Distance
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 . 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 required to move from one position to another.
Encyclopedia
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 . 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 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 and is given by
This formula could also be used as follows:
Similarly, given points and 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 and applying the
Pythagorean theorem.
In the study of complicated geometries, we call this 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 . Other distances, based on other norms, are sometimes used instead.
For a point and a point , the
Minkowski distance of order p 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 .
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 = 0, and d = 0 if and only if x = y.
- It is symmetric: d = d.
- It satisfies the triangle inequality: d = d + d. .
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|. 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 = 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 definition 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 , 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
References
See also
External links
