# Dictionary Definition

proximity

### Noun

1 the property of being close together [syn:
propinquity]

2 the region close around a person or thing

3 a Gestalt principle of organization holding
that (other things being equal) objects or events that are near to
one another (in space or time) are perceived as belonging together
as a unit [syn: law of
proximity]

# User Contributed Dictionary

## English

### Noun

#### Translations

- Finnish: läheisyys

#### Derived terms

# Extensive Definition

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

### Geometry

In neutral geometry, the minimum distance between two points 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

- d=\sqrt=\sqrt.\,

Similarly, given points (x1, y1, z1) and (x2, y2,
z2) in
three-space, the distance between them is

- d=\sqrt=\sqrt.

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: 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 represents the distance kings 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 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).

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 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, 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 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

Distance cannot be negative.
Distance is a scalar
quantity, containing only a magnitude,
whereas displacement
is an equivalent vector
quantity containing both magnitude and
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 is used in coding theory.
- Levenshtein distance
- Chebyshev distance

## See also

- Taxicab geometry
- astronomical units of length
- cosmic distance ladder
- comoving distance
- distance geometry
- distance (graph theory)
- Distance in military affairs
- Dijkstra's algorithm
- 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
- proxemics - physical distance between people

## References

- E. Deza, M.M. Deza, Dictionary of Distances, Elsevier (2006) ISBN 0-444-52087-2

proximity in Arabic: مسافة

proximity in Bulgarian: Разстояние

proximity in Catalan: Distància

proximity in Czech: Vzdálenost

proximity in Danish: Distance

proximity in German: Abstand

proximity in Modern Greek (1453-):
Απόσταση

proximity in Spanish: Distancia

proximity in Esperanto: Distanco

proximity in Basque: Luzera

proximity in French: Distance
(mathématiques)

proximity in Galician: Distancia

proximity in Korean: 거리

proximity in Ido: Disto

proximity in Indonesian: Jarak

proximity in Interlingua (International
Auxiliary Language Association): Distantia

proximity in Icelandic: Fjarlægðarformúlan

proximity in Italian: Distanza

proximity in Luxembourgish: Ofstand

proximity in Malay (macrolanguage): Jarak

proximity in Dutch: Afstand

proximity in Japanese: 距離

proximity in Polish: Odległość

proximity in Portuguese: Distância

proximity in Russian: Расстояние

proximity in Simple English: Distance

proximity in Slovak: Vzdialenosť

proximity in Slovenian: Razdalja

proximity in Swedish: Avstånd

proximity in Thai: ระยะทาง

proximity in Vietnamese: Khoảng cách

proximity in Urdu: فاصلہ (ریاضی)

proximity in Yiddish: ווייטקייט

proximity in Chinese: 距离

proximity in Finnish: Välimatka

# Synonyms, Antonyms and Related Words

accord,
addition, adjacency, adjunct, affairs, affiliation, affinity, alliance, approach, approximation, assemblage, association, bond, closeness, combination, confines, connectedness, connection, contiguity, contiguousness, contrariety, convergence, dealings, deduction, disjunction, environs, filiation, foreground, homology, immediacy, immediate
foreground, intercourse, intimacy, junction, juxtaposition, liaison, link, linkage, linking, mutual attraction,
nearness, neighborhood, nighness, precinct, propinquity, purlieus, rapport, relatedness, relation, relations, relationship, similarity, sympathy, tie, tie-in, togetherness, union, vicinage, vicinity