# Distance

The distance between two points is the length of a straight line segment between them. In the case of two locations on Earth, usually the distance along the surface is meant: either "as the crow flies" (along a great circle) or by road, railroad, etc. Distance is sometimes expressed in terms of the time to cover it, for example walking or by car. Sometimes a distance thus indicated is ambiguous because the means of transport is neither mentioned nor obvious.

Distance as mentioned above is sometimes not symmetric, hence not a metric (see below): this applies to distance by car in the case of one-way streets, and also in the case the distance is expressed in terms of the time to cover it (a road may be more crowded in one direction than in the other, for a ship upstream and downstream makes a difference).

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.

In the study of complicated geometries, we call the most common type of distance Euclidean distance, as we define it from the Pythagorean theorem.

## Distance covered

File:Distancedisplacement.png
Distance along a path compared with displacement

The distance covered by a vehicle (often recorded by a 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.

## Formal definition

A distance between two points P and Q in a metric space is d(P,Q), where d is the distance function that defines the given metric space.

We can also define the distance between two sets A and B in a metric space as being the minimum (or infimum) of distances between any two points P in A and Q in B.

Alternatively, the distance between sets may indicate "how different they are", by taking the supremum over one set of the distance from a point in that set to the other set, and conversely, and taking the larger of the two values (Hausdorff distance).

### The distance formula

The (Euclidean) distance, d, between two points expressed in Cartesian coordinates equals the square root of the sum of the squares of the changes of each coordinate. Thus, in a two-dimensional space

$\displaystyle d = \sqrt{(\Delta x)^2 + (\Delta y)^2},$

and in a three-dimensional space:

$\displaystyle d = \sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2}.$

Here, "Δ" (delta) refers to the change in a variable. Thus, Δx is the change in x, pronounced as such, or as "delta-x". In mathematical terms, $\displaystyle \Delta x = \left|x_1 - x_0\right|$ , and so $\displaystyle (\Delta x)^2 = (x_1 - x_2)^2$ .

This distance formula can be seen as a specialized form of the Pythagorean theorem; it can also be expanded into the arc-length formula.

### Generalized distance in arbitrary dimensions: Norms

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

p need not be an integer, but it cannot be less than 1, because then the triangle inequality does not hold. There is no such thing as a negative distance.

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, queens, and bishops must travel between two squares on a chessboard.

Also, if you measure the strength of each of the n links in a chain (where larger numbers mean weaker links), then because a chain is only as strong as its weakest link, the strength of the chain will be the infinity-norm distance from the list of measurements to the origin.

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.