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Hyperplane (geometry)
In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is one less than that of the ambient space. Two lower-dimensional examples of hyperplanes are one-dimensional lines in a plane and zero-dimensional points on a line. Most commonly, the ambient space is -dimensional Euclidean space, in which case the hyperplanes are the -dimensional "flats", each of which separates the space into two half spaces. A reflection across a hyperplane is a kind of motion (geometric transformation preserving distance between points), and the group of all motions is generated by the reflections. A convex polytope is the intersection (set theory), intersection of half-spaces. In non-Euclidean geometry, the ambient space might be the n-sphere, -dimensional sphere or hyperbolic space, or more generally a pseudo-Riem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersecting Planes
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their intersection is the point at which they meet. More generally, in set theory, the intersection of sets is defined to be the set of elements which belong to all of them. Unlike the Euclidean definition, this does not presume that the objects under consideration lie in a common space. It simply means the overlapping area of two or more objects or geometries. Intersection is one of the basic concepts of geometry. An intersection can have various geometric shapes, but a point is the most common in a plane geometry. Incidence geometry defines an intersection (usually, of flats) as an object of lower dimension that is incident to each of the original objects. In this approach an intersection can be sometimes undefined, such as for parallel l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Half-space (geometry)
In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space. If the space is two-dimensional, then a half-space is called a ''half-plane'' (open or closed). A half-space in a one-dimensional space is called a ''half-line'' or ray''.'' More generally, a half-space is either of the two parts into which a hyperplane divides an n-dimensional space. That is, the points that are not incident to the hyperplane are partitioned into two convex sets (i.e., half-spaces), such that any subspace connecting a point in one set to a point in the other must intersect the hyperplane. A half-space can be either ''open'' or ''closed''. An open half-space is either of the two open sets produced by the subtraction of a hyperplane from the affine space. A closed half-space is the union of an open half-space and the hyperplane that defines it. The open (closed) ''upper half-space'' is the half-space of all (''x''1, ''x''2, ..., ''x''''n'') suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Space Form
In mathematics, a space form is a complete Riemannian manifold ''M'' of constant sectional curvature ''K''. The three most fundamental examples are Euclidean ''n''-space, the ''n''-dimensional sphere, and hyperbolic space, although a space form need not be simply connected. Reduction to generalized crystallography The Killing–Hopf theorem of Riemannian geometry states that the universal cover of an ''n''-dimensional space form M^n with curvature K = -1 is isometric to H^n, hyperbolic space, with curvature K = 0 is isometric to R^n, Euclidean ''n''-space, and with curvature K = +1 is isometric to S^n, the n-dimensional sphere of points distance 1 from the origin in R^. By rescaling the Riemannian metric on H^n, we may create a space M_K of constant curvature K for any K 0. Thus the universal cover of a space form M with constant curvature K is isometric to M_K. This reduces the problem of studying space forms to studying discrete groups of isometries \Gamma of M_K which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo-Riemannian Manifold
In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed. Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space. A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as timelike, null, and spacelike. Introduction Manifolds In differential geometry, a differentiable manifold is a space that is locally similar to a Euclidean space. In an ''n''-dimensional Euclidean space any point can be specified by ''n'' real numbers. These are called the coordinates of the point. An ''n''-dimensional differentiable manifold is a generalisation of ''n''-dimensional Euclidean space. In a manifold it may only be possible to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Space
In mathematics, hyperbolic space of dimension ''n'' is the unique simply connected, ''n''-dimensional Riemannian manifold of constant sectional curvature equal to −1. It is homogeneous, and satisfies the stronger property of being a symmetric space. There are many ways to construct it as an open subset of \mathbb R^n with an explicitly written Riemannian metric; such constructions are referred to as models. Hyperbolic 2-space, H2, which was the first instance studied, is also called the hyperbolic plane. It is also sometimes referred to as Lobachevsky space or Bolyai–Lobachevsky space after the names of the author who first published on the topic of hyperbolic geometry. Sometimes the qualificative "real" is added to distinguish it from complex hyperbolic spaces. Hyperbolic space serves as the prototype of a Gromov hyperbolic space, which is a far-reaching notion including differential-geometric as well as more combinatorial spaces via a synthetic approach to negati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-sphere
In mathematics, an -sphere or hypersphere is an - dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer . The circle is considered 1-dimensional and the sphere 2-dimensional because a point within them has one and two degrees of freedom respectively. However, the typical embedding of the 1-dimensional circle is in 2-dimensional space, the 2-dimensional sphere is usually depicted embedded in 3-dimensional space, and a general -sphere is embedded in an -dimensional space. The term ''hyper''sphere is commonly used to distinguish spheres of dimension which are thus embedded in a space of dimension , which means that they cannot be easily visualized. The -sphere is the setting for -dimensional spherical geometry. Considered extrinsically, as a hypersurface embedded in -dimensional Euclidean space, an -sphere is the locus of points at equal distance (the ''radius'') from a given '' center'' point. Its interior, consisting of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-Euclidean Geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries that have also been called non-Euclidean geometry. Principles The essential difference between the metric geometries is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line and a point ''A'', which is not on , there is exactly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersection (set Theory)
In set theory, the intersection of two Set (mathematics), sets A and B, denoted by A \cap B, is the set containing all elements of A that also belong to B or equivalently, all elements of B that also belong to A. Notation and terminology Intersection is written using the symbol "\cap" between the terms; that is, in infix notation. For example: \\cap\=\ \\cap\=\varnothing \Z\cap\N=\N \\cap\N=\ The intersection of more than two sets (generalized intersection) can be written as: \bigcap_^n A_i which is similar to capital-sigma notation. For an explanation of the symbols used in this article, refer to the table of mathematical symbols. Definition The intersection of two sets A and B, denoted by A \cap B, is the set of all objects that are members of both the sets A and B. In symbols: A \cap B = \. That is, x is an element of the intersection A \cap B if and only if x is both an element of A and an element of B. For example: * The intersection of the sets and is . * The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others''Mathematical Programming'', by Melvyn W. Jeter (1986) p. 68/ref> (including this article) allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. In the influential textbooks of Grünbaum and Ziegler on the subject, as well as in many other texts in discrete geometry, convex polytopes are often simply called "polytopes". Grünbaum points out that this is solely to avoid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Set Of A Group
In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group (mathematics), group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their Inverse element, inverses. In other words, if S is a subset of a group G, then \langle S\rangle, the ''subgroup generated by S'', is the smallest subgroup of G containing every element of S, which is equal to the intersection over all subgroups containing the elements of S; equivalently, \langle S\rangle is the subgroup of all elements of G that can be expressed as the finite product of elements in S and their inverses. (Note that inverses are only needed if the group is infinite; in a finite group, the inverse of an element can be expressed as a power of that element.) If G=\langle S\rangle, then we say that S ''generates'' G, and the elements in S are called ''generators'' or ''group generators''. If S is the empty set, then \langle S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set with an Binary operation, operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is Associative property, associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition, addition operation form a group. The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry, groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distance (mathematics)
Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). The term is also frequently used metaphorically to mean a measurement of the amount of difference between two similar objects (such as statistical distance between probability distributions or edit distance between string (computer science), strings of text) or a degree of separation (as exemplified by distance (graph theory), distance between people in a social network). Most such notions of distance, both physical and metaphorical, are formalized in mathematics using the notion of a metric space. In the social sciences, distance can refer to a qualitative measurement of separation, such as social distance or psychological distance. Distances in physics and geometry The distance between physical locations can be defined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
