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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 (''Land'') of Baden-Württemberg after its capital of 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 near the French border, between the Mannheim/Ludwigshafen conurbation to the north and Strasbourg/Kehl to the south, Karlsruhe is Germany's legal center, being home to the Federal Constitutional Court (''Bundesverfassungsgericht''), the Federal Court of Justice (''Bundesgerichtshof'') and the Public Prosecutor General of the Federal Court of Justice (''Generalbundesanwalt beim Bundesgerichtshof''). 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Geometry
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most 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 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 branches of mathematics. Many types of mathematical objects have a natural notion of distance and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most 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 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 branches of mathematics. Many types of mathematical objects have a natural notion of distance and t ... [...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. Ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karlsruhe
Karlsruhe ( , , ; South Franconian: ''Kallsruh'') is the third-largest city of the German state (''Land'') of Baden-Württemberg after its capital of 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 near the French border, between the Mannheim/ Ludwigshafen conurbation to the north and Strasbourg/Kehl to the south, Karlsruhe is Germany's legal center, being home to the Federal Constitutional Court (''Bundesverfassungsgericht''), the Federal Court of Justice (''Bundesgerichtshof'') and the Public Prosecutor General of the Federal Court of Justice (''Generalbundesanwalt beim Bundesgerichtshof''). 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), th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Coordinate System
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the origin of a Cartesian coordinate system) is called the ''pole'', and the ray from the pole in the reference direction is the ''polar axis''. 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''. Angles in polar notation are generally expressed in either degrees or radians (2 rad being equal to 360°). Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-17th century, though the actual term "polar coordinates" has been attributed to Gregorio Fontana in the 18th century. The initial motivation for the introduction of the polar system was the study of circula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angular Distance
Angular distance \theta (also known as angular separation, apparent distance, or apparent separation) is the angle between the two sightlines, or between two point objects as viewed from an observer. Angular distance appears in mathematics (in particular geometry and trigonometry) and all natural sciences (e.g. 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 couple of stars observed from Earth). Measurement Since the angular distance (or separation) is conceptually identical to an angle, it is measured in the same units, such as degrees or radians, using instruments such as goniometers or optical instruments specially designed to point in well-defined directions and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manhattan Distance
A taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or Metric (mathematics), metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian coordinates. The taxicab metric is also known as rectilinear distance, ''L''1 distance, ''L''1 distance or \ell_1 norm (see Lp space, ''Lp'' space), Snake (video game), snake distance, city block distance, Manhattan distance or Manhattan length. The latter names refer to the rectilinear street layout on the island of Manhattan, where the shortest path a taxi travels between two points is the sum of the absolute values of distances that it travels on avenues and on streets. The geometry has been used in regression analysis since the 18th century, and is often referred to as Lasso (statistics), LASSO. The geometric interpretation dates to non-Euclidean geometry of the 19th century and is due to Hermann Minkowski. In \mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamming Distance
In information theory, the Hamming distance between two strings 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 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 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, among other possibilities. For example, the Hamming distance between: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
