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. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. g. "two counties over"). In mathematics, distance must meet more rigorous criteria. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and

In most cases there is symmetry and "distance from A to B" is interchangeable with "distance between B and A".

Mathematics

See also: Metric (mathematics)

Geometry

In neutral geometry, the minimum distance between two points is the length of the line segment between them. In Mathematics, a metric or distance function is a function which defines a Distance between elements of a set. Absolute geometry is a Geometry based on an Axiom system that does not assume the Parallel postulate or any of its alternatives

In analytic geometry, the distance between two points of the xy-plane can be found using the distance formula. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane 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

Which is easily proven 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 Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane A hypotenuse is the longest side of a Right triangle, the side opposite of the Right angle. In Mathematics, two Vectors are orthogonal if they are Perpendicular, i In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry

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. In Mathematics, the Euclidean distance or Euclidean metric is the "ordinary" Distance between two points that one would measure with a ruler In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry This distance formula can also be expanded into the arc-length formula. In Mathematics and in the Sciences a formula (plural formulae, formulæ or formulas) is a concise way of expressing information Determining the length of an irregular arc segment — also called Rectification of a Curve — was historically difficult

Distance in Euclidean space

In the Euclidean space Rⁿ, the distance between two points is usually given by the Euclidean distance (2-norm distance). In Mathematics, the Euclidean distance or Euclidean metric is the "ordinary" Distance between two points that one would measure with a ruler Other distances, based on other norms, are sometimes used instead. In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length

For a point (x₁, x₂, . . . ,x_n) and a point (y₁, y₂, . . . ,y_n), 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. In Mathematics, the triangle inequality states that for any Triangle, the length of a given side must be less than or equal to the sum of the other two sides but greater

The 2-norm distance is the Euclidean distance, a generalization of the Pythagorean theorem to more than two coordinates. In Mathematics, the Euclidean distance or Euclidean metric is the "ordinary" Distance between two points that one would measure with a ruler In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point It is what would be obtained if the distance between two points were measured with a ruler: the "intuitive" idea of distance. A ruler, or rule, is an instrument used in Geometry, Technical drawing and engineering/building to measure distances and/or to rule straight

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). Taxicab geometry, considered by Hermann Minkowski in the 19th century is a form of Geometry in which the usual metric of Euclidean geometry

The infinity norm distance is also called Chebyshev distance. In Mathematics, Chebyshev distance (or Tchebychev distance) or L∞ metric is a metric defined on a Vector space where In 2D it represents the distance kings must travel between two squares on a chessboard. In Chess, the King (♔ ♚ is the most important piece. The object of the game is to trap the opponent's king so that it would not be able to avoid capture A chessboard is the type of Checkerboard used in the Game of Chess, and consists of 64 squares (eight rows and eight columns arranged in two alternating

The p-norm is rarely used for values of p other than 1, 2, and infinity, but see; super ellipse. The superellipse (or Lamé curve) is the geometric figure defined in the Cartesian coordinate system as the set of all points ( x, y) with

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. In Physics, a rigid body is an idealization of a solid body of finite size in which Deformation is neglected A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of 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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, the real numbers may be described informally in several different ways ↔ (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). In Mathematics, a Binary relation R over a set X is symmetric if it holds for all a and b in X that (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). In Mathematics, the triangle inequality states that for any Triangle, the length of a given side must be less than or equal to the sum of the other two sides but greater (The distance between two points is the shortest distance along any path).

Such a distance function is known as a metric. In Mathematics, a metric or distance function is a function which defines a Distance between elements of a set. Together with the set, it makes up a metric space. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined

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. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a 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. In Topology, a discrete space is a particularly simple example of a Topological space or similar structure one in which the points are " isolated "

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. A Low Earth Orbit (LEO is generally defined as an Orbit within the locus extending from the Earth’s surface up to an altitude of 2000 km 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. In Mathematics the infimum of a Subset of some set is the Greatest element, not necessarily in the subset that is less than or equal to all elements of This is a symmetric prametric. In Topology, a prametric space generalizes the concept of a Metric space by not requiring the conditions of symmetry indiscernability and the triangle inequality 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. In Mathematics, a hemimetric space is a generalization of a Metric space, obtained by removing the requirements of Identity of indiscernibles and of e. , the triangle inequality does not hold, except in special cases. In Mathematics, the triangle inequality states that for any Triangle, the length of a given side must be less than or equal to the sum of the other two sides but greater Therefore only in special cases this distance makes a collection of sets a metric space. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined
• 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. The Hausdorff distance, or Hausdorff metric, measures how far two compact non-empty Subsets of a Metric space are from each other This distance makes the set of non-empty compact subsets of a metric space itself a metric space. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined

The distance between a point and a sethttp://en.wikipedia.org../../../../articles/m/e/t/Metric_space.html#Distance_between_points_and_sets is the infimum of the distances between the point and those in the set. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined 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 along a path compared with displacement

Distance cannot be negative. A negative number is a Number that is less than zero, such as −2 Distance is a scalar quantity, containing only a magnitude, whereas displacement is an equivalent vector quantity containing both magnitude and direction. In Physics, a scalar is a simple Physical quantity that is not changed by Coordinate system rotations or translations (in Newtonian mechanics or The magnitude of a mathematical object is its size a property by which it can be larger or smaller than other objects of the same kind in technical terms an Ordering In Physics, displacement is the vector that specifies the position of a point or a particle in reference to a previous position or to the origin of the chosen Direction is the information contained in the relative position of one point with respect to another point without the Distance information

The distance covered by a vehicle (often recorded by an odometer), person, animal, object, etc. An odometer (often known colloquially as a mileometer or milometer) is a device used for indicating Distance traveled by an Automobile or other 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. In Statistics, Mahalanobis distance is a Distance measure introduced by P Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data.
• Hamming distance and Lee distance are used in coding theory. Examples The Hamming distance between 1011101 and 1001001 Definition and basic properties In Coding theory, the Lee distance is a Distance between two strings x_1 x_2. Coding theory is one of the most important and direct applications of Information theory.
• Levenshtein distance
• Chebyshev distance

See also

References

• E. In Information theory and Computer science, the Levenshtein distance is a metric for measuring the amount of difference between two sequences (i In Mathematics, Chebyshev distance (or Tchebychev distance) or L∞ metric is a metric defined on a Vector space where Taxicab geometry, considered by Hermann Minkowski in the 19th century is a form of Geometry in which the usual metric of Euclidean geometry Astronomers use a number of different length units for different objects The cosmic distance ladder (also known as the Extragalactic Distance Scale) is the succession of methods by which astronomers determine the Distances to celestial In standard cosmology, ' comoving' distance and ' proper distance' are two closely related distance measures used by cosmologists to define distances between Distance geometry is the characterization and study of sets of points based only on given values of the Distances between member pairs In the mathematical field of Graph theory, the distance between two vertices in a graph is the number of edges in a shortest path Geographic Distance is a key factor in military affairs. The shorter the distance the greater the ease with which force can be brought to bear upon an opponent Dijkstra's algorithm, conceived by Dutch Computer scientist Edsger Dijkstra in 1959 is a Graph search algorithm that solves the single-source Shortest An exit number is a number assigned to a Road junction, usually an exit from a Freeway. Distance Measuring Equipment (DME is a transponder-based radio navigation technology that measures distance by timing the propagation delay of VHF or UHF radio signals The great-circle distance is the shortest Distance between any two points on the surface of a Sphere measured along a path on the surface of the sphere (as Length is the long Dimension of any object The length of a thing is the distance between its ends its linear extent as measured from end to end In Mathematics, a metric or distance function is a function which defines a Distance between elements of a set. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined To help compare different Orders of magnitude, the following list describes various Lengths between 1 In relativistic Physics, proper Length is an invariant quantity which is the rod Distance between Spacelike In Mathematics, a distance matrix is a matrix (two-dimensional Array) containing the Distances taken pairwise of a set of points Examples The Hamming distance between 1011101 and 1001001 Definition and basic properties In Coding theory, the Lee distance is a Distance between two strings x_1 x_2. The term proxemics was introduced by anthropologist Edward T Hall in 1966 to describe set measurable distances between people as they interact Deza, M. M. Deza, Dictionary of Distances, Elsevier (2006) ISBN 0-444-52087-2