distance

Dictionary


  • the property created by the space between two objects or points a distant region
  • "I could see it in the distance" size of the gap between two places
  • "the distance from New York to Chicago"
  • "he determined the length of the shortest line segment joining the two points" indifference by personal withdrawal
  • "emotional distance" the interval between two times
  • "the distance from birth to death"
  • "it all happened in the space of 10 minutes" a remote point in time
  • "if that happens it will be at some distance in the future"
  • "at a distance of ten years he had forgotten many of the details" keep at a distance
  • "we have to distance ourselves from these events in order to continue living" go far ahead of
  • "He outdistanced the other runners"

    • Wikipedia


    This article is about distance in the mathematicsmathematical / physicsphysical sense. For other uses, see distance (disambiguation).''The distance between two points is the length of a straight line segment between them. In the case of two locations on Earth, usually the distance along the surface is meant: either "as the crow flies" (along a great circle) or by road, railroad, etc. Distance is sometimes expressed in terms of the time to cover it, for example walking or by car. Sometimes a distance thus indicated is ambiguous because the means of transport is neither mentioned nor obvious.Distance as mentioned above is sometimes not symmetric, hence not a metric (see below): this applies to distance by car in the case of one-way streets, and also in the case the distance is expressed in terms of the time to cover it (a road may be more crowded in one direction than in the other, for a ship upstream and downstream makes a difference).As opposed to a position coordinate, a distance can not be negative numbernegative. Distance is a scalar quantity, containing only a Magnitude (mathematics)magnitude, whereas displacement is an equivalent vector (spatial)vector quantity containing both magnitude and direction.In the study of complicated geometries, we call the most common type of distance Euclidean distance, as we define it from the Pythagorean theorem.

    Distance covered - The distance covered by a vehicle (often recorded by a odometer), person, animal, object, etc. should be distinguished from the distance from starting point to end point, even if latter is taken to mean e.g. the shortest distance along the road, because a detour could be made, and the end point can even coincide with the starting point.

    Formal definition - A distance between two points ''P'' and ''Q'' in a metric space is ''d''(''P'',''Q''), where ''d'' is the distance function that defines the given metric space.We can also define the distance between two sets ''A'' and ''B'' in a metric space as being the minimum (or infimum) of distances between any two points ''P'' in ''A'' and ''Q'' in ''B''.Alternatively, the distance between sets may indicate "how different they are", by taking the supremum over one set of the distance from a point in that set to the other set, and conversely, and taking the larger of the two values (Hausdorff distance).

    The distance formula - The Euclidean distance(Euclidean) distance, ''d'', between two points expressed in Cartesian coordinate systemCartesian coordinates equals the square root of the sum of the squares of the changes of each coordinate.Thus, in a two-dimensional space:d = \sqrt ,and in a three-dimensional space::d = \sqrt .Here, "Δ" (''delta'') refers to the change in a variable. Thus, Δ''x'' is the change in ''x'', pronounced as such, or as "delta-''x''". In mathematical terms, \Delta x = \leftx_1 - x_0\right, and so (\Delta x)^2 = (x_1 - x_2)^2.This distance formula can be seen as a specialized form of the Pythagorean theorem; it can also be expanded into the length of an arcarc-length formula.

    Generalized distance in arbitrary dimensions: Norms - In the Euclidean space Rn, the distance between two points is usually given by the Euclidean distance (2-norm distance). Other distances, based on other norm (mathematics)norms, are sometimes used instead.For a point !(''x''1 ,? ''x''2, !...,''x''''n'')? and a point !(''y''1 ,? ''y''2, !...,''y''''n''),? the Minkowski distance of order p (p-norm distance) is defined as: ^n \left x_i - y_i \right- 2-norm distance = \left( \sum_ ^n \left x_i - y_i \right^2 \right)^ - ''p''-norm distance = \left( \sum_ ^n \left x_i - y_i \right^p \right)^ - infinity norm distance = \lim_ \left( \sum_ ^n \left x_i - y_i \right^p \right)^ - = \max \left(x_1 - y_1, x_2 - y_2, \ldots, x_n - y_n \right).}''p'' need not be an integer, but it cannot be less than 1, because then the triangle inequality does not hold. There is no such thing as a negative distance.The 2-norm distance is the Euclidean distance, a generalization of the Pythagorean theorem to more than two coordinates. It is what would be obtained if the distance between two points were measured with a ruler: the "intuitive" idea of distance.The 1-norm distance is more colourfully called the ''taxicab norm'' or ''taxicab geometryManhattan distance'', because it is the distance a car would drive in a city laid out in square blocks (if there are no one-way streets).The infinity norm distance is also called Chebyshev distance. In 2D it represents the distance king (chess)kings, queen (chess)queens, and bishop (chess)bishops must travel between two squares on a chessboard.Also, if you measure the strength of each of the ''n'' links in a chain (where larger numbers mean weaker links), then because a chain is only as strong as its weakest link, the strength of the chain will be the infinity-norm distance from the list of measurements to the origin.The ''p''-norm is rarely used for values of ''p'' other than 1, 2, and infinity, but see super ellipse.In physical space the Euclidean distance is in a way the most natural one, because in this case the length of a rigid body does not change with rotation.

    Distances in other spaces -
  • Mahalanobis distance is used in statistics.
  • Hamming distance is used in coding theory.
  • Levenshtein distance

    See also -
  • astronomical units of length
  • cosmic distance ladder
  • comoving distance
  • distance geometry
  • distance (graph theory)
  • exit number#Distance-based numbersdistance-based road exit numbers
  • Distance Measuring Equipment (DME)
  • great-circle distance
  • length
  • milestone
  • Metric (mathematics)
  • Metric space
  • neighborhood
  • orders of magnitude !(length)Category:LengthCategor y:Elementary? !mathematicsde:Abstandes:Distan ciaeo:Distancofr:Distance? !(mathématiques)ia:Distantiais :Fjarlægðarformúlanja:ū 17;離nl:Afstandpl:Odleg& #322;ośćpt:Distânci asimple:Distancevi:Khoản g? !cáchzh-cn:距离

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