Euclidean Distance Information

In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space (or even any inner product space) becomes a metric space. The associated norm is called the Euclidean norm. Older literature refers to the metric as Pythagorean metric.

Definition

The Euclidean distance between points p and q is the length of the line segment . In Cartesian coordinates, if p = (p₁, p₂,..., p_n) and q = (q₁, q₂,..., q_n) are two points in Euclidean n-space, then the distance from p to q is given by:

(1)

The Euclidean norm measures the distance of a point to the origin of Euclidean space:

where the last equation involves the dot product. This is the length of p, when regarded as a Euclidean vector from the origin. The distance itself is given by

(2)

Special cases

In one dimension, the distance between two points on the real line is the absolute value of their numerical difference. Thus if x and y are two points on the real line, then the distance between them is computed as

In one dimension, there is a single homogeneous, translation-invariant metric (in other words, a distance that is induced by a norm), up to a scale factor of length, which is the Euclidean distance. In higher dimensions there are other possible norms.

In the Euclidean plane, if p = (p₁, p₂) and q = (q₁, q₂) then the distance is given by

Alternatively, it follows from (2) that if the polar coordinates of the point p are (r₁, θ₁) and those of q are (r₂, θ₂), then the distance between the points is

In three-dimensional Euclidean space, the distance is

and so on.

See also

• Mahalanobis distance normalizes based on a covariance matrix to make the distance metric scale-invariant.
• Manhattan distance measures distance following only axis-aligned directions.
• Chebyshev distance measures distance assuming only the most significant dimension is relevant.
• Minkowski distance is a generalization that unifies Euclidean distance, Manhattan distance, and Chebyshev distance.
• Metric
• Pythagorean addition

References

• Elena Deza & Michel Marie Deza (2009) Encyclopedia of Distances, page 94, Springer.

Categories: Metric geometry | Length

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