distance
American Heritage Dictionary:
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dis·tance
(dĭs'təns) 
n.
n.
- The extent of space between two objects or places; an intervening space.
- The fact or condition of being apart in space; remoteness.
- Mathematics. The length or numerical value of a straight line or curve.
- The extent of space between points on a measured course.
- The length of a race, especially of a horserace.
- A point or area that is far away: "Telephone poles stretched way into a distance I couldn't quite see" (Leigh Allison Wilson).
- A depiction of a such a point or area.
- A stretch of space without designation of limit; an expanse: a land of few hills and great distances.
- The extent of time between two events; an intervening period.
- A point removed in time: At a distance of 11 years, his memory of the crime was blurry.
- The full period or length of a contest or game: The challenger had never attempted the distance of 12 rounds.
- An amount of progress: The curriculum committee is a distance from where it was last month.
- Difference or disagreement: The candidates could not be at a greater distance on this issue.
- Emotional separateness or reserve; aloofness.
- To place or keep at or as if at a distance: "To understand Russian strategy ... it is necessary for us to distance ourselves from our own myths and to enter into theirs" (Freeman J. Dyson).
- To cause to appear at a distance.
- To leave far behind; outrun.
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distance
Roget's Thesaurus:
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distance
noun
- An extent, measured or unmeasured, of linear space: length, space, stretch. Informal piece, way. See big/small/amount.
- The fact or condition of being far removed or apart: farness, remoteness. See big/small/amount, near/far/distance.
- A wide and open area, as of land, sky, or water: expanse, expansion, extent, reach, space, spread, stretch, sweep. See place.
- Degree of separation, especially in time: remove. See near/far/distance.
- Dissociation from one's surroundings or worldly affairs: aloofness, detachment, remoteness. See attitude/good attitude/bad attitude/neutral attitude, concern/unconcern, include/exclude, near/far/distance.
Antonyms by Answers.com:
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distance
n
Definition: aloofness
Antonyms: affection, friendliness, sympathy, warmth
v
Definition: dissociate oneself; leave behind
Antonyms: associate, be friendly, go to
Oxford Dictionary of Geography:
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distance
Absolute distance is expressed in physical units such as kilometres and is unchangeable. Relative distance includes any other kind of distance such as time-distance, which is measured in hours and minutes and changes with varying technology. Thus, a location 6 hours away by train is only 90 minutes away by air. Cost distance is expressed in terms of currency and varies with the transport mode, the volume and type of traffic and goods, and their destination. Convenience distance expresses the ease of travel.
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A scalar measurement of the extent of a body's motion, irrespective of the direction in which it has travelled. Thus, when a body moves from one location to another, the distance through which it moves is the length of path it follows. In a race around a 400 m track, the distance travelled is 400 m. Compare displacement.
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(DOD) 1. The space between adjacent individual ships or boats measured in any direction between foremasts. 2. The space between adjacent men, animals, vehicles, or units in a formation measured from front to rear. 3. The space between known reference points or a ground observer and a target, measured in meters (artillery), in yards (naval gunfire), or in units specified by the observer. See also interval.
Devil's Dictionary:
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distance
A cynical view of the world by Ambrose Bierce
n.
The only thing that the rich are willing for the poor to call theirs, and keep.
Word Tutor:
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distance
IN BRIEF: Space between two points.
Anywhere is walking distance if you have the time.
— Steven Wright, Canadian comedian.
LearnThatWord.com is a free vocabulary and spelling program where you only pay for results!
Saunders Veterinary Dictionary:
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distance
The measure of space intervening between two objects or two points of reference.
- critical d. — see flight distance.
- focal–film d. — the distance between the anode of the x-ray tube and the film; an important exposure value.
- flight d. — see flight distance.
- guard d. — see flight distance.
- interocclusal d. — the distance between the occluding surfaces of the maxillary and mandibular teeth with the mandible in physiological rest position.
- interocular d. — the distance between the eyes, usually used in reference to the interpupillary distance (the distance between the two pupils when the visual axes are parallel).
Mosby's Dental Dictionary:
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distance
n
The measure of space intervening between two objects or two points of reference.
Random House Word Menu:
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categories related to 'distance'
- Principles of Mechanics, Waves, and Measurement - distance: spatial separation of two points measured by length of hypothetical straight line joining them
- Horse and Dog Racing
- Dimensions and Directions - distance: extent of space between two points, lines, surfaces, or objects, usu. along shortest path
See crossword solutions for the clue Wikipedia on Answers.com:
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Distance
This article is about distance in the mathematical or physical sense. For other senses of the term, see distance (disambiguation).
"Proximity" redirects here. For the 2001 film, see Proximity (film).
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This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (March 2008) |
Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria (e.g. "two counties over"). In mathematics, a distance function or metric is a generalization of the concept of physical distance. A metric is a function that behaves according to a specific set of rules, and provides a concrete way of describing what it means for elements of some space to be "close to" or "far away from" each other.
In most cases, "distance from A to B" is interchangeable with "distance between B and A".
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Contents
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Mathematics
See also: Metric (mathematics)
Geometry
In neutral geometry, the distance between (x1.) and (x2) is the length of the line segment between them:
In analytic geometry, the distance between two points of the xy-plane can be found using the distance formula. The distance between (x1, y1) and (x2, y2) is given by:
Similarly, given points (x1, y1, z1) and (x2, y2, z2) in three-space, the distance between them is:
These formulae are easily derived by constructing a right triangle with a leg on the hypotenuse of another (with the other leg orthogonal to the 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 geometries. This distance formula can also be expanded into the arc-length formula.
Distance in Euclidean space
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 norms, are sometimes used instead.
For a point (x1, x2, ...,xn) and a point (y1, y2, ...,yn), the Minkowski distance of order p (p-norm distance) is defined as:
| 1-norm distance |  |
| 2-norm distance |  |
| p-norm distance |  |
| infinity norm distance |  |
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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 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 is the minimum number of moves kings require to 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.
Variational formulation of distance
The Euclidean distance between two points in space (
and 
) may be written in a variational form where the distance is the minimum value of an integral:
Here 
is the trajectory (path) between the two points. The value of the integral (D) represents the length of this trajectory. The distance is the minimal value of this integral and is obtained when 
where 
is the optimal trajectory. In the familiar Euclidean case (the above integral) this optimal trajectory is simply a straight line. It is well known that the shortest path between two points is a straight line. Straight lines can formally be obtained by solving the Euler-Lagrange equations for the above functional. In non-Euclidean manifolds (curved spaces) where the nature of the space is represented by a metric 
the integrand has be to modified to 
, where Einstein summation convention has been used.
Generalization to higher-dimensional objects
The Euclidean distance between two objects may also be generalized to the case where the objects are no longer points but are higher-dimensional manifolds, such as space curves, so in addition to talking about distance between two points one can discuss concepts of distance between two strings. Since the new objects that are dealt with are extended objects (not points anymore) additional concepts such as non-extensibility, curvature constraints, and non-local interactions that enforce non-crossing become central to the notion of distance. The distance between the two manifolds is the scalar quantity that results from minimizing the generalized distance functional, which represents a transformation between the two manifolds:
![\mathcal {D} = \int_0^L\int_0^T \left \{ \sqrt{\left({\partial \vec{r}(s,t) \over \partial t}\right)^2} + \lambda \left[\sqrt{\left({\partial \vec{r}(s,t) \over \partial s}\right)^2} - 1\right] \right\} \, ds \, dt](http://wpcontent.answcdn.com/wikipedia/en/math/1/7/0/170eaa5bf615ea25ad27fae117f04d9c.png)
The above double integral is the generalized distance functional between two plymer conformation. 
is a spatial parameter and 
is pseudo-time. This means that 
is the polymer/string conformation at time 
and is parameterized along the string length by . Similarly 
is the trajectory of an infinitesimal segment of the string during transformation of the entire string from conformation 
to conformation 
. The term with cofactor 
is a Lagrange multiplier and its role is to ensure that the length of the polymer remains the same during the transformation. If two discrete polymers are inextensible, then the minimal-distance transformation between them no longer involves purely straight-line motion, even on a Euclidean metric. There is a potential application of such generalized distance to the problem of protein folding[1][2] This generalized distance is analogous to the Nambu-Goto action in string theory, however there is no exact correspondence because the Euclidean distance in 3-space is inequivalent to the space-time distance minimized for the classical relativistic string.
Algebraic distance
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The algebraic distance is a metric often used in computer vision that can be minimized by least squares estimation. [1][2] For curves or surfaces given by the equation 
(such as a conic in homogeneous coordinates), the algebraic distance from the point 
to the curve is simply 
. It may serve as an "initial guess" for geometric distance to refine estimations of the curve by more accurate methods, such as non-linear least squares.
General case
In mathematics, in particular geometry, a distance function on a given set M is a function d: M×M → R, where R denotes the set of real numbers, 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: 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. 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
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 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 subsets 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 premetric. 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, 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 subsets of a metric space itself a metric space.
The 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.
Graph theory
In graph theory the distance between two vertices is the length of the shortest path between those vertices.
Distance versus directed distance and displacement
Distance cannot be negative and distance travelled never decreases. Distance is a scalar quantity or a magnitude, whereas displacement is a vector quantity with both magnitude and direction.
The distance covered by a vehicle (for example as recorded by an odometer), person, animal, or object along a curved path from a point A to a point B should be distinguished from the straight line distance from A to B. For example whatever the distance covered during a round trip from A to B and back to A, the displacement is zero as start and end points coincide. In general the straight line distance does not equal distance travelled, except for journeys in a straight line.
Directed distance
Directed distances are distances with a direction or sense. They can be determined along straight lines and along curved lines. A directed distance along a straight line from A to B is a vector joining any two points in a n-dimensional Euclidean vector space. A directed distance along a curved line is not a vector and is represented by a segment of that curved line defined by endpoints A and B, with some specific information indicating the sense (or direction) of an ideal or real motion from one endpoint of the segment to the other (see figure). For instance, just labelling the two endpoints as A and B can indicate the sense, if the ordered sequence (A, B) is assumed, which implies that A is the starting point.
A displacement (see above) is a special kind of directed distance defined in mechanics. A directed distance is called displacement when it is the distance along a straight line (minimum distance) from A and B, and when A and B are positions occupied by the same particle at two different instants of time. This implies motion of the particle. displace is a vector quantity.
Another kind of directed distance is that between two different particles or point masses at a given time. For instance, the distance from the center of gravity of the Earth A and the center of gravity of the Moon B (which does not strictly imply motion from A to B).Shortest path length may be equal to displacement or may not be equal to.Distance from starting point is always equal to magnitude of displacement. For same particle distance travelled is always greater than or equal to magnitude of displacement. Shortest path length is not necessary always displacement.Diplacement may increase or decrease but distance travelled never decreases.
Other "distances"
- E-statistics, or energy statistics, are functions of distances between statistical observations.
- Mahalanobis distance is used in statistics.
- Hamming distance and Lee distance are used in coding theory.
- Levenshtein distance
- Chebyshev distance
- Canberra distance
Circular distance is the distance traveled by a wheel. The circumference of the wheel is 2π × radius, and assuming the radius to be 1, then each revolution of the wheel is equivalent of the distance 2π radians. In engineering ω = 2πƒ is often used, where ƒ is the frequency.
See also
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References
- Deza, E.; Deza, M. (2006), Dictionary of Distances, Elsevier, ISBN 0-444-52087-2.
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This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)
Translations:
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Distance
Dansk (Danish)
n. - afstand, mellemrum, distance, fjerntliggende punkt, afstandtagen, reserverthed, opløbsfelt, fastlagt kamplængde
v. tr. - distancere, holde på afstand, lægge afstand til, overgå
idioms:
- at a distance noget borte, på afstand
- distance is no object afstand er ikke noget problem
- distance oneself from holde sig på afstand af
- go the distance stå distancen, gå kampen ud
- keep one's distance være reserveret, holde afstand
Nederlands (Dutch)
afstand, verte, tijdsduur, afstandelijkheid, geplande duur van een bokswedstrijd, distance (lengtemaat bij paardenrennen)
Français (French)
n. - distance, à distance, à l'écart, intervalle, écart
v. tr. - (Sport) distancer, (fig) se distancier de qch
idioms:
- at a distance assez loin, à quelque distance
- distance is no object la distance n'est pas un problème
- distance oneself from se distancier de
- go the distance tenir la distance
- keep one's distance garder ses distances
Deutsch (German)
n. - Abstand, Distanz, Entfernung, Ferne
v. - hinter sich lassen
idioms:
- at a distance auf einige Entfernung, in einiger Entfernung, in kurzer Entfernung
- distance is no object Entfernung spielt keine Rolle
- distance oneself from Abstand nehmen von
- go the distance über die volle Distanz gehen, (bis zum Schluß) durchhalten
- keep one's distance Abstand halten
Ελληνική (Greek)
n. - απόσταση, διάστημα, (μτφ.) επιφυλακτικότητα ή ψυχρότητα (συμπεριφοράς)
v. - απομακρύνω/-ομαι
idioms:
- at a distance σε (κάποια) απόσταση
- distance is no object δεν έχει σημασία η απόσταση
- distance oneself from απομακρύνομαι από
- go the distance (καθομ.) (για πυγμάχους) αντέχω σε όλους τους γύρους
- keep one's distance κρατώ τις αποστάσεις
idioms:
- at a distance a distanza
- distance oneself from distanziarsi da
- go the distance reggere fino in fondo
- keep one's distance mantenere le distanze
Português (Portuguese)
n. - distância (f), afastamento (m)
v. - distanciar
idioms:
- at a distance à distância
- distance oneself from distanciar-se de
- go the distance continuar competindo até o fim
- keep one's distance evitar envolvimento
Русский (Russian)
расстояние, дистанция
idioms:
- at a distance на расстоянии
- distance oneself from отдалиться от
- go the distance дойти до конца
- keep one's distance держаться на расстоянии
Español (Spanish)
n. - distancia, lejanía, separación, diferencia
v. tr. - distanciar
idioms:
- at a distance a distancia, de lejos, después de
- distance is no object la distancia no importa
- distance oneself from distanciarse de, desvincularse de
- go the distance completar el recorrido, llegar al final
- keep one's distance guardar las distancias, no acercarse
Svenska (Swedish)
n. - avstånd, väg
v. - lämna bakom sig, distansera
中文（简体）(Chinese (Simplified))
距离, 远处, 路程, 冷淡, 疏远, 把...远远甩在后面, 使疏远
idioms:
- at a distance 在远处
- distance is no object 不管多远
- distance oneself from 跟...疏远
- go the distance 不换人地打全场, 赛足回合
- keep one's distance 保持疏远
中文（繁體）(Chinese (Traditional))
n. - 距離, 遠處, 路程, 冷淡, 疏遠
v. tr. - 把...遠遠甩在後面, 使疏遠
idioms:
- at a distance 在遠處
- distance is no object 不管多遠
- distance oneself from 跟...疏遠
- go the distance 不換人地打全場, 賽足回合
- keep one's distance 保持疏遠
한국어 (Korean)
n. - 원 거리, 사이, 넓이, 음정
v. tr. - 거리를 두다, 앞서다
idioms:
- at a distance 좀 떨어져서
- distance oneself from ~로부터 멀어지다
- go the distance 끝까지 해내다
- keep one's distance 멀리 하다
日本語 (Japanese)
n. - 距離, 遠距離, 隔たり, 相違, よそよそしさ, 広がり, 遠景
v. - 追い抜く, 引き離す
idioms:
- at a distance ある距離をおいて
- distance oneself from から距離をおく
- go the distance 最後までやり抜く, 完投する
- spitting distance 短い距離
العربيه (Arabic)
(الاسم) مسافه, بعد, برود ( في المشاعر), فترة (فعل) يبعد, يقصي, يسبق
עברית (Hebrew)
n. - מרחק, מרווח, הימנעות מיחסים קרובים, פרק-זמן
v. tr. - השאיר מאחור, הרחיק
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The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved.
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Roget's II: The New Thesaurus, Third Edition by the Editors of the American Heritage® Dictionary Copyright © 1995 byHoughton Mifflin Company. Published by Houghton Mifflin Company. All rights reserved.
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Oxford Dictionary of Geography
A Dictionary of Geography. Copyright © Susan Mayhew 1992, 1997, 2004. All rights reserved.
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The Oxford Dictionary of Sports Science & Medicine. Copyright © Michael Kent 1998, 2006, 2007. All rights reserved.
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US Defense Department Military Dictionary
US Department of Defense Dictionary of Military and Associated Words, 2003.
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Saunders Comprehensive Veterinary Dictionary 3rd Edition. Copyright © 2007 by D.C. Blood, V.P. Studdert and C.C. Gay, Elsevier. All rights reserved.
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Mosby's Dental Dictionary. Copyright © 2004 by Elsevier, Inc. All rights reserved.
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This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article Distance.
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