by Harcourt General and became an imprint of the Academic Press, a subsidiary of Harcourt. Harcourt was acquired by Reed Elsevier in 2001; Morgan Kaufmann is now ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensional line segments. A triangle has three internal angles, each one bounded by a pair of adjacent edges; the sum of angles of a triangle always equals a straight angle (180 degrees or π radians). The triangle is a plane figure and its interior is a planar region. Sometimes an arbitrary edge is chosen to be the ''base'', in which case the opposite vertex is called the ''apex''; the shortest segment between the base and apex is the ''height''. The area of a triangle equals one-half the product of height and base length. In Euclidean geometry, any two points determine a unique line segment situated within a unique straight line, and any three points that do not all lie on the same straight line determine a unique triangle situated w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Self-intersecting Polygons
Self-intersecting polygons, crossed polygons, or self-crossing polygons are polygons some of whose edges cross each other. They contrast with simple polygons, whose edges never cross. Some types of self-intersecting polygons are: *the crossed quadrilateral, with four edges **the antiparallelogram, a crossed quadrilateral with alternate edges of equal length ***the crossed rectangle, an antiparallelogram whose edges are two opposite sides and the two diagonals of a rectangle, hence having two edges parallel * Star polygons ** pentagram, with five edges ** hexagram, with six edges ** heptagram, with seven edges ** octagram, with eight edges ** enneagram or nonagram, with nine edges ** decagram, with ten edges ** hendecagram, with eleven edges **dodecagram, with twelve edges ** icositetragram, with twenty four edges ** 257-gram, with two hundred and fifty seven edges See also * *Complex polygon The term ''complex polygon'' can mean two different things: * In geometry, a polyg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Width
In geometry, the mean width is a measure of the "size" of a body; see Hadwiger's theorem for more about the available measures of bodies. In n dimensions, one has to consider (n-1)-dimensional hyperplanes perpendicular to a given direction \hat in S^, where S^n is the n-sphere (the surface of a (n+1)-dimensional sphere). The "width" of a body in a given direction \hat is the distance between the closest pair of such planes, such that the body is entirely in between the two hyper planes (the planes only intersect with the boundary of the body). The mean width is the average of this "width" over all \hat in S^. More formally, define a compact body B as being equivalent to set of points in its interior plus the points on the boundary (here, points denote elements of \mathbb^n). The support function of body B is defined as : h_B(n)=\max\ where n is a direction and \langle,\rangle denotes the usual inner product on \mathbb^n. The mean width is then : b(B)=\frac \int_ h_B(\hat)+h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homothetic Transformation
In mathematics, a homothety (or homothecy, or homogeneous dilation) is a Transformation (mathematics), transformation of an affine space determined by a point called its ''center'' and a nonzero number called its ''ratio'', which sends point to a point by the rule, : \overrightarrow=k\overrightarrow for a fixed number k\ne 0. Using position vectors: :\mathbf x'=\mathbf s + k(\mathbf x -\mathbf s). In case of S=O (Origin): :\mathbf x'=k\mathbf x, which is a uniform scaling and shows the meaning of special choices for k: :for k=1 one gets the ''identity'' mapping, :for k=-1 one gets the ''reflection'' at the center, For 1/k one gets the ''inverse'' mapping defined by k. In Euclidean geometry homotheties are the Similarity (geometry), similarities that fix a point and either preserve (if k>0) or reverse (if k<0) the direction of all vectors. Together with the Translation (geometry), translations, all homotheties of an affine (or Euclidean) space form a group (mathematics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallelogram
In Euclidean geometry, a parallelogram is a simple polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence (geometry), congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations. By comparison, a quadrilateral with at least one pair of parallel sides is a trapezoid in American English or a trapezium in British English. The three-dimensional counterpart of a parallelogram is a parallelepiped. The word "parallelogram" comes from the Greek παραλληλό-γραμμον, ''parallēló-grammon'', which means "a shape of parallel lines". Special cases *Rectangle – A par ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperplane Separation Theorem
In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in ''n''-dimensional Euclidean space. There are several rather similar versions. In one version of the theorem, if both these sets are closed and at least one of them is compact, then there is a hyperplane in between them and even two parallel hyperplanes in between them separated by a gap. In another version, if both disjoint convex sets are open, then there is a hyperplane in between them, but not necessarily any gap. An axis which is orthogonal to a separating hyperplane is a separating axis, because the orthogonal projections of the convex bodies onto the axis are disjoint. The hyperplane separation theorem is due to Hermann Minkowski. The Hahn–Banach separation theorem generalizes the result to topological vector spaces. A related result is the supporting hyperplane theorem. In the context of support-vector machines, the ''optimally separating hyperplane'' or ''maximum-margin hy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krein–Milman Theorem
In the mathematical theory of functional analysis, the Krein–Milman theorem is a proposition about compact convex sets in locally convex topological vector spaces (TVSs). This theorem generalizes to infinite-dimensional spaces and to arbitrary compact convex sets the following basic observation: a convex (i.e. "filled") triangle, including its perimeter and the area "inside of it", is equal to the convex hull of its three vertices, where these vertices are exactly the extreme points of this shape. This observation also holds for any other convex polygon in the plane \R^2. Statement and definitions Preliminaries and definitions Throughout, X will be a real or complex vector space. For any elements x and y in a vector space, the set , y:= \ is called the or closed interval between x and y. The or open interval between x and y is (x, y) := \varnothing when x = y while it is (x, y) := \ when x \neq y; it satisfies (x, y) = , y\setminus \ and , y= (x, y) \cup \. The points ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helly's Theorem
Helly's theorem is a basic result in discrete geometry on the intersection of convex sets. It was discovered by Eduard Helly in 1913,. but not published by him until 1923, by which time alternative proofs by and had already appeared. Helly's theorem gave rise to the notion of a Helly family. Statement Let be a finite collection of convex subsets of \R^d, with n\geq d+1. If the intersection of every d+1 of these sets is nonempty, then the whole collection has a nonempty intersection; that is, :\bigcap_^n X_j\ne\varnothing. For infinite collections one has to assume compactness: Let \ be a collection of compact convex subsets of \R^d, such that every subcollection of cardinality at most d+1 has nonempty intersection. Then the whole collection has nonempty intersection. Proof We prove the finite version, using Radon's theorem as in the proof by . The infinite version then follows by the finite intersection property characterization of compactness: a collection of closed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fan Triangulation
In computational geometry, a fan triangulation is a simple way to Polygon triangulation, triangulate a polygon by choosing a Vertex (geometry), vertex and drawing Edge (geometry), edges to all of the other vertices of the polygon. Not every polygon can be triangulated this way, so this method is usually only used for convex polygons. Properties Aside from the properties of all triangulations, fan triangulations have the following properties: * All convex polygons, but not all polygons, can be fan triangulated. * Polygons with only one concave vertex can always be fan triangulated, as long as the diagonals are drawn from the concave vertex. * It can be known if a polygon can be fan triangulated by solving the Art gallery problem, in order to determine whether there is at least one vertex that is visible from every point in the polygon. * The triangulation of a polygon with n vertices uses n - 3 diagonals, and generates n - 2 triangles. * Generating the list of triangles is trivi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
