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Ceva's Theorem
In Euclidean geometry, Ceva's theorem is a theorem about triangles. Given a triangle , let the lines be drawn from the vertices to a common point (not on one of the sides of ), to meet opposite sides at respectively. (The segments are known as cevians.) Then, using signed lengths of segments, :\frac \cdot \frac \cdot \frac = 1. In other words, the length is taken to be positive or negative according to whether is to the left or right of in some fixed orientation of the line. For example, is defined as having positive value when is between and and negative otherwise. Ceva's theorem is a theorem of affine geometry, in the sense that it may be stated and proved without using the concepts of angles, areas, and lengths (except for the ratio of the lengths of two line segments that are collinear). It is therefore true for triangles in any affine plane over any field. A slightly adapted converse is also true: If points are chosen on respectively so that : \frac \cd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yusuf Al-Mu'taman Ibn Hud
Abu Amir Yusuf ibn Ahmad ibn Hud ( ar, أبو عامر يوسف إبن أحمد إبن هود, Abū ʿĀmir Yūsuf ibn Aḥmad ibn Hūd; died ), more commonly known as al-Mu'taman, was a mathematician, and also one of the kings of the Taifa of Zaragoza. The name al-Mu'taman is itself a shortening of his full regnal name al-Mu'taman Billah ( ar, المؤتمن بالله, al-Mūʾtaman bi-ʾLlāh, Trustee through God). Al-Mu'taman was the third king of the Banu Hud dynasty, reigning from 1081 to 1085, at the height of power of Muslim Zaragoza, following the thriving period of his father Ahmad al-Muqtadir. He continued his father's efforts and created around him a court of intellectuals, living in the beautiful palace of Aljafería, nicknamed as "the palace of joy". As king, Al-Mu'taman was a patron of science, philosophy and arts, and was himself a scholar of considerable accomplishment. He knew astrology, philosophy, and especially mathematics, a discipline in which he wrote the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mass Distribution
In physics and mechanics, mass distribution is the spatial distribution of mass within a solid body. In principle, it is relevant also for gases or liquids, but on Earth their mass distribution is almost homogeneous. Astronomy In astronomy mass distribution has decisive influence on the development e.g. of nebulae, stars and planets. The mass distribution of a solid defines its center of gravity and influences its dynamical behaviour - e.g. the oscillations and eventual rotation. Mathematical modelling A mass distribution can be modeled as a measure. This allows point masses, line masses, surface masses, as well as masses given by a volume density function. Alternatively the latter can be generalized to a distribution. For example, a point mass is represented by a delta function defined in 3-dimensional space. A surface mass on a surface given by the equation may be represented by a density distribution , where g/\left, \nabla f\ is the mass per unit area. The mathematical mode ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Facet (mathematics)
In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself. More specifically: * In three-dimensional geometry, a facet of a polyhedron is any polygon whose corners are vertices of the polyhedron, and is not a ''face''. To ''facet'' a polyhedron is to find and join such facets to form the faces of a new polyhedron; this is the reciprocal process to '' stellation'' and may also be applied to higher-dimensional polytopes. * In polyhedral combinatorics and in the general theory of polytopes, a facet (or hyperface) of a polytope of dimension ''n'' is a face that has dimension ''n'' − 1. Facets may also be called (''n'' − 1)-faces. In three-dimensional geometry, they are often called "faces" without qualification. * A facet of a simplicial complex is a maximal simplex, that is a simplex that is not a face of another simplex of the complex.. For (boundary complexes of) sim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barycentric Coordinates (mathematics)
In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex (a triangle for points in a plane, a tetrahedron for points in three-dimensional space, etc.). The barycentric coordinates of a point can be interpreted as masses placed at the vertices of the simplex, such that the point is the center of mass (or ''barycenter'') of these masses. These masses can be zero or negative; they are all positive if and only if the point is inside the simplex. Every point has barycentric coordinates, and their sum is not zero. Two tuples of barycentric coordinates specify the same point if and only if they are proportional; that is to say, if one tuple can be obtained by multiplying the elements of the other tuple by the same non-zero number. Therefore, barycentric coordinates are either considered to be defined up to multiplication by a nonzero constant, or normalized for summing to unity. Barycentric coordinates ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, * a 0-dimensional simplex is a point, * a 1-dimensional simplex is a line segment, * a 2-dimensional simplex is a triangle, * a 3-dimensional simplex is a tetrahedron, and * a 4-dimensional simplex is a 5-cell. Specifically, a ''k''-simplex is a ''k''-dimensional polytope which is the convex hull of its ''k'' + 1 vertices. More formally, suppose the ''k'' + 1 points u_0, \dots, u_k \in \mathbb^ are affinely independent, which means u_1 - u_0,\dots, u_k-u_0 are linearly independent. Then, the simplex determined by them is the set of points : C = \left\ This representation in terms of weighted vertices is known as the barycentric coordinate system. A regular simplex is a simplex that is also a regular polytope. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plane (geometry)
In mathematics, a plane is a Euclidean (flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with one of a room's walls, infinitely extended, or they may enjoy an independent existence in their own right, as in the setting of two-dimensional Euclidean geometry. Sometimes the word ''plane'' is used more generally to describe a two-dimensional surface, for example the hyperbolic plane and elliptic plane. When working exclusively in two-dimensional Euclidean space, the definite article is used, so ''the'' plane refers to the whole space. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory, and graphing are performed in a two-dimensional space, often in the plane. Euclidean geometry Euclid set forth the first great landmark of mathematical thought, an axiomatic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Left-hand Side
In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. The two sides have the same value, expressed differently, since equality is symmetric. definition and example of abbreviation More generally, these terms may apply to an or ; the right-hand side is ever ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector (geometry)
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a '' directed line segment'', or graphically as an arrow connecting an ''initial point'' ''A'' with a ''terminal point'' ''B'', and denoted by \overrightarrow . A vector is what is needed to "carry" the point ''A'' to the point ''B''; the Latin word ''vector'' means "carrier". It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from ''A'' to ''B''. Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operations whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments. In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead ''displacement vectors'', also called ''translation'' vectors or simply ''translations'', between two points of the space. Thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point translated from the starting point by that vector. Any vector space may be viewed as an affine spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alfred S
Alfred may refer to: Arts and entertainment *''Alfred J. Kwak'', Dutch-German-Japanese anime television series * ''Alfred'' (Arne opera), a 1740 masque by Thomas Arne * ''Alfred'' (Dvořák), an 1870 opera by Antonín Dvořák *"Alfred (Interlude)" and "Alfred (Outro)", songs by Eminem from the 2020 album ''Music to Be Murdered By'' Business and organisations * Alfred, a radio station in Shaftesbury, England *Alfred Music, an American music publisher *Alfred University, New York, U.S. *The Alfred Hospital, a hospital in Melbourne, Australia People * Alfred (name) includes a list of people and fictional characters called Alfred * Alfred the Great (848/49 – 899), or Alfred I, a king of the West Saxons and of the Anglo-Saxons Places Antarctica * Mount Alfred (Antarctica) Australia * Alfredtown, New South Wales * County of Alfred, South Australia Canada * Alfred and Plantagenet, Ontario * Alfred Island, Nunavut * Mount Alfred, British Columbia United States * Alfred, Maine, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duality (projective Geometry)
In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and (plane) duality is the formalization of this concept. There are two approaches to the subject of duality, one through language () and the other a more functional approach through special mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries that is called a ''duality''. Such a map can be constructed in many ways. The concept of plane duality readily extends to space duality and beyond that to duality in any finite-dimensional projective geometry. Principle of duality A projective plane may be defined axiomatically as an incidence structure, in terms of a set of ''points'', a set of ''lines'', and an incidence relation that determines which points lie on which lines. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
