View source for Euclidean distance

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{{short description|Length of a line segment}}
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[[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.

== Distance formulas ==
=== One dimension ===
The distance between any two points on the [[real line]] is the [[absolute value]] of the numerical difference of their coordinates, their [[absolute difference]]. Thus if <math>p</math> and <math>q</math> are two points on the real line, then the distance between them is given by:<ref name=smith>{{citation|title=Precalculus: A Functional Approach to Graphing and Problem Solving|first=Karl|last=Smith|publisher=Jones & Bartlett Publishers|year=2013|isbn=978-0-7637-5177-7|page=8|url=https://books.google.com/books?id=ZUJbVQN37bIC&pg=PA8}}</ref>

<math display=block>d(p,q) = |p-q|.</math>

A more complicated formula, giving the same value, but generalizing more readily to higher dimensions, is:<ref name=smith />

<math display=block>d(p,q) = \sqrt{(p-q)^2}.</math>

In this formula, [[square (algebra)|squaring]] and then taking the [[square root]] leaves any positive number unchanged, but replaces any negative number by its absolute value.<ref name=smith />

=== Two dimensions ===
In the [[Euclidean plane]], let point <math>p</math> have [[Cartesian coordinates]] <math>(p_1,p_2)</math> and let point <math>q</math> have coordinates <math>(q_1,q_2)</math>. Then the distance between <math>p</math> and <math>q</math> is given by:<ref name=cohen>{{citation|title=Precalculus: A Problems-Oriented Approach|first=David|last=Cohen|edition=6th|publisher=Cengage Learning|year=2004|isbn=978-0-534-40212-9|page=698|url=https://books.google.com/books?id=_6ukev29gmgC&pg=PA698}}</ref>

<math display=block>d(p,q) = \sqrt{(p_1-q_1)^2 + (p_2-q_2)^2}.</math>

This can be seen by applying the [[Pythagorean theorem]] to a [[right triangle]] with horizontal and vertical sides, having the line segment from <math>p</math> to <math>q</math> as its [[hypotenuse]]. The two squared formulas inside the square root give the areas of squares on the horizontal and vertical sides, and the outer square root converts the area of the square on the hypotenuse into the length of the hypotenuse.<ref>{{citation|title=College Trigonometry|first1=Richard N.|last1=Aufmann|first2=Vernon C.|last2=Barker|first3=Richard D.|last3=Nation|edition=6th|publisher=Cengage Learning|year=2007|isbn=978-1-111-80864-8|page=17|url=https://books.google.com/books?id=kZ8HAAAAQBAJ&pg=PA17}}</ref> In terms of the [[Pythagorean addition]] operation <math>\oplus</math>, available in many [[software library|software libraries]] as <code>hypot</code>, the same formula can be expressed as:<ref>{{citation|title=Java Script Notes for Professionals|first=Rohit|last=Manglik|publisher=EduGorilla|year=2024|isbn=9789367840320|contribution=Section 14.22: Math.hypot|page=144|contribution-url=https://books.google.com/books?id=jwU6EQAAQBAJ&pg=PA144}}</ref>

<math display=block>d(p,q) = (p_1-q_1) \oplus (p_2-q_2) = \mathsf{hypot}(p_1-q_1,p_2-q_2).</math>

It is also possible to compute the distance for points given by [[Polar coordinate system|polar coordinates]]. If the polar coordinates of <math>p</math> are <math>(r,\theta)</math> and the polar coordinates of <math>q</math> are <math>(s,\psi)</math>, then their distance is<ref name=cohen /> given by the [[law of cosines]]:

<math display=block>d(p,q)=\sqrt{r^2 + s^2 - 2rs\cos(\theta-\psi)}.</math>

When <math>p</math> and <math>q</math> are expressed as [[complex number]]s in the [[complex plane]], the same formula for one-dimensional points expressed as real numbers can be used, although here the absolute value sign indicates the [[complex norm]]:<ref>{{citation|title=Complex Numbers from A to ... Z|first1=Titu|last1=Andreescu|first2=Dorin|last2=Andrica|publisher=Birkhäuser|year=2014|edition=2nd|isbn=978-0-8176-8415-0|contribution=3.1.1 The Distance Between Two Points|pages=57–58}}</ref>

<math display=block>d(p,q)=|p-q|.</math>

=== Higher dimensions ===
[[File:Euclidean distance 3d 2 cropped.png|thumb|upright=1.2|Deriving the <math>n</math>-dimensional Euclidean distance formula by repeatedly applying the Pythagorean theorem]]

In three dimensions, for points given by their Cartesian coordinates, the distance is

<math display=block>d(p,q)=\sqrt{(p_1-q_1)^2 + (p_2-q_2)^2 + (p_3-q_3)^2}.</math>

In general, for points given by Cartesian coordinates in <math>n</math>-dimensional Euclidean space, the distance is<ref>{{citation|title=Geometry: The Language of Space and Form|series=Facts on File math library|first=John|last=Tabak|publisher=Infobase Publishing|year=2014|isbn=978-0-8160-6876-0|page=150|url=https://books.google.com/books?id=r0HuPiexnYwC&pg=PA150}}</ref>

<math display=block>d(p,q) = \sqrt{(p_1- q_1)^2 + (p_2 - q_2)^2+\cdots+(p_n - q_n)^2}.</math>

The Euclidean distance may also be expressed more compactly in terms of the [[Euclidean norm]] of the [[Euclidean vector]] difference:

<math display=block>d(p,q) = \| p - q \|.</math>

=== Objects other than points ===
For pairs of objects that are not both points, the distance can most simply be defined as the smallest distance between any two points from the two objects, although more complicated generalizations from points to sets such as [[Hausdorff distance]] are also commonly used.<ref>{{citation|title=Metric Spaces|series=Springer Undergraduate Mathematics Series|first=Mícheál|last=Ó Searcóid|publisher=Springer|year=2006|isbn=978-1-84628-627-8|contribution=2.7 Distances from Sets to Sets|pages=29–30|url=https://books.google.com/books?id=aP37I4QWFRcC&pg=PA29}}</ref> Formulas for computing distances between different types of objects include:
*The [[distance from a point to a line]], in the Euclidean plane<ref name=baljer>{{citation|last1=Ballantine|first1=J. P.|last2=Jerbert|first2=A. R.|date=April 1952|department=Classroom notes|doi=10.2307/2306514|issue=4|journal=[[American Mathematical Monthly]]|jstor=2306514|pages=242–243|title=Distance from a line, or plane, to a point|volume=59}}</ref>
*The [[distance from a point to a plane]] in three-dimensional Euclidean space<ref name=baljer />
*The [[Skew lines#Distance|distance between two lines]] in three-dimensional Euclidean space<ref>{{citation|last=Bell|first=Robert J. T.|author-link=Robert J. T. Bell|edition=2nd|contribution=49. The shortest distance between two lines|contribution-url=https://archive.org/details/elementarytreati00bell/page/56/mode/2up|pages=57–61|publisher=Macmillan|title=An Elementary Treatise on Coordinate Geometry of Three Dimensions|year=1914}}</ref>
The distance from a point to a [[curve]] can be used to define its [[parallel curve]], another curve all of whose points have the same distance to the given curve.<ref>{{citation | last = Maekawa | first = Takashi | date = March 1999 | doi = 10.1016/s0010-4485(99)00013-5 | issue = 3 | journal = Computer-Aided Design | pages = 165–173 | title = An overview of offset curves and surfaces | volume = 31}}</ref>

== Properties ==
The Euclidean distance is the prototypical example of the distance in a [[metric space]],<ref>{{citation|title=Easy as π?: An Introduction to Higher Mathematics|first=Oleg A.|last=Ivanov|publisher=Springer|year=2013|isbn=978-1-4612-0553-1|page=140|url=https://books.google.com/books?id=reALBwAAQBAJ&pg=PA140}}</ref> and obeys all the defining properties of a metric space:<ref name=strichartz>{{citation|title=The Way of Analysis|first=Robert S.|last=Strichartz|publisher=Jones & Bartlett Learning|year=2000|isbn=978-0-7637-1497-0|page=357|url=https://books.google.com/books?id=Yix09oVvI1IC&pg=PA357}}</ref>
*It is ''symmetric'', meaning that for all points <math>p</math> and <math>q</math>, <math>d(p,q)=d(q,p)</math>. That is (unlike road distance with one-way streets) the distance between two points does not depend on which of the two points is the start and which is the destination.<ref name=strichartz />
*It is ''positive'', meaning that the distance between every two distinct points is a [[positive number]], while the distance from any point to itself is zero.<ref name=strichartz />
*It obeys the [[triangle inequality]]: for every three points <math>p</math>, <math>q</math>, and <math>r</math>, <math>d(p,q)+d(q,r)\ge d(p,r)</math>. Intuitively, traveling from <math>p</math> to <math>r</math> via <math>q</math> cannot be any shorter than traveling directly from <math>p</math> to <math>r</math>.<ref name=strichartz />

Another property, [[Ptolemy's inequality]], concerns the Euclidean distances among four points <math>p</math>, <math>q</math>, <math>r</math>, and <math>s</math>. It states that

<math display=block>d(p,q)\cdot d(r,s)+d(q,r)\cdot d(p,s)\ge d(p,r)\cdot d(q,s).</math>

For points in the plane, this can be rephrased as stating that for every [[quadrilateral]], the products of opposite sides of the quadrilateral sum to at least as large a number as the product of its diagonals. However, Ptolemy's inequality applies more generally to points in Euclidean spaces of any dimension, no matter how they are arranged.<ref>{{citation|title=Rays, Waves, and Scattering: Topics in Classical Mathematical Physics|series=Princeton Series in Applied Mathematics|first=John A.|last=Adam|publisher=Princeton University Press|year=2017|isbn=978-1-4008-8540-4|pages=26–27|chapter-url=https://books.google.com/books?id=DnygDgAAQBAJ&pg=PA26|chapter=Chapter 2. Introduction to the "Physics" of Rays|doi=10.1515/9781400885404-004}}</ref> For points in metric spaces that are not Euclidean spaces, this inequality may not be true. Euclidean [[distance geometry]] studies properties of Euclidean distance such as Ptolemy's inequality, and their application in testing whether given sets of distances come from points in a Euclidean space.<ref>{{citation|title=Euclidean Distance Geometry: An Introduction|series=Springer Undergraduate Texts in Mathematics and Technology|first1=Leo|last1=Liberti|first2=Carlile|last2=Lavor|publisher=Springer|year=2017|isbn=978-3-319-60792-4|page=xi|url=https://books.google.com/books?id=jOQ2DwAAQBAJ&pg=PP10}}</ref>

According to the [[Beckman–Quarles theorem]], any transformation of the Euclidean plane or of a higher-dimensional Euclidean space that preserves unit distances must be an [[isometry]], preserving all distances.<ref>{{citation
 | last1 = Beckman | first1 = F. S.
 | last2 = Quarles | first2 = D. A. Jr.
 | doi = 10.2307/2032415 | doi-access = free
 | journal = [[Proceedings of the American Mathematical Society]]
 | mr = 0058193
 | pages = 810–815
 | title = On isometries of Euclidean spaces
 | volume = 4
 | year = 1953| issue = 5
 | jstor = 2032415
 }}</ref>

== Squared Euclidean distance ==
{{multiple image
|image1=3d-function-5.svg
|caption1=A [[cone]], the [[Graph of a function|graph]] of Euclidean distance from the origin in the plane
|image2=3d-function-2.svg
|caption2=A [[paraboloid]], the graph of squared Euclidean distance from the origin
}}

In many applications, and in particular when comparing distances, it may be more convenient to omit the final square root in the calculation of Euclidean distances, as the square root does not change the order (<math>d_1^2 > d_2^2</math> if and only if <math>d_1 > d_2</math>). The value resulting from this omission is the [[Square (algebra)|square]] of the Euclidean distance, and is called the '''squared Euclidean distance'''.<ref name=spencer /> For instance, the [[Euclidean minimum spanning tree]] can be determined using only the ordering between distances, and not their numeric values. Comparing squared distances produces the same result but avoids an unnecessary square-root calculation and sidesteps issues of numerical precision.<ref>{{citation
 | last = Yao | first = Andrew Chi Chih | author-link = Andrew Yao
 | doi = 10.1137/0211059
 | issue = 4
 | journal = [[SIAM Journal on Computing]]
 | mr = 677663
 | pages = 721–736
 | title = On constructing minimum spanning trees in {{mvar|k}}-dimensional spaces and related problems
 | volume = 11
 | year = 1982}}</ref> As an equation, the squared distance can be expressed as a [[sum of squares]]:

<math display=block>d^2(p,q) = (p_1 - q_1)^2 + (p_2 - q_2)^2+\cdots+(p_n - q_n)^2.</math>

Beyond its application to distance comparison, squared Euclidean distance is of central importance in [[statistics]], where it is used in the method of [[least squares]], a standard method of fitting statistical estimates to data by minimizing the average of the squared distances between observed and estimated values,<ref>{{citation|title=Basic Statistics in Multivariate Analysis|series=Pocket Guide to Social Work Research Methods|first1=Karen A.|last1=Randolph|author1-link=Karen Randolph|first2=Laura L.|last2=Myers|publisher=Oxford University Press|year=2013|isbn=978-0-19-976404-4|page=116|url=https://books.google.com/books?id=WgSnudjEsrMC&pg=PA116}}</ref> and as the simplest form of [[divergence (statistics)|divergence]] to compare [[probability distribution]]s.<ref>{{citation
 | last = Csiszár | first = I. | author-link = Imre Csiszár
 | doi = 10.1214/aop/1176996454
 | journal = [[Annals of Probability]]
 | jstor = 2959270
 | mr = 365798
 | pages = 146–158
 | title = {{mvar|I}}-divergence geometry of probability distributions and minimization problems
 | volume = 3
 | year = 1975| issue = 1 | doi-access = free
 }}</ref> The addition of squared distances to each other, as is done in least squares fitting, corresponds to an operation on (unsquared) distances called [[Pythagorean addition]].<ref>{{citation |author=Moler, Cleve and Donald Morrison |title=Replacing Square Roots by Pythagorean Sums |journal=IBM Journal of Research and Development |volume=27 |issue=6 |pages=577–581 |year=1983 |url=http://www.research.ibm.com/journal/rd/276/ibmrd2706P.pdf |doi=10.1147/rd.276.0577 | citeseerx = 10.1.1.90.5651 }}</ref> In [[cluster analysis]], squared distances can be used to strengthen the effect of longer distances.<ref name=spencer>{{citation|title=Essentials of Multivariate Data Analysis|first=Neil H.|last=Spencer|publisher=CRC Press|year=2013|isbn=978-1-4665-8479-2|contribution=5.4.5 Squared Euclidean Distances|page=95|contribution-url=https://books.google.com/books?id=EG3SBQAAQBAJ&pg=PA95}}</ref>

Squared Euclidean distance does not form a metric space, as it does not satisfy the triangle inequality.<ref>{{citation|last1=Mielke|first1=Paul W.|last2=Berry|first2=Kenneth J.|editor1-last=Brown|editor1-first=Timothy J.|editor2-last=Mielke|editor2-first=Paul W. Jr.|contribution=Euclidean distance based permutation methods in atmospheric science|doi=10.1007/978-1-4757-6581-6_2|pages=7–27|publisher=Springer|title=Statistical Mining and Data Visualization in Atmospheric Sciences|year=2000|isbn=978-1-4419-4974-5 }}</ref> However it is a smooth, strictly [[convex function]] of the two points, unlike the distance, which is non-smooth (near pairs of equal points) and convex but not strictly convex. The squared distance is thus preferred in [[optimization theory]], since it allows [[convex analysis]] to be used. Since squaring is a [[monotonic function]] of non-negative values, minimizing squared distance is equivalent to minimizing the Euclidean distance, so the optimization problem is equivalent in terms of either, but easier to solve using squared distance.<ref>{{citation|title=Maxima and Minima with Applications: Practical Optimization and Duality|volume=51|series=Wiley Series in Discrete Mathematics and Optimization|first=Wilfred|last=Kaplan|publisher=John Wiley & Sons|year=2011|isbn=978-1-118-03104-9|page=61|url=https://books.google.com/books?id=bAo6KNZcUP0C&pg=PA61}}</ref>

The collection of all squared distances between pairs of points from a finite set may be stored in a [[Euclidean distance matrix]], and is used in this form in distance geometry.<ref>{{citation|title=Euclidean Distance Matrices and Their Applications in Rigidity Theory|first=Abdo Y.|last=Alfakih|publisher=Springer|year=2018|isbn=978-3-319-97846-8|page=51|url=https://books.google.com/books?id=woJyDwAAQBAJ&pg=PA51}}</ref>

== 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}}</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>

== History ==
Euclidean distance is the distance in [[Euclidean space]]. Both concepts are named after ancient Greek mathematician [[Euclid]], whose [[Euclid's Elements|''Elements'']] became a standard textbook in geometry for many centuries.<ref>{{citation|title=Visualization for Information Retrieval|first=Jin|last=Zhang|publisher=Springer|year=2007|isbn=978-3-540-75148-9}}</ref> Concepts of [[length]] and [[distance]] are widespread across cultures, can be dated to the earliest surviving "protoliterate" bureaucratic documents from [[Sumer]] in the fourth millennium BC (far before Euclid),<ref>{{citation|last=Høyrup|first=Jens|author-link=Jens Høyrup|editor1-last=Jones|editor1-first=Alexander|editor2-last=Taub|editor2-first=Liba|editor2-link=Liba Taub|contribution=Mesopotamian mathematics|contribution-url=https://akira.ruc.dk/~jensh/Publications/2018%7Bj%7D_Mesopotamian%20Mathematics_S.pdf|pages=58–72|publisher=Cambridge University Press|title=The Cambridge History of Science, Volume 1: Ancient Science|year=2018|access-date=October 21, 2020|archive-date=May 17, 2021|archive-url=https://web.archive.org/web/20210517124414/http://akira.ruc.dk/~jensh/Publications/2018%7Bj%7D_Mesopotamian%20Mathematics_S.pdf|url-status=dead}}</ref> and have been hypothesized to develop in children earlier than the related concepts of speed and time.<ref>{{citation|last1=Acredolo|first1=Curt|last2=Schmid|first2=Jeannine|doi=10.1037/0012-1649.17.4.490|issue=4|journal=[[Developmental Psychology (journal)|Developmental Psychology]]|pages=490–493|title=The understanding of relative speeds, distances, and durations of movement|volume=17|year=1981}}</ref> But the notion of a distance, as a number defined from two points, does not actually appear in Euclid's ''Elements''. Instead, Euclid approaches this concept implicitly, through the [[Congruence (geometry)|congruence]] of line segments, through the comparison of lengths of line segments, and through the concept of [[Proportionality (mathematics)|proportionality]].<ref>{{citation|last=Henderson|first=David W.|author-link=David W. Henderson|journal=[[Bulletin of the American Mathematical Society]]|pages=563–571|title=Review of ''Geometry: Euclid and Beyond'' by Robin Hartshorne|url=https://www.ams.org/journals/bull/2002-39-04/S0273-0979-02-00949-7|volume=39|year=2002|doi=10.1090/S0273-0979-02-00949-7|doi-access=free}}</ref>

The [[Pythagorean theorem]] is also ancient, but it could only take its central role in the measurement of distances after the invention of [[Cartesian coordinates]] by [[René Descartes]] in 1637. The distance formula itself was first published in 1731 by [[Alexis Clairaut]].<ref>{{citation|last=Maor|first=Eli|author-link=Eli Maor|isbn=978-0-691-19688-6|pages=133–134|publisher=Princeton University Press|title=The Pythagorean Theorem: A 4,000-Year History|url=https://books.google.com/books?id=XuWZDwAAQBAJ&pg=PA133|year=2019}}</ref> Because of this formula, Euclidean distance is also sometimes called Pythagorean distance.<ref>{{citation|last1=Rankin|first1=William C.|last2=Markley|first2=Robert P.|last3=Evans|first3=Selby H.|date=March 1970|doi=10.3758/bf03210143|issue=2|journal=[[Perception & Psychophysics]]|pages=103–107|title=Pythagorean distance and the judged similarity of schematic stimuli|volume=7|doi-access=free }}</ref> Although accurate measurements of long distances on the Earth's surface, which are not Euclidean, had again been studied in many cultures since ancient times (see [[history of geodesy]]), the idea that Euclidean distance might not be the only way of measuring distances between points in mathematical spaces came even later, with the 19th-century formulation of [[non-Euclidean geometry]].<ref>{{citation|last=Milnor|first=John|author-link=John Milnor|doi=10.1090/S0273-0979-1982-14958-8|issue=1|journal=[[Bulletin of the American Mathematical Society]]|mr=634431|pages=9–24|title=Hyperbolic geometry: the first 150 years |url=https://www.ams.org/journals/bull/1982-06-01/S0273-0979-1982-14958-8/S0273-0979-1982-14958-8.pdf |volume=6|year=1982|doi-access=free}}</ref> The definition of the Euclidean norm and Euclidean distance for geometries of more than three dimensions also first appeared in the 19th century, in the work of [[Augustin-Louis Cauchy]].<ref>{{citation|title=Foundations of Hyperbolic Manifolds|volume=149|series=[[Graduate Texts in Mathematics]]|first=John G.|last=Ratcliffe|edition=3rd|publisher=Springer|year=2019|isbn=978-3-030-31597-9|page=32|url=https://books.google.com/books?id=yMO4DwAAQBAJ&pg=PA32}}</ref>

==References==

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[[Category:Distance]]
[[Category:Length]]
[[Category:Metric geometry]]
[[Category:Pythagorean theorem]]
[[Category:Euclid|distance]]