Distance

From Free net encyclopedia

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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.

Contents 1 Formal treatment 1.1 Mathematical analysis 1.1.1 General case 1.1.2 Distances between sets 1.1.3 Distance in Euclidean space 1.2 Geometry 1.3 Physics 1.4 Other uses 2 Informal treatment 3 See also 4 External links

Formal treatment

Mathematical analysis

General case

In mathematical analysis, distance is a function d: M x M -> R defined on a given set M which meets the following criteria:

• One can find the distance between any two points.
• That distance is a distinct real number.
• It is positive definite. d(x,y) ≥ 0, and d(x,y) = 0 if and only if x = y. (Distance is always positive, and it is zero precisely when measuring the distance from a point to itself).
• It is symmetric. d(x,y) = d(y,x). (The distance from x to y is the same as that from y to x).
• It satisfies the triangle inequality, d(x,z) ≤ d(x,y) + d(y,z). (The shortest distance between two points is on a straight line).

Such a function is known as a metric. Together with the set, it makes up a metric space. For example, one might define the distance between two real numbers x and y as d(x,y) = |x - y|.

Distances between sets

One can also find the distance between sets by finding the infimum of the distances between any two of their respective points. 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. This is called the Hausdorff metric.

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	x_i - y_i \right|</math>
2-norm distance	x_i - y_i \right|^2 \right)^{1/2}</math>
p-norm distance	x_i - y_i \right|^p \right)^{1/p}</math>
infinity norm distance	x_i - y_i \right|^p \right)^{1/p}</math>
	x_1 - y_1|, |x_2 - y_2|, \ldots, |x_n - y_n| \right).</math>

p need not be an integer, but it cannot be less than 1, because then 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 and queens 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.

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

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

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

<math>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}.</math>

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.

Physics

Image:Distancedisplacement.png

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.

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 uses

• Mahalanobis distance is used in statistics.
• Hamming distance is used in coding theory.
• Levenshtein distance

Informal treatment

In day-to-day discussion, distance often refers to the length of a straight line between objects. For 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, these informal treatments do not meet the criterion for a metric. For example, if one measures distance by car and there are one-way streets, then that distance probably won't be symmetric. Measuring distance by time might also not be symmetric, as a road may be more crowded in one direction than in the other, for instance, and a ship can go faster downstream than up.

Even in a given direction, though, time to go from A to B may depend on day and time of departure:

• this determines how crowded the roads are, downtown might be an hour from home in good traffic and five hours in bad.
• in the case of public transport, travel time depends on the timetable.

In both cases there is also an element of chance, as e.g. accidents cause delay.

If the time to go from A to B is defined as the shortest time over all times of departure then the triangle equality need not hold, especially in public transport: it may be a 5 minutes rides from A to B, and ditto from B to C, but a much longer ride from A to C due to waiting time in B.

If the time to go from A to B is defined as the longest time, including waiting, over all times of starting at A, then the triangle equality does hold.

See also

• Taxicab geometry
• astronomical units of length
• cosmic distance ladder
• comoving distance
• distance geometry
• distance (graph theory)
• distance-based road exit numbers
• Distance Measuring Equipment (DME)
• great-circle distance
• length
• milestone
• Metric (mathematics)
• Metric space
• orders of magnitude (length)
• distance matrix

External links

• Length Unit Converter - convert between various units of length, such as meter, mile, yard, and so on

• Interactive Length Conversion Table - convert selected unit to all other units of lengthar:مسافة

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