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Euclidean distance
 
 
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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the Euclidean distance or Euclidean metric is the "ordinary" distance
Distance

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 ....
 between two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem
Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a triangle#Types of triangles....
. By using this formula as distance, Euclidean space becomes a metric space
Metric space

In mathematics, a metric space is a Set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space....
 (even a Hilbert space
Hilbert space

The mathematics concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces....
). The associated norm
Norm (mathematics)

In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector....
 is called the Euclidean norm.

Older literature refers to this metric as Pythagorean metric.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the Euclidean distance or Euclidean metric is the "ordinary" distance
Distance

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 ....
 between two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem
Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a triangle#Types of triangles....
. By using this formula as distance, Euclidean space becomes a metric space
Metric space

In mathematics, a metric space is a Set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space....
 (even a Hilbert space
Hilbert space

The mathematics concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces....
). The associated norm
Norm (mathematics)

In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector....
 is called the Euclidean norm.

Older literature refers to this metric as Pythagorean metric. The technique has been rediscovered numerous times throughout history, as it is a logical extension of the Pythagorean theorem.

Definition


The Euclidean distance between points and , in Euclidean n-space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
, is defined as:

One-dimensional distance

For two 1D points, and , the distance is computed as:

The absolute value signs are used since distance is normally considered to be an unsigned scalar value.

In one dimension, there is a single homogeneous, translation-invariant metric
Metric (mathematics)

In mathematics, a metric or distance function is a function which defines a distance between elements of a Set . A set with a metric is called a metric space....
 (in other words, a distance that is induced by a norm
Norm (mathematics)

In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector....
), up to a scale factor of length, which is the Euclidean distance. In higher dimensions there are other possible norms.

Two-dimensional distance

For two 2D points, and , the distance is computed as:

Alternatively, expressed in polar coordinates, using and , the distance can be computed as:

Three-dimensional distance


For two 3D points, and , the distance is computed as

N-dimensional distance


For two N-D points, and , the distance is computed as

See also


  • Mahalanobis distance
    Mahalanobis distance

    In statistics, Mahalanobis distance is a distance measure introduced by P. C. Mahalanobis in 1936. It is based on correlations between variables by which different patterns can be identified and analyzed....
  • Manhattan distance
  • Metric
    Metric (mathematics)

    In mathematics, a metric or distance function is a function which defines a distance between elements of a Set . A set with a metric is called a metric space....
  • Pythagorean addition
    Pythagorean addition

    In mathematics, Pythagorean addition is the following binary operation:The name recalls the Pythagorean theorem, which states that the length of the hypotenuse of a right triangle is where a and b are the lengths of the other sides....