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{{Use American English|date = February 2019}}{{short description|Length of a line segment}}{{Use mdy dates|date = February 2019}}{{good article}}(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 points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is occasionally called the Pythagorean distance.These names come from the ancient Greek mathematicians Euclid and Pythagoras. In the Greek deductive geometry exemplified by Euclid’s 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 tool used to draw a circle, whose points all have the same distance from a common 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 spaces, and other distances than Euclidean have been studied. In some applications in statistics and 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 p and q are two points on the real line, then the distance between them is given by:{{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}}d(p,q) = |p-q|.A more complicated formula, giving the same value, but generalizing more readily to higher dimensions, is:d(p,q) = sqrt{(p-q)^2}.In this formula, squaring and then taking the square root leaves any positive number unchanged, but replaces any negative number by its absolute value.

Two dimensions

In the Euclidean plane, let point p have Cartesian coordinates (p_1,p_2) and let point q have coordinates (q_1,q_2). Then the distance between p and q is given by:{{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}}d(p,q) = sqrt{(p_1-q_1)^2 + (p_2-q_2)^2}.This can be seen by applying the Pythagorean theorem to a right triangle with horizontal and vertical sides, having the line segment from p to q 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.{{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}}It is also possible to compute the distance for points given by polar coordinates. If the polar coordinates of p are (r,theta) and the polar coordinates of q are (s,psi), then their distance is given by the law of cosines:d(p,q)=sqrt{r^2 + s^2 - 2rscos(theta-psi)}.When p and q are expressed as complex numbers 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:{{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}}d(p,q)=|p-q|.

Higher dimensions

(File:Euclidean distance 3d 2 cropped.png|thumb|upright=1.2|Deriving the n-dimensional Euclidean distance formula by repeatedly applying the Pythagorean theorem)In three dimensions, for points given by their Cartesian coordinates, the distance isd(p,q)=sqrt{(p_1-q_1)^2 + (p_2-q_2)^2 + (p_3-q_3)^2}.In general, for points given by Cartesian coordinates in n-dimensional Euclidean space, the distance is{{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}}d(p,q) = sqrt{(p_1- q_1)^2 + (p_2 - q_2)^2+cdots+(p_n - q_n)^2}.The Euclidean distance may also be expressed more compactly in terms of the Euclidean norm of the Euclidean vector difference:d(p,q) = | p - q |.

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.{{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}} Formulas for computing distances between different types of objects include:

• The distance from a point to a line, in the Euclidean plane{{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}}
• The distance from a point to a plane in three-dimensional Euclidean space
• The distance between two lines in three-dimensional Euclidean space{{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}}

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.{{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}}

Properties

The Euclidean distance is the prototypical example of the distance in a metric space,{{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}} and obeys all the defining properties of a metric space:{{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}}

• It is symmetric, meaning that for all points p and q, d(p,q)=d(q,p). 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.
• 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.
• It obeys the triangle inequality: for every three points p, q, and r, d(p,q)+d(q,r)ge d(p,r). Intuitively, traveling from p to r via q cannot be any shorter than traveling directly from p to r.

Another property, Ptolemy’s inequality, concerns the Euclidean distances among four points p, q, r, and s. It states thatd(p,q)cdot d(r,s)+d(q,r)cdot d(p,s)ge d(p,r)cdot d(q,s).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.{{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}} 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.{{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}}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.{{citation

Squared Euclidean distance

{{multiple image|image1=3d-function-5.svg

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 (d_1^2 > d_2^2 if and only if d_1 > d_2). The value resulting from this omission is the square of the Euclidean distance, and is called the squared Euclidean distance. 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.{{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}} As an equation, the squared distance can be expressed as a sum of squares: d^2(p,q) = (p_1 - q_1)^2 + (p_2 - q_2)^2+cdots+(p_n - q_n)^2.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,{{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}} and as the simplest form of divergence to compare probability distributions.{{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 }} 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.{{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 }} In cluster analysis, squared distances can be used to strengthen the effect of longer distances.{{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}} Squared Euclidean distance does not form a metric space, as it does not satisfy the triangle inequality.{{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}} 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.{{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}}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.{{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}} Generalizations In more advanced areas of mathematics, when viewing Euclidean space as a vector space, its distance is associated with a norm called the Euclidean norm, defined as the distance of each vector from the 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.{{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}} 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 theonly norm with this property.{{citation|last=MatouÅ¡ek|first=Jiří|author-link=Jiří MatouÅ¡ek (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}} It can be extended to infinite-dimensional vector spaces as the {{math|L2}} norm or {{math|L2}} distance.{{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}} The Euclidean distance gives Euclidean space the structure of a topological space, the Euclidean topology, with the open balls (subsets of points at less than a given distance from a given point) as its neighborhoods.{{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}}(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 spaces and function spaces:{{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}} Chebyshev distance ({{math|L∞}} distance), which measures distance as the maximum of the distances in each coordinate. Taxicab distance ({{math|L1}} distance), also called Manhattan distance, which measures distance as the sum of the distances in each coordinate. Minkowski distance ({{math|Lp}} 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.{{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}} History Euclidean distance is the distance in Euclidean space. Both concepts are named after ancient Greek mathematician Euclid, whose Elements became a standard textbook in geometry for many centuries.{{citation|title=Visualization for Information Retrieval|first=Jin|last=Zhang|publisher=Springer|year=2007|isbn=978-3-540-75148-9}} 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),{{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/20210517124414akira.ruc.dk/~jensh/Publications/2018%7Bj%7D_Mesopotamian%20Mathematics_S.pdf|url-status=dead}} and have been hypothesized to develop in children earlier than the related concepts of speed and time.{{citation|last1=Acredolo|first1=Curt|last2=Schmid|first2=Jeannine|doi=10.1037/0012-1649.17.4.490|issue=4|journal=Developmental Psychology|pages=490–493|title=The understanding of relative speeds, distances, and durations of movement|volume=17|year=1981}} 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 of line segments, through the comparison of lengths of line segments, and through the concept of proportionality.{{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}}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.{{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}} Because of this formula, Euclidean distance is also sometimes called Pythagorean distance.{{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|s2cid=144797925|doi-access=free}} 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.{{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|volume=6|year=1982|doi-access=free}} 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.{{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}} References {{reflist}}{{Lp spaces}}{{Machine learning evaluation metrics|state=collapsed}}{{Authority control|state=collapsed}}