Editing Euclidean distance (section)

== 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>