Distance

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Distance_Learning :: Distance_Learning_and_Training :: Distance_Programs

Distance is a numerical description of how far apart things lie. 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.

Physics

According to special relativity, distances through reality's time and space can only be measured as a spacetime interval. See the introduction to special relativity.

When not including time, distances through space are equal to the geometric formulas given below. These types of distances are also equal to the amount of (speed x time) required to move from one position to another. These formulas for distance could roughly be described as the relationship between dimensional differences, and force. For example, Gravity reduces proportionately to distance. It can, generally, be described as "how far apart things lie", because of this relationship.

Formulas for distance define what the shortest route between two points is. For example, if you traveled a distance of 10 on one axis, and then travelled a distance of 5 on a perpendicular axis, you would have travelled a total distance of 15. However, if you moved by both of the amounts at the same time, you would have only travelled a distance of ≈ 11.18.

Geometry

In neutral geometry, the distance between two points is the length of the line segment between them.

In algebraic geometry, one can find the distance between two points of the xy-plane 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_1-x_2)^2+(y_1-y_2)^2}.

This formula could also be used as follows:

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_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}.

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 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.

Mathematics

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, ...,x_n) and a point (y₁, y₂, ...,y_n), 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
p-norm distance	= \left( \sum_{i=1}^n \left > x_i - y_i \right
infinity norm distance	= \lim_{p \to \infty} \left( \sum_{i=1}^n \left > x_i - y_i \right
	= \max \left( >x_1 - y_1

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).

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.

Distances between non-empty sets

One might attempt to define the distance between two non-empty subsets of a given set as the infimum of the distances between any two of their respective points, which would agree with the every-day use of the word. However, this does not define a metric, since the distance between two different but overlapping sets will be found to be equal. A defintion that does work defines the distance as 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 is a metric, called the Hausdorff metric.

Distinguish

As opposed to a position coordinate, a distance can not be negative. Distance is a scalar quantity, containing only a magnitude, whereas displacement is an equivalent vector quantity containing both magnitude and direction.

Informal

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

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