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Hypercone
In geometry, a hypercone (or spherical cone) is the figure in the 4-dimensional Euclidean space represented by the equation :x^2 + y^2 + z^2 - w^2 = 0. It is a quadric surface, and is one of the possible 3- manifolds which are 4-dimensional equivalents of the conical surface in 3 dimensions. It is also named "spherical cone" because its intersections with hyperplanes perpendicular to the ''w''-axis are spheres. A four-dimensional right hypercone can be thought of as a sphere which expands with time, starting its expansion from a single point source, such that the center of the expanding sphere remains fixed. An oblique hypercone would be a sphere which expands with time, again starting its expansion from a point source, but such that the center of the expanding sphere moves with a uniform velocity. Parametric form A right spherical hypercone can be described by the function : \vec \sigma (\phi, \theta, t) = (t s \cos \theta \cos \phi, t s \cos \theta \sin \phi, t s \sin \th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Light Cone
In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take through spacetime. Details If one imagines the light confined to a two-dimensional plane, the light from the flash spreads out in a circle after the event E occurs, and if we graph the growing circle with the vertical axis of the graph representing time, the result is a cone, known as the future light cone. The past light cone behaves like the future light cone in reverse, a circle which contracts in radius at the speed of light until it converges to a point at the exact position and time of the event E. In reality, there are three space dimensions, so the light would actually form an expanding or contracting sphere in three-dimensional (3D) space rather than a circle in 2D, and the light cone would actually be a four-dimensional version of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Light Cone
In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take through spacetime. Details If one imagines the light confined to a two-dimensional plane, the light from the flash spreads out in a circle after the event E occurs, and if we graph the growing circle with the vertical axis of the graph representing time, the result is a cone, known as the future light cone. The past light cone behaves like the future light cone in reverse, a circle which contracts in radius at the speed of light until it converges to a point at the exact position and time of the event E. In reality, there are three space dimensions, so the light would actually form an expanding or contracting sphere in three-dimensional (3D) space rather than a circle in 2D, and the light cone would actually be a four-dimensional version of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Cone Lines
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings. Basic terminology As mentioned earlier is the sphere's r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isosceles Triangle
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids. The mathematical study of isosceles triangles dates back to ancient Egyptian mathematics and Babylonian mathematics. Isosceles triangles have been used as decoration from even earlier times, and appear frequently in architecture and design, for instance in the pediments and gables of buildings. The two equal sides are called the legs and the third side is called the base of the triangle. The other dimensions of the triangle, such as its height, area, and perimeter, can be calculated by simple formulas from the lengths of the legs an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Four-dimensional Geometry
A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called ''dimensions'', to describe the sizes or locations of objects in the everyday world. For example, the volume of a rectangular box is found by measuring and multiplying its length, width, and height (often labeled ''x'', ''y'', and ''z''). The idea of adding a fourth dimension began with Jean le Rond d'Alembert's "Dimensions" being published in 1754, was followed by Joseph-Louis Lagrange in the mid-1700s, and culminated in a precise formalization of the concept in 1854 by Bernhard Riemann. In 1880, Charles Howard Hinton popularized these insights in an essay titled "s:What is the Fourth Dimension?, What is the Fourth Dimension?", which explained the concept of a "Tesseract, four-dimensional cube" with a step-by-step generalization of the properties of line ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean space, an affine space or a projective space. Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation, at least locally (near every point), and sometimes globally. A hypersurface in a (Euclidean, affine, or projective) space of dimension two is a plane curve. In a space of dimension three, it is a surface. For example, the equation :x_1^2+x_2^2+\cdots+x_n^2-1=0 defines an algebraic hypersurface of dimension in the Euclidean space of dimension . This hypersurface is also a smooth manifold, and is called a hypersphere or an -sphere. Smooth hypersurface A hypersurface that is a smooth manifold is called a ''smooth hypersurface''. In , a smooth hypersurface is orienta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dimensional lines. This notion can be used in any general space in which the concept of the dimension of a subspace is defined. In different settings, hyperplanes may have different properties. For instance, a hyperplane of an -dimensional affine space is a flat subset with dimension and it separates the space into two half spaces. While a hyperplane of an -dimensional projective space does not have this property. The difference in dimension between a subspace and its ambient space is known as the codimension of with respect to . Therefore, a necessary and sufficient condition for to be a hyperplane in is for to have codimension one in . Technical description In geometry, a hyperplane of an ''n''-dimensi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in ''n'' dimensions is equal to \sqrt. An ''n''-dimensional hypercube is more commonly referred to as an ''n''-cube or sometimes as an ''n''-dimensional cube. The term measure polytope (originally from Elte, 1912) is also used, notably in the work of H. S. M. Coxeter who also labels the hypercubes the γn polytopes. The hypercube is the special case of a hyperrectangle (also called an ''n-orthotope''). A ''unit hypercube'' is a hypercube whose side has length one unit. Often, the hypercube whose corners (or ''vertices'') are the 2''n'' points in R''n'' with each coordinate equal to 0 or 1 is called ''the'' unit hypercube. Construction A hyp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensions is an ordinary sphere (or 2-sphere, a two-dimensional surface), the boundary of a ball in four dimensions is a 3-sphere (an object with three dimensions). A 3-sphere is an example of a 3-manifold and an ''n''-sphere. Definition In coordinates, a 3-sphere with center and radius is the set of all points in real, 4-dimensional space () such that :\sum_^3(x_i - C_i)^2 = ( x_0 - C_0 )^2 + ( x_1 - C_1 )^2 + ( x_2 - C_2 )^2+ ( x_3 - C_3 )^2 = r^2. The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere and is usually denoted : :S^3 = \left\. It is often convenient to regard as the space with 2 complex dimensions () or the quaternions (). The unit 3-sphere is then given by :S^3 = \left\ or :S^3 = \left\. This ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rita G
Rita may refer to: People * Rita (given name) * Rita (Indian singer) (born 1984) * Rita (Israeli singer) (born 1962) * Rita (Japanese singer) * Eliza Humphreys (1850–1938), wrote under the pseudonym Rita Places * Djarrit, also known as Rita, a community in the Marshall Islands * 1180 Rita, an asteroid * Rita, West Virginia * Santa Rita, California (other), several places Film, television, and theater * ''Rita'' (1959 film), a 1959 Australian television play * ''Rita'' (2009 Italian film), a 2009 Italian film * ''Rita'' (2009 Indian film), a 2009 Marathi film directed by Renuka Shahane * ''Rita'' (TV series), a Danish television show * RITA Award, an award for romantic fiction * ''Educating Rita'', a 1980 stage play by Willy Russel ** ''Educating Rita'' (film), a 1983 British film based on that play *Rita Santos, an adult mermaid on the TV series ''Mako Mermaids'' Music * ''Rita'' (opera), an 1841 opera by Gaetano Donizetti Albums * ''Rita'' (Rita Yahan-Farouz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Wave
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics. Single mechanical or electromagnetic waves propagating in a pre-defined direction can also be described with the first-order one-way wave equation which is much easier to solve and also valid for inhomogenious media. Introduction The (two-way) wave equation is a second-order partial differential equation describing waves, including traveling and standing waves; the latter can be considered as linear superpositions of waves traveling in opposite directions. This article mostly focuses on the scalar wave equation describing waves in scalars by scalar functions of a time variable (a variable representing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
