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Coordinate Singularity
A coordinate singularity occurs when an apparent singularity or discontinuity occurs in one coordinate frame that can be removed by choosing a different frame. An example is the apparent (longitudinal) singularity at the 90 degree latitude in spherical coordinates. An object moving due north (for example, along the line 0 degrees longitude) on the surface of a sphere will suddenly experience an instantaneous change in longitude at the pole (i.e., jumping from longitude 0 to longitude 180 degrees). In fact, longitude is not uniquely defined at the poles. This discontinuity, however, is only apparent; it is an artifact of the coordinate system chosen, which is singular at the poles. A different coordinate system would eliminate the apparent discontinuity, e.g. by replacing the latitude/longitude representation with an -vector representation. Stephen Hawking aptly summed this up, when once asking the question, "What lies north of the North Pole?". [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Singularity
In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity. For example, the real function : f(x) = \frac has a singularity at x = 0, where the numerical value of the function approaches \pm\infty so the function is not defined. The absolute value function g(x) = , x, also has a singularity at x = 0, since it is not differentiable there. The algebraic curve defined by \left\ in the (x, y) coordinate system has a singularity (called a cusp) at (0, 0). For singularities in algebraic geometry, see singular point of an algebraic variety. For singularities in differential geometry, see singularity theory. Real analysis In real analysis, singularities are either discontinuities, or discontinuities of the derivative (sometimes also discontinuities of higher order derivatives). There are four kinds of discon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greenwich Meridian
The historic prime meridian or Greenwich meridian is a geographical reference line that passes through the Royal Observatory, Greenwich, Royal Observatory, Greenwich, in London, England. The modern IERS Reference Meridian widely used today is based on the Greenwich meridian, but differs slightly from it. This prime meridian (at the time, one of prime meridian, many) was first established by George Biddell Airy, Sir George Airy in 1851, and by 1884, over two-thirds of all ships and tonnage used it as the reference Meridian (geography), meridian on their Nautical chart, charts and maps. In October of that year, at the behest of President of the United States, US President Chester A. Arthur, 41 delegates from 25 nations met in Washington, D.C., United States, for the International Meridian Conference. This conference selected the meridian passing through Greenwich as the world standard prime meridian due to its popularity. However, France abstained from the vote, and French maps ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-vector
The ''n''-vector representation (also called geodetic normal or ellipsoid normal vector) is a three-parameter non-singular representation well-suited for replacing geodetic coordinates (latitude and longitude) for horizontal position representation in mathematical calculations and computer algorithms. Geometrically, the ''n''-vector for a given position on an ellipsoid is the outward-pointing unit vector that is normal in that position to the ellipsoid. For representing horizontal positions on Earth, the ellipsoid is a reference ellipsoid and the vector is decomposed in an Earth-centered Earth-fixed coordinate system. It behaves smoothly at all Earth positions, and it holds the mathematical one-to-one property. More in general, the concept can be applied to representing positions on the boundary of a strictly convex bounded subset of ''k''-dimensional Euclidean space, provided that that boundary is a differentiable manifold. In this general case, the ''n''-vector consist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chronometric Singularity
A chronometric singularity (also called a temporal or horological singularity) is a point at which time cannot be measured or described. An example involves a time at a coordinate singularity, e.g.a geographical pole. Since time on Earth is measured through longitudes, and no unique longitude exists at a pole, time is not defined uniquely at this point. There is a clear connection with coordinate singularities, as can be seen from this example. In relativity, similar singularities can be found in the case of Schwarzschild coordinates. Stephen Hawking once compared by a talk-show guest's question about "before the beginning of time" to asking "what's north of the north pole". wiseGEEK.com. Accessed 15 Feb 2013. In a related discussion, he mentions this again [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Imaginary Time
Imaginary time is a mathematical representation of time which appears in some approaches to special relativity and quantum mechanics. It finds uses in connecting quantum mechanics with statistical mechanics and in certain cosmological theories. Mathematically, imaginary time is real time which has undergone a Wick rotation so that its coordinates are multiplied by the imaginary unit ''i''. Imaginary time is ''not'' imaginary in the sense that it is unreal or made-up (any more than, say, irrational numbers defy logic), it is simply expressed in terms of what mathematicians call imaginary numbers. Origins In mathematics, the imaginary unit i is the square root of -1, such that i^2 is defined to be -1. A number which is a direct multiple of i is known as an imaginary number. In certain physical theories, periods of time are multiplied by i in this way. Mathematically, an imaginary time period \tau may be obtained from real time t via a Wick rotation by \pi/2 in the complex plan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Singularity
In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity. For example, the real function : f(x) = \frac has a singularity at x = 0, where the numerical value of the function approaches \pm\infty so the function is not defined. The absolute value function g(x) = , x, also has a singularity at x = 0, since it is not differentiable there. The algebraic curve defined by \left\ in the (x, y) coordinate system has a singularity (called a cusp) at (0, 0). For singularities in algebraic geometry, see singular point of an algebraic variety. For singularities in differential geometry, see singularity theory. Real analysis In real analysis, singularities are either discontinuities, or discontinuities of the derivative (sometimes also discontinuities of higher order derivatives). There are four kinds of discon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
