![Distance board in Vizag](https://upload.wikimedia.org/wikipedia/commons/6/6c/Distance_board_in_Vizag.jpg)
Distance is a numerical or occasionally qualitative
measurement
Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events.
In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
of how far apart objects or points are. In
physics or everyday usage, distance may refer to a physical
length
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
or an estimation based on other criteria (e.g. "two counties over"). Since
spatial cognition
Spatial cognition is the acquisition, organization, utilization, and revision of knowledge about spatial environments. It is most about how animals including humans behave within space and the knowledge they built around it, rather than space itse ...
is a rich source of
conceptual metaphors in
human thought, the term is also frequently used metaphorically to mean a measurement of the amount of difference between two similar objects (such as
statistical distance
In statistics, probability theory, and information theory, a statistical distance quantifies the distance between two statistical objects, which can be two random variables, or two probability distributions or samples, or the distance can be be ...
between
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
s or
edit distance between
strings of text) or a degree of separation (as exemplified by
distance between people in a
social network). Most such notions of distance, both physical and metaphorical, are formalized in
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
using the notion of a
metric space.
In the
social sciences, distance can refer to a qualitative measurement of separation, such as
social distance or
psychological distance.
Distances in physics and geometry
The distance between physical locations can be defined in different ways in different contexts.
Straight-line or Euclidean distance
The distance between two points in physical
space is the
length
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
of a
straight line between them, which is the shortest possible path. This is the usual meaning of distance in
classical physics
Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...
, including
Newtonian mechanics
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motion ...
.
Straight-line distance is formalized mathematically as the
Euclidean distance in
two- and
three-dimensional space. In
Euclidean geometry, the distance between two points and is often denoted
. In
coordinate geometry, Euclidean distance is computed using the
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
. The distance between points and in the plane is given by:
Similarly, given points (''x''
1, ''y''
1, ''z''
1) and (''x''
2, ''y''
2, ''z''
2) in three-dimensional space, the distance between them is:
This idea generalizes to higher-dimensional
Euclidean spaces.
Measurement
There are many ways of
measuring straight-line distances. For example, it can be done directly using a
ruler, or indirectly with a
radar (for long distances) or
interferometry
Interferometry is a technique which uses the ''interference'' of superimposed waves to extract information. Interferometry typically uses electromagnetic waves and is an important investigative technique in the fields of astronomy, fiber opt ...
(for very short distances). The
cosmic distance ladder is a set of ways of measuring extremely long distances.
Shortest-path distance on a curved surface
![Greatcircle Jetstream routes](https://upload.wikimedia.org/wikipedia/commons/7/79/Greatcircle_Jetstream_routes.svg)
The straight-line distance between two points on the surface of the Earth is not very useful for most purposes, since we cannot tunnel straight through the
Earth's mantle. Instead, one typically measures the shortest path along the
surface of the Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surface ...
,
as the crow flies. This is approximated mathematically by the
great-circle distance
The great-circle distance, orthodromic distance, or spherical distance is the distance along a great circle.
It is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a ...
on a sphere.
More generally, the shortest path between two points along a
curved surface is known as a
geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
. The
arc length of geodesics gives a way of measuring distance from the perspective of an
ant or other flightless creature living on that surface.
Effects of relativity
In the
theory of relativity, because of phenomena such as
length contraction
Length contraction is the phenomenon that a moving object's length is measured to be shorter than its proper length, which is the length as measured in the object's own rest frame. It is also known as Lorentz contraction or Lorentz–FitzGerald ...
and the
relativity of simultaneity
In physics, the relativity of simultaneity is the concept that ''distant simultaneity'' – whether two spatially separated events occur at the same time – is not absolute, but depends on the observer's reference frame. This possi ...
, distances between objects depend on a choice of
inertial frame of reference. On galactic and larger scales, the measurement of distance is also affected by the
expansion of the universe
The expansion of the universe is the increase in distance between any two given gravitationally unbound parts of the observable universe with time. It is an intrinsic expansion whereby the scale of space itself changes. The universe does not exp ...
. In practice, a number of
distance measures are used in
cosmology to quantify such distances.
Other spatial distances
![Manhattan distance](https://upload.wikimedia.org/wikipedia/commons/0/08/Manhattan_distance.svg)
Unusual definitions of distance can be helpful to model certain physical situations, but are also used in theoretical mathematics:
* In practice, one is often interested in the travel distance between two points along roads, rather than as the crow flies. In a
grid plan
In urban planning, the grid plan, grid street plan, or gridiron plan is a type of city plan in which streets run at right angles to each other, forming a grid.
Two inherent characteristics of the grid plan, frequent intersections and orthogona ...
, the travel distance between street corners is given by the
Manhattan distance: the number of east–west and north–south blocks one must traverse to get between those two points.
* Chessboard distance, formalized as
Chebyshev distance, is the minimum number of moves a
king must make on a
chessboard
A chessboard is a used to play chess. It consists of 64 squares, 8 rows by 8 columns, on which the chess pieces are placed. It is square in shape and uses two colours of squares, one light and one dark, in a chequered pattern. During play, the bo ...
in order to travel between two squares.
Metaphorical distances
Many abstract notions of distance used in mathematics, science and engineering represent a degree of difference or separation between similar objects. This page gives a few examples.
Statistical distances
In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
and
information geometry
Information geometry is an interdisciplinary field that applies the techniques of differential geometry to study probability theory and statistics. It studies statistical manifolds, which are Riemannian manifolds whose points correspond to prob ...
,
statistical distance
In statistics, probability theory, and information theory, a statistical distance quantifies the distance between two statistical objects, which can be two random variables, or two probability distributions or samples, or the distance can be be ...
s measure the degree of difference between two
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
s. There are many kinds of statistical distances, typically formalized as
divergences; these allow a set of probability distributions to be understood as a
geometrical object called a
statistical manifold. The most elementary is the
squared Euclidean distance, which is minimized by the
least squares
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the res ...
method; this is the most basic
Bregman divergence. The most important in
information theory
Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by the works of Harry Nyquist a ...
is the
relative entropy (
Kullback–Leibler divergence), which allows one to analogously study
maximum likelihood estimation geometrically; this is an example of both an
''f''-divergence and a Bregman divergence (and in fact the only example which is both). Statistical manifolds corresponding to Bregman divergences are
flat manifold In mathematics, a Riemannian manifold is said to be flat if its Riemann curvature tensor is everywhere zero. Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles of ...
s in the corresponding geometry, allowing an analog of the
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
(which holds for squared Euclidean distance) to be used for
linear inverse problem
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, sound source reconstruction, source reconstruction in ac ...
s in inference by
optimization theory.
Other important statistical distances include the
Mahalanobis distance The Mahalanobis distance is a measure of the distance between a point ''P'' and a distribution ''D'', introduced by P. C. Mahalanobis in 1936. Mahalanobis's definition was prompted by the problem of identifying the similarities of skulls based ...
and the
energy distance
Energy distance is a statistical distance between probability distributions. If X and Y are independent random vectors in ''R''d with cumulative distribution functions (cdf) F and G respectively, then the energy distance between the distribution ...
.
Edit distances
In
computer science, an
edit distance or
string metric between two
strings
String or strings may refer to:
*String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects
Arts, entertainment, and media Films
* ''Strings'' (1991 film), a Canadian anim ...
measures how different they are. For example, the words "dog" and "dot", which differ by just one letter, are closer than "dog" and "cat", which have no letters in common. This idea is used in
spell checker In software, a spell checker (or spelling checker or spell check) is a software feature that checks for misspellings in a text. Spell-checking features are often embedded in software or services, such as a word processor, email client, electronic di ...
s and in
coding theory, and is mathematically formalized in a number of different ways, including
Levenshtein distance,
Hamming distance,
Lee distance In coding theory, the Lee distance is a distance between two strings x_1 x_2 \dots x_n and y_1 y_2 \dots y_n of equal length ''n'' over the ''q''-ary alphabet of size . It is a metric defined as
\sum_^n \min(, x_i - y_i, ,\, q - , x_i - y_i, ).
I ...
, and
Jaro–Winkler distance.
Distance in graph theory
In a
graph, the
distance between two vertices is measured by the length of the shortest
edge path between them. For example, if the graph represents a
social network, then the idea of
six degrees of separation can be interpreted mathematically as saying that the distance between any two vertices is at most six. Similarly, the
Erdős number and the
Bacon number—the number of collaborative relationships away a person is from prolific mathematician
Paul Erdős
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...
and actor
Kevin Bacon, respectively—are distances in the graphs whose edges represent mathematical or artistic collaborations.
In the social sciences
In
psychology,
human geography, and the
social sciences, distance is often theorized not as an objective numerical measurement, but as a qualitative description of a subjective experience. For example,
psychological distance is "the different ways in which an object might be removed from" the self along dimensions such as "time, space, social distance, and hypotheticality".
In
sociology,
social distance describes the separation between individuals or
social groups in
society along dimensions such as
social class
A social class is a grouping of people into a set of Dominance hierarchy, hierarchical social categories, the most common being the Upper class, upper, Middle class, middle and Working class, lower classes. Membership in a social class can for ...
,
race/
ethnicity
An ethnic group or an ethnicity is a grouping of people who identify with each other on the basis of shared attributes that distinguish them from other groups. Those attributes can include common sets of traditions, ancestry, language, history, ...
,
gender or
sexuality
Human sexuality is the way people experience and express themselves sexually. This involves biological, psychological, physical, erotic, emotional, social, or spiritual feelings and behaviors. Because it is a broad term, which has varied ...
.
Mathematical formalization
Most of the notions of distance between two points or objects described above are examples of the mathematical idea of a
metric. A ''metric'' or ''distance function'' is a
function which takes pairs of points or objects to
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
and satisfies the following rules:
# The distance between an object and itself is always zero.
# The distance between distinct objects is always positive.
# Distance is
symmetric: the distance from to is always the same as the distance from to .
# Distance satisfies the
triangle inequality: if , , and are three objects, then
This condition can be described informally as "intermediate stops can't speed you up."
As an exception, many of the
divergences used in statistics are not metrics.
Distance between sets
![Distance between sets](https://upload.wikimedia.org/wikipedia/commons/9/97/Distance_between_sets.svg)
There are multiple ways of measuring the physical distance between objects that
consist of more than one point:
* One may measure the distance between representative points such as the
center of mass
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
; this is used for astronomical distances such as the
Earth–Moon distance.
* One may measure the distance between the closest points of the two objects; in this sense, the
altitude of an airplane or spacecraft is its distance from the Earth. The same sense of distance is used in Euclidean geometry to define
distance from a point to a line,
distance from a point to a plane, or, more generally,
perpendicular distance In geometry, the perpendicular distance between two objects is the distance from one to the other, measured along a line that is perpendicular to one or both.
The distance from a point to a line is the distance to the nearest point on that line. Th ...
between
affine subspaces.
: Even more generally, this idea can be used to define the distance between two
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s of a metric space. The distance between sets and is the
infimum of the distances between any two of their respective points:
This does not define a metric on the set of such subsets: the distance between overlapping sets is zero, and this distance does not satisfy the triangle inequality for any metric space with two or more points (consider the triple of sets consisting of two distinct singletons and their union).
* The
Hausdorff distance In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a metric ...
between two subsets of a metric space can be thought of as measuring how far they are from perfectly overlapping. Somewhat more precisely, the Hausdorff distance between and is either the distance from to the farthest point of , or the distance from to the farthest point of , whichever is larger. (Here "farthest point" must be interpreted as a supremum.) The Hausdorff distance defines a metric on the set of
compact subsets of a metric space.
Related ideas
The word distance is also used for related concepts that are not encompassed by the description "a numerical measurement of how far apart points or objects are".
Distance travelled
The distance travelled by an object is the length of a specific path travelled between two points,
such as the distance walked while navigating a
maze
A maze is a path or collection of paths, typically from an entrance to a goal. The word is used to refer both to branching tour puzzles through which the solver must find a route, and to simpler non-branching ("unicursal") patterns that lea ...
. This can even be a closed distance along a
closed curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
which starts and ends at the same point, such as a ball thrown straight up, or the Earth when it completes one
orbit. This is formalized mathematically as the
arc length of the curve.
The distance travelled may also be
signed: a "forward" distance is positive and a "backward" distance is negative.
Circular distance is the distance traveled by a point on the circumference of a
wheel
A wheel is a circular component that is intended to rotate on an axle Bearing (mechanical), bearing. The wheel is one of the key components of the wheel and axle which is one of the Simple machine, six simple machines. Wheels, in conjunction wi ...
, which can be useful to consider when designing vehicles or mechanical gears (see also
odometry). The circumference of the wheel is ; if the radius is 1, each revolution of the wheel causes a vehicle to travel radians.
Displacement and directed distance
![Distancedisplacement](https://upload.wikimedia.org/wikipedia/commons/4/40/Distancedisplacement.svg)
The
displacement
Displacement may refer to:
Physical sciences
Mathematics and Physics
*Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
in classical physics measures the change in position of an object during an interval of time. While distance is a
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
quantity, or a
magnitude, displacement is a
vector quantity with both magnitude and
direction. In general, the vector measuring the difference between two locations (the
relative position
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or ...
) is sometimes called the directed distance.
For example, the directed distance from the
New York City Main Library flag pole to the
Statue of Liberty
The Statue of Liberty (''Liberty Enlightening the World''; French: ''La Liberté éclairant le monde'') is a List of colossal sculpture in situ, colossal neoclassical sculpture on Liberty Island in New York Harbor in New York City, in the U ...
flag pole has:
* A starting point: library flag pole
* An ending point: statue flag pole
* A direction: -38°
* A distance: 8.72 km
Signed distance
See also
*
Absolute difference
*
Astronomical system of units
*
Color difference
In color science, color difference or color distance is the separation between two colors. This metric allows quantified examination of a notion that formerly could only be described with adjectives. Quantification of these properties is of great ...
*
Closeness (mathematics)
Closeness is a basic concept in topology and related areas in mathematics. Intuitively we say two sets are close if they are arbitrarily near to each other. The concept can be defined naturally in a metric space where a notion of distance betwee ...
*
Distance geometry problem Distance geometry is the branch of mathematics concerned with characterizing and studying sets of points based ''only'' on given values of the distances between pairs of points. More abstractly, it is the study of semimetric spaces and the isom ...
*
Dijkstra's algorithm
*
Distance matrix
*
Distance set
In geometry, the distance set of a collection of points is the set of distances between distinct pairs of points. Thus, it can be seen as the generalization of a difference set, the set of distances (and their negations) in collections of numbers. ...
*
Engineering tolerance
*
Multiplicative distance
*
Optical path length In optics, optical path length (OPL, denoted ''Λ'' in equations), also known as optical length or optical distance, is the product of the geometric length of the optical path followed by light and the refractive index of homogeneous medium through ...
*
Orders of magnitude (length)
*
Proper length
*
Proxemics – physical distance between people
*
Signed distance function
*
Similarity measure
*
Social distancing
*
Vertical distance
Library support
*
Python (programming language)
*
Interspace-A package for finding the distance between two vectors, numbers and strings.
*
-Distance computations (
scipy.spatial.distance
)
*
Julia (programming language)
Julia Statistics Distance-A Julia package for evaluating distances (metrics) between vectors.
References
Bibliography
*
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