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Karlsruhe Metric
In metric geometry, the Karlsruhe metric is a measure of distance that assumes travel is only possible along rays through the origin and circular arcs centered at the origin. The name alludes to the layout of the city of Karlsruhe, which has radial streets and circular avenues around a central point. This metric is also called Moscow metric. In this metric, there are two types of shortest paths. One possibility, when the two points are on nearby rays, combines a circular arc through the nearer to the origin of the two points and a segment of a ray through the farther of the two points. Alternatively, for points on rays that are nearly opposite, it is shorter to follow one ray all the way to the origin and then follow the other ray back out. Therefore, the Karlsruhe distance between two points d_k(p_1,p_2) is the minimum of the two lengths that would be obtained for these two types of path. That is, it equals d_k(p_1,p_2)= \begin \min(r_1,r_2) \cdot \delta(p_1,p_2) +, r_1-r_2, ,&\t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karlsruhe Bodenehr 1721 Koloriert
Karlsruhe ( ; ; ; South Franconian: ''Kallsruh'') is the third-largest city of the German state of Baden-Württemberg, after its capital Stuttgart and Mannheim, and the 22nd-largest city in the nation, with 308,436 inhabitants. It is also a former capital of Baden, a historic region named after Hohenbaden Castle in the city of Baden-Baden. Located on the right bank of the Rhine (Upper Rhine) near the French border, between the Mannheim-Ludwigshafen conurbation to the north and Strasbourg to the south, Karlsruhe is Germany's legal center, being home to the Federal Constitutional Court, the Federal Court of Justice and the Public Prosecutor General. Karlsruhe was the capital of the Margraviate of Baden-Durlach (Durlach: 1565–1718; Karlsruhe: 1718–1771), the Margraviate of Baden (1771–1803), the Electorate of Baden (1803–1806), the Grand Duchy of Baden (1806–1918), and the Republic of Baden (1918–1945). Its most remarkable building is Karlsruhe Palace, which was bui ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Geometry
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), function called a metric or distance function. Metric spaces are a general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a Conceptual metaphor , metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different bra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter ''O'', used as a fixed point of reference for the geometry of the surrounding space. In physical problems, the choice of origin is often arbitrary, meaning any choice of origin will ultimately give the same answer. This allows one to pick an origin point that makes the mathematics as simple as possible, often by taking advantage of some kind of geometric symmetry. Cartesian coordinates In a Cartesian coordinate system, the origin is the point where the axes of the system intersect.. The origin divides each of these axes into two halves, a positive and a negative semiaxis. Points can then be located with reference to the origin by giving their numerical coordinates—that is, the positions of their projections along each axis, either in the positive or negative direction. The coordinates of the origin are always all zero, for example (0,0) in two dimensions and (0,0,0) in three. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karlsruhe
Karlsruhe ( ; ; ; South Franconian German, South Franconian: ''Kallsruh'') is the List of cities in Baden-Württemberg by population, third-largest city of the States of Germany, German state of Baden-Württemberg, after its capital Stuttgart and Mannheim, and the List of cities in Germany by population, 22nd-largest city in the nation, with 308,436 inhabitants. It is also a former capital of Baden, a historic region named after Hohenbaden Castle in the city of Baden-Baden. Located on the right bank of the Rhine (Upper Rhine) near the French border, between the Rhine-Neckar Metropolitan Region, Mannheim-Ludwigshafen conurbation to the north and Strasbourg to the south, Karlsruhe is Germany's legal center, being home to the Federal Constitutional Court, the Federal Court of Justice and the Public Prosecutor General (Germany), Public Prosecutor General. Karlsruhe was the capital of the Margraviate of Baden-Durlach (Durlach: 1565–1718; Karlsruhe: 1718–1771), the Margraviate of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Coordinate System
In mathematics, the polar coordinate system specifies a given point in a plane by using a distance and an angle as its two coordinates. These are *the point's distance from a reference point called the ''pole'', and *the point's direction from the pole relative to the direction of the ''polar axis'', a ray drawn from the pole. The distance from the pole is called the ''radial coordinate'', ''radial distance'' or simply ''radius'', and the angle is called the ''angular coordinate'', ''polar angle'', or ''azimuth''. The pole is analogous to the origin in a Cartesian coordinate system. Polar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point in a plane, such as spirals. Planar physical systems with bodies moving around a central point, or phenomena originating from a central point, are often simpler and more intuitive to model using polar coordinates. The polar coordinate system i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angular Distance
Angular distance or angular separation is the measure of the angle between the orientation (geometry), orientation of two straight lines, ray (geometry), rays, or vector (geometry), vectors in three-dimensional space, or the central angle subtended by the radius, radii through two points on a sphere. When the rays are Line of sight, lines of sight from an observer to two points in space, it is known as the apparent distance or apparent separation. Angular distance appears in mathematics (in particular geometry and trigonometry) and all natural sciences (e.g., kinematics, astronomy, and geophysics). In the classical mechanics of rotating objects, it appears alongside angular velocity, angular acceleration, angular momentum, moment of inertia and torque. Use The term ''angular distance'' (or ''separation'') is technically synonymous with ''angle'' itself, but is meant to suggest the linear distance between objects (for instance, a pair of stars observed from Earth). Measurement Sin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manhattan Distance
Taxicab geometry or Manhattan geometry is geometry where the familiar Euclidean distance is ignored, and the distance between two point (geometry), points is instead defined to be the sum of the absolute differences of their respective Cartesian coordinates, a distance function (or Metric (mathematics), metric) called the ''taxicab distance'', ''Manhattan distance'', or ''city block distance''. The name refers to the island of Manhattan, or generically any planned city with a rectangular grid of streets, in which a taxicab can only travel along grid directions. In taxicab geometry, the distance between any two points equals the length of their shortest grid path. This different definition of distance also leads to a different definition of the length of a curve, for which a line segment between any two points has the same length as a grid path between those points rather than its Euclidean length. The taxicab distance is also sometimes known as ''rectilinear distance'' or distanc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamming Distance
In information theory, the Hamming distance between two String (computer science), strings or vectors of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number of ''substitutions'' required to change one string into the other, or equivalently, the minimum number of ''errors'' that could have transformed one string into the other. In a more general context, the Hamming distance is one of several string metrics for measuring the edit distance between two sequences. It is named after the American mathematician Richard Hamming. A major application is in coding theory, more specifically to block codes, in which the equal-length strings are Vector space, vectors over a finite field. Definition The Hamming distance between two equal-length strings of symbols is the number of positions at which the corresponding symbols are different. Examples The symbols may be letters, bits, or decimal digits, am ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
