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From Wikipedia, the free encyclopedia

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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Mathematics

Geometry

In neutral geometry, the distance between (x1) and (x2) is the length of the line segment between them:

d=\sqrt{(\Delta x)^2}=\sqrt{(x_2-x_1)^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:

d=\sqrt{(\Delta x)^2+(\Delta y)^2}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.\,

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

d=\sqrt{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}.

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  = \sum_{i=1}^n \left| x_i - y_i \right|
2-norm distance  = \left( \sum_{i=1}^n \left| x_i - y_i \right|^2 \right)^{1/2}
p-norm distance  = \left( \sum_{i=1}^n \left| x_i - y_i \right|^p \right)^{1/p}
infinity norm distance  = \lim_{p \to \infty} \left( \sum_{i=1}^n \left| x_i - y_i \right|^p \right)^{1/p}
 = \max \left(|x_1 - y_1|, |x_2 - y_2|, \ldots, |x_n - y_n| \right).

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 (A = \vec{r}(0) and B = \vec{r}(T)) may be written in a variational form where the distance is the minimum value of an integral:

 D = \int_0^T dt \sqrt{\left({\partial \vec{r}(t) \over \partial t}\right)^2}

Here \vec{r}(t) 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 r = r * where r * 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 gab the integrand has be to modified to \sqrt{g^{ac}\dot{r}_{c}g_{ab}\dot{r}^{b}}, 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 ds dt \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\}

The above double integral is the generalized distance functional between two plymer conformation. s is a spatial parameter and t is pseudo-time. This means that \vec{r}(s,t=t_i) is the polymer/string conformation at time ti and is parameterized along the string length by s. Similarly \vec{r}(s=S,t) is the trajectory of an infinitesimal segment of the string during transformation of the entire string from conformation \vec{r}(s,0) to conformation \vec{r}(s,T). The term with cofactor λ 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

The algebraic distance is a metric often used in computer vision that that can be minimized by least squares estimation. [1][2] For curves or surfaces given by the equation xTCx = 0 (such as a conic in homogeneous coordinates), the algebraic distance from the point x' to the curve is simply x'TCx'. 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) = |xy|. 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

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 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 along a path compared with displacement

Distance cannot be negative. Distance is a scalar quantity, containing only a magnitude, whereas a displacement vector is a vector quantity characterized by both magnitude and direction.

The distance covered by a vehicle (often 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 respective displacement (the distance along a straight line from A to B). For instance, the distance covered during a round trip from A to B and back to A may be very long, while the displacement is always zero (because starting and ending points coincide).

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.

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).

Other "distances"

Circular distance is the distance traveled by a wheel. The circumference of the wheel is 2*(pi)*(radius), and assuming the radius to be 1, then each revolution of the wheel is equivalent of the distance 2*(pi) radians. In engineering (omega)=2*(pi)*f is used a lot, where f is the frequency.

See also

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 0444520872 .

Quotes

Up to date as of January 14, 2010

From Wikiquote

Distance is the quality of objects being physically separated from one another.

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Quotes

  • Yon foaming flood seems motionless as ice;
    Its dizzy turbulence eludes the eye,
    Frozen by distance.
  • By this time, like one who had set out on his way by night, and travelled through a region of smooth or idle dreams, our history now arrives on the confines, where daylight and truth meet us with a clear dawn, representing to our view, though at a far distance, true colours and shapes.
  • Life is like a landscape. You live in the midst of it but can describe it only from the vantage point of distance.
    • Charles Lindbergh, as quoted in Lindbergh: Flight's Enigmatic Hero (2002) by Von Hardesty

Distance makes things more sweet

  • Distance sometimes endears friendship, and absence sweeteneth it.
  • In notes by distance made more sweet.
  • Sweetest melodies.
    Are those that are by distance made more sweet.

Distant things appear better

  • Glories, like glow-worms, afar off shine bright,
    But look'd too near have neither heat nor light.
  • Love is like a landscape which doth stand
    Smooth at a distance, rough at hand.
    • Robert Hegge, On Love, reported in Bartlett's Familiar Quotations, 10th ed. (1919).
  • We're charm'd with distant views of happiness,
    But near approaches make the prospect less.
    • Thomas Yalden, Against Enjoyment, reported in Bartlett's Familiar Quotations, 10th ed. (1919).
  • As distant prospects please us, but when near
    We find but desert rocks and fleeting air.
  • 'T is distance lends enchantment to the view,
    And robes the mountain in its azure hue.

External link

Wikipedia
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Look up distance in Wiktionary, the free dictionary

Simple English

Simple English Wiktionary has the word meaning for:

Distance is how far one thing is from another thing. Distance is a measure of the space between two things.

=Examples of Distance

=

  • One ball is 5 feet from another ball. The distance between the two balls is 5 feet.
  • John walks north at 6 kilometres per hour for one hour. John has walked a distance of 6 kilometres.
  • A circle is a curved line. Each point on the circle is the same distance from the centre of the circle.

Displacement

Distance is different from displacement. Displacement is the difference between a starting point and a finishing point.

Example of Displacement

A and B are 1 metre apart. Tom walks from A to B. Then Tom walks from B back to A. The distance Tom walked is 2 metres. Tom's displacement is 0 metres, because Tom started at A and finished at A.








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