Dilation (metric space)

In mathematics, a dilation is a function $f$ from a metric space $M$ into itself that satisfies the identity

:$d(f(x),f(y))=rd(x,y)$

for all points $x,\; y\; \backslash in\; M$, where $d(x,\; y)$ is the distance from $x$ to $y$ and $r$ is some positive real number.{{citation

| last = Montgomery | first = Richard

| isbn = 0-8218-1391-9

| mr = 1867362

| page = 122

| publisher = American Mathematical Society, Providence, RI

| series = Mathematical Surveys and Monographs

| title = A tour of subriemannian geometries, their geodesics and applications

| url = https://books.google.com/books?id=DYAt3gVB7Q4C&pg=PA122

| volume = 91

| year = 2002}}.

In Euclidean space, such a dilation is a similarity of the space.{{citation

| last = King

| first = James R.

| editor1-last = King

| editor1-first = James R.

| editor2-last = Schattschneider

| editor2-first = Doris

| editor2-link = Doris Schattschneider

| contribution = An eye for similarity transformations

| isbn = 9780883850992

| pages = [https://archive.org/details/geometryturnedon0000unse/page/109 109–120]

| publisher = Cambridge University Press

| series = Mathematical Association of America Notes

| title = Geometry Turned On: Dynamic Software in Learning, Teaching, and Research

| volume = 41

| year = 1997

| url = https://archive.org/details/geometryturnedon0000unse/page/109

}}. See in particular [https://books.google.com/books?id=lR0SDnl2bPwC&pg=PA110 p. 110]. Dilations change the size but not the shape of an object or figure.

Every dilation of a Euclidean space that is not a congruence has a unique fixed point{{citation|title=Geometry|series=Universitext|first=Michele|last=Audin|publisher=Springer|year=2003|isbn=9783540434986|at=Proposition 3.5, pp. 80–81|url=https://books.google.com/books?id=U_cTJMCIzdUC&pg=PA80}}. that is called the center of dilation.{{citation|title=The Facts on File Geometry Handbook|first=Catherine A.|last=Gorini|publisher=Infobase Publishing|year=2009|isbn=9781438109572|page=49|url=https://books.google.com/books?id=PlYCcvgLJxYC&pg=PA49}}. Some congruences have fixed points and others do not.{{citation|title=Abstract Algebra: Applications to Galois Theory, Algebraic Geometry and Cryptography|first1=Celine|last1=Carstensen|first2=Benjamin|last2=Fine|first3=Gerhard|last3=Rosenberger|publisher=Walter de Gruyter|year=2011|isbn=9783110250091|page=140|url=https://books.google.com/books?id=X1SJ_ywbgy8C&pg=PA140}}.