Distance 39378 217586375 2008-06-06T18:20:57Z Ramses68 7153337 Lee distance {{Citations missing|date=March 2008}} {{dablink|This article is about distance in the [[mathematics|mathematical]] or [[physics|physical]] sense. For other senses of the term, see [[distance (disambiguation)]].}} '''Distance''' is a numerical description of how far apart objects are. In [[physics]] or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria (e.g. "two counties over"). In [[mathematics]], distance must meet more rigorous criteria. In most cases there is symmetry and "distance from A to B" is interchangeable with "distance between B and A". ==Mathematics== {{see also|Metric (mathematics)}} ===Geometry=== In [[neutral geometry]], the minimum distance between two points is the length of the [[Line (mathematics)|line segment]] between them. In [[analytic geometry]], the distance between two points of the [[Cartesian coordinate system|xy-plane]] can be found using the distance formula. The distance between (''x''<sub>1</sub>, ''y''<sub>1</sub>) and (''x''<sub>2</sub>, ''y''<sub>2</sub>) is given by :<math>d=\sqrt{(\Delta x)^2+(\Delta y)^2}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.\,</math> Similarly, given points (''x''<sub>1</sub>, ''y''<sub>1</sub>, ''z''<sub>1</sub>) and (''x''<sub>2</sub>, ''y''<sub>2</sub>, ''z''<sub>2</sub>) in [[Cartesian coordinate system|three-space]], the distance between them is :<math>d=\sqrt{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}.</math> Which is easily proven by constructing a right triangle with a leg on the [[hypotenuse]] of another (with the other leg [[orthogonal]] to the [[Plane (mathematics)|plane]] that contains the 1st triangle) and applying the [[Pythagorean theorem]]. In the study of complicated geometries, we call this (most common) type of distance [[Euclidean distance]], as it is derived from the [[Pythagorean theorem]], which does not hold in [[Non-Euclidean geometry|Non-Euclidean geometries]]. This distance [[formula]] can also be expanded into the [[arc length|arc-length formula]]. ===Distance in Euclidean space=== In the [[Euclidean space]] '''R'''<sup>n</sup>, 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''<sub>1</sub>, ''x''<sub>2</sub>, ...,''x''<sub>''n''</sub>) and a point (''y''<sub>1</sub>, ''y''<sub>2</sub>, ...,''y''<sub>''n''</sub>), the '''Minkowski distance''' of order p ('''p-norm distance''') is defined as: {| cellpadding="2" | 1-norm distance || <math> = \sum_{i=1}^n \left| x_i - y_i \right|</math> |- | 2-norm distance || <math> = \left( \sum_{i=1}^n \left| x_i - y_i \right|^2 \right)^{1/2}</math> |- | ''p''-norm distance ||<math> = \left( \sum_{i=1}^n \left| x_i - y_i \right|^p \right)^{1/p}</math> |- | infinity norm distance ||<math> = \lim_{p \to \infty} \left( \sum_{i=1}^n \left| x_i - y_i \right|^p \right)^{1/p}</math> |- | || <math> = \max \left(|x_1 - y_1|, |x_2 - y_2|, \ldots, |x_n - y_n| \right).</math> |} ''p'' need not be an integer, but it cannot be less than 1, because otherwise the [[triangle inequality]] does not hold. 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 geometry|Manhattan 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)|king]]s must travel between two squares on a [[chessboard]]. 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]]. ===General case=== In [[mathematics]], in particular [[geometry]], a distance function on a given [[set]] ''M'' is a [[Function (mathematics)|function]] d: ''M''×''M''&nbsp;→&nbsp;'''R''', where '''R''' denotes the set of [[real number]]s, that satisfies the following conditions: *d(''x'',''y'') ≥ 0, and d(''x'',''y'') = 0 [[if and only if]] ''x'' = ''y''. (Distance is positive between two different points, and is zero precisely from a point to itself.) *It is [[Symmetric relation|symmetric]]: d(''x'',''y'') = d(''y'',''x''). (The distance between ''x'' and ''y'' is the same in either direction.) *It satisfies the [[triangle inequality]]: d(''x'',''z'') ≤ d(''x'',''y'') + d(''y'',''z''). (The distance between two points is the shortest distance along any path). Such a distance function is known as a [[Metric (mathematics)|metric]]. Together with the set, it makes up a [[metric space]]. For example, the usual definition of distance between two real numbers ''x'' and ''y'' is: d(''x'',''y'') = |''x'' − ''y''|. This definition satisfies the three conditions above, and corresponds to the standard [[topology]] of the [[real line]]. But distance on a given set is a definitional choice. Another possible choice is to define: d(''x'',''y'') = 0 if ''x'' = ''y'', and 1 otherwise. This also defines a metric, but gives a completely different topology, the "[[discrete topology]]"; with this definition numbers cannot be arbitrarily close. ===Distances between sets and between a point and a set === [[Image:Distância entre conjuntos.png‎|thumb| d(A,B)>d(A,C)+d(C,B)]] Various distance definitions are possible between objects. For example, between celestial bodies one should not confuse the surface-to-surface distance and the center-to-center distance. If the former is much less than the latter, as for a [[Low Earth orbit|LEO]], the first tends to be quoted (altitude), otherwise, e.g. for the Earth-Moon distance, the latter. There are two common definitions for the distance between two non-empty [[subset]]s of a given set: *One version of distance between two non-empty sets is the [[infimum]] of the distances between any two of their respective points, which is the every-day meaning of the word. This is a symmetric [[prametric space|prametric]]. On a collection of sets of which some touch or overlap each other, it is not "separating", because the distance between two different but touching or overlapping sets is zero. Also it is not [[hemimetric space|hemimetric]], i.e., the [[triangle inequality]] does not hold, except in special cases. Therefore only in special cases this distance makes a collection of sets a [[metric space]]. *The [[Hausdorff distance]] is the larger of two values, one being the [[supremum]], for a point ranging over one set, of the infimum, for a second point ranging over the other set, of the distance between the points, and the other value being likewise defined but with the roles of the two sets swapped. This distance makes the set of non-empty [[compact space|compact]] subsets of a metric space itself a [[metric space]]. The {{ml|Metric_space|Distance_between_points_and_sets|distance between a point and a set}} is the infimum of the distances between the point and those in the set. This corresponds to the distance, according to the first-mentioned definition above of the distance between sets, from the set containing only this point to the other set. In terms of this, the definition of the Hausdorff distance can be simplified: it is the larger of two values, one being the supremum, for a point ranging over one set, of the distance between the point and the set, and the other value being likewise defined but with the roles of the two sets swapped. == Distance versus displacement == [[Image:Distancedisplacement.svg|thumb|Distance along a path compared with displacement]] Distance cannot be [[negative number|negative]]. Distance is a [[scalar (physics)|scalar]] quantity, containing only a [[Magnitude (mathematics)|magnitude]], whereas [[displacement (vector)|displacement]] is an equivalent [[vector (spatial)|vector]] quantity containing both magnitude and [[Direction (geometry, geography)|direction]]. The distance covered by a vehicle (often recorded by an [[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. ==Other "distances"== *[[Mahalanobis distance]] is used in [[statistics]]. *[[Hamming distance]] and [[Lee distance]] are used in [[coding theory]]. *[[Levenshtein distance]] *[[Chebyshev distance]] ==See also== <div style="-moz-column-count:3; column-count:3;"> *[[Taxicab geometry]] *[[astronomical units of length]] *[[cosmic distance ladder]] *[[comoving distance]] *[[distance geometry]] *[[distance (graph theory)]] *[[Distance in military affairs]] *[[Dijkstra's algorithm]] *[[exit number#Distance-based numbers|distance-based road exit numbers]] *[[Distance Measuring Equipment]] (DME) *[[great-circle distance]] *[[length]] *[[milestone]] *[[Metric (mathematics)]] *[[Metric space]] *[[orders of magnitude (length)]] *[[Proper length]] *[[distance matrix]] *[[Hamming distance]] *[[Lee distance]] *[[proxemics]] - physical distance between people </div> ==References== * E. Deza, M.M. Deza, ''Dictionary of Distances'', [[Elsevier]] (2006) ISBN 0-444-52087-2 [[Category:Length]] [[Category:Elementary mathematics]] [[ar:مسافة]] [[bg:Разстояние]] [[ca:Distància]] [[cs:Vzdálenost]] [[da:Distance]] [[de:Abstand]] [[el:Απόσταση]] [[es:Distancia]] [[eo:Distanco]] [[eu:Luzera]] [[fr:Distance (mathématiques)]] [[gl:Distancia]] [[ko:거리]] [[io:Disto]] [[id:Jarak]] [[ia:Distantia]] [[is:Fjarlægðarformúlan]] [[it:Distanza]] [[lb:Ofstand]] [[ms:Jarak]] [[nl:Afstand]] [[ja:距離]] [[pl:Odległość]] [[pt:Distância]] [[ru:Расстояние]] [[simple:Distance]] [[sk:Vzdialenosť]] [[sl:Razdalja]] [[sv:Avstånd]] [[th:ระยะทาง]] [[vi:Khoảng cách]] [[ur:فاصلہ (ریاضی)]] [[yi:ווייטקייט]] [[zh:距离]] [[fi:Välimatka]]