Editing Euclidean distance (section)

{{Short description|Length of a line segment}}
{{Good article}}
{{Use American English|date = February 2019}}
{{Use mdy dates|date = February 2019}}
[[File:Euclidean distance 2d.svg|thumb|upright=1.35|Using the Pythagorean theorem to compute two-dimensional Euclidean distance]]

In [[mathematics]], the '''Euclidean distance''' between two [[Point (geometry)|points]] in [[Euclidean space]] is the [[length]] of the [[line segment]] between them. It can be calculated from the [[Cartesian coordinate]]s of the points using the [[Pythagorean theorem]], and therefore is occasionally called the '''Pythagorean distance'''.

These names come from the ancient [[Greek mathematics|Greek mathematicians]] [[Euclid]] and [[Pythagoras]]. In the Greek [[deductive]] [[geometry]] exemplified by Euclid's [[Euclid's Elements|''Elements'']], distances were not represented as numbers but line segments of the same length, which were considered "equal". The notion of distance is inherent in the [[compass (drawing tool)|compass]] tool used to draw a [[circle]], whose points all have the same distance from a common [[center (geometry)|center point]]. The connection from the Pythagorean theorem to distance calculation was not made until the 18th century.

The distance between two objects that are not points is usually defined to be the smallest distance among pairs of points from the two objects. Formulas are known for computing distances between different types of objects, such as the [[distance from a point to a line]]. In advanced mathematics, the concept of distance has been generalized to abstract [[metric space]]s, and other distances than Euclidean have been studied. In some applications in [[statistics]] and [[Mathematical optimization|optimization]], the square of the Euclidean distance is used instead of the distance itself.