Editing Euclidean distance (section)

== Generalizations ==
In more advanced areas of mathematics, when viewing Euclidean space as a [[vector space]], its distance is associated with a [[Norm (mathematics)|norm]] called the [[Norm (mathematics)#Euclidean norm|Euclidean norm]], defined as the distance of each vector from the [[Origin (mathematics)|origin]]. One of the important properties of this norm, relative to other norms, is that it remains unchanged under arbitrary rotations of space around the origin.<ref>{{citation|title=Relativistic Celestial Mechanics of the Solar System|first1=Sergei|last1=Kopeikin|first2=Michael|last2=Efroimsky|first3=George|last3=Kaplan|publisher=John Wiley & Sons|year=2011|isbn=978-3-527-63457-6|page=106|url=https://books.google.com/books?id=uN5_DQWSR14C&pg=PA106}}</ref> By [[Dvoretzky's theorem]], every finite-dimensional [[normed vector space]] has a high-dimensional subspace on which the norm is approximately Euclidean; the Euclidean norm is the
only norm with this property.<ref>{{citation|last=Matoušek|first=Jiří|author-link=Jiří Matoušek (mathematician)|isbn=978-0-387-95373-1|page=349|publisher=Springer|series=[[Graduate Texts in Mathematics]]|title=Lectures on Discrete Geometry|url=https://books.google.com/books?id=K0fhBwAAQBAJ&pg=PA349|year=2002}}</ref> It can be extended to infinite-dimensional vector spaces as the [[Lp space|{{math|''L''<sup>2</sup>}} norm]] or {{math|''L''<sup>2</sup>}} distance.<ref>{{citation|title=Linear and Nonlinear Functional Analysis with Applications|first=Philippe G.|last=Ciarlet|publisher=Society for Industrial and Applied Mathematics|year=2013|isbn=978-1-61197-258-0|page=173|url=https://books.google.com/books?id=AUlWAQAAQBAJ&pg=PA173}}</ref> The Euclidean distance gives Euclidean space the structure of a [[topological space]], the [[Euclidean topology]], with the [[open ball]]s (subsets of points at less than a given distance from a given point) as its [[Neighbourhood (mathematics)|neighborhoods]].<ref>{{citation|title=General Topology: An Introduction|publisher=De Gruyter|first=Tom|last=Richmond|year=2020|isbn=978-3-11-068657-9|page=32|url=https://books.google.com/books?id=jPgdEAAAQBAJ&pg=PA32}}</ref>

[[File:Minkowski_distance_examples.svg|thumb|Comparison of Chebyshev, Euclidean and taxicab distances for the hypotenuse of a 3-4-5 triangle on a chessboard]]
Other common distances in [[real coordinate space]]s and [[function space]]s:<ref>{{citation|last=Klamroth|first=Kathrin|author-link=Kathrin Klamroth|contribution=Section 1.1: Norms and Metrics|doi=10.1007/0-387-22707-5_1|pages=4–6|publisher=Springer|series=Springer Series in Operations Research|title=Single-Facility Location Problems with Barriers|year=2002|isbn=978-1-4419-3027-9 }}</ref>
*[[Chebyshev distance]] ({{math|''L''<sup>∞</sup>}} distance), which measures distance as the maximum of the distances in each coordinate.
*[[Taxicab distance]] ({{math|''L''<sup>1</sup>}} distance), also called Manhattan distance, which measures distance as the sum of the distances in each coordinate.
*[[Minkowski distance]] ({{math|''L''<sup>''p''</sup>}} distance), a generalization that unifies Euclidean distance, taxicab distance, and Chebyshev distance.

For points on surfaces in three dimensions, the Euclidean distance should be distinguished from the [[geodesic]] distance, the length of a shortest curve that belongs to the surface. In particular, for measuring great-circle distances on the Earth or other spherical or near-spherical surfaces, distances that have been used include the [[haversine distance]] giving great-circle distances between two points on a sphere from their longitudes and latitudes, and [[Vincenty's formulae]] also known as "Vincent distance" for distance on a spheroid.<ref>{{citation|title=Computing in Geographic Information Systems|first=Narayan|last=Panigrahi|publisher=CRC Press|year=2014|isbn=978-1-4822-2314-9|contribution=12.2.4 Haversine Formula and 12.2.5 Vincenty's Formula|pages=212–214|url=https://books.google.com/books?id=kjj6AwAAQBAJ&pg=PA212}}</ref>