1. Distance (or farness) 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 is 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".
Contents
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1 Mathematics
o
1.1 Geometry
o
1.2 Distance in Euclidean space
o
1.3 Variational formulation of distance
o
1.4 Generalization to higher-dimensional objects
o
1.5 Algebraic distance
o
1.6 General case
o
1.7 Distances between sets and between a point
and a set
o
1.8 Graph theory
2 Distance versus directed distance and displacement
o
2.1 Directed distance
3 Other "distances"
4 See also
5 References
[edit]Mathematics
See also: Metric (mathematics)
[edit]Geometry

2. 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
thePythagorean 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.
[edit]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
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).

3. 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.
[edit]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
used.
, where Einstein summation convention has been
[edit]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:
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

4. 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.
[edit]Algebraic
distance
This section requires expansion.
(December 2008)
This 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 nonlinear least squares.
[edit]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.
[edit]Distances
between sets and between a point and a set

5. 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 centerto-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, i.e.
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.

6. 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.
[edit]Graph
theory
In graph theory the distance between two vertices is the length of the
shortest path between those vertices.
[edit]Distance
versus directed distance and
displacement
Distance along a path compared with displacement
Distance cannot be negative and distance travelled never decreases. Distance
is a scalar quantity or a magnitude, whereas displacement is a vectorquantity
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 Bshould
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.
[edit]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 ndimensional 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.

7. 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 gravityof 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. Displacement may increase or decrease but distance
travelled never decreases.
[edit]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.
[edit]See
also
•
Taxicab geometry
•
Astronomical units of length
•
Cosmic distance ladder
•
L
•
Distance measures (cosmology)
•
M
•
Comoving distance
•
M

8. •
Distance geometry
•
M
•
Distance (graph theory)
•
O
•
Dijkstra's algorithm
•
P
•
Distance-based road exit numbers
•
D
•
Distance measuring equipment (DME)
•
H
•
Engineering tolerance
•
L
•
Great-circle distance
•
P
•
M
[edit]References
1.
^ SS Plotkin, PNAS.2007; 104: 14899–14904,
2.
^ AR Mohazab, SS Plotkin,"Minimal Folding Pathways for Coarse-Grained
Biopolymer Fragments" Biophysical Journal, Volume 95, Issue 12, Pages
5496–5507
•
Deza, E.; Deza, M. (2006), Dictionary of Distances, Elsevier, ISBN 0-44452087-2.