Dilation (metric space)

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a dilation is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> 
 
 
 
 f
 
 
 {\displaystyle f}
 
 from a <a href="/facts/Metric_space/gnBBRXtc">metric space</a> 
 
 
 
 M
 
 
 {\displaystyle M}
 
 into itself that satisfies the identity

d
        (
        f
        (
        x
        )
        ,
        f
        (
        y
        )
        )
        =
        r
        d
        (
        x
        ,
        y
        )
      
    
    {\displaystyle d(f(x),f(y))=rd(x,y)}

for all points 
 
 
 
 x
 ,
 y
 ∈
 M
 
 
 {\displaystyle x,y\in M}
 
, where 
 
 
 
 d
 (
 x
 ,
 y
 )
 
 
 {\displaystyle d(x,y)}
 
 is the distance from 
 
 
 
 x
 
 
 {\displaystyle x}
 
 to 
 
 
 
 y
 
 
 {\displaystyle y}
 
 and 
 
 
 
 r
 
 
 {\displaystyle r}
 
 is some positive <a href="/facts/Real_number/R02gw5Pb">real number</a>.
In <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>, such a dilation is a <a href="/facts/Similarity_(geometry)/Sbd6hEAN">similarity</a> of the space. Dilations change the size but not the shape of an object or figure.
Every dilation of a Euclidean space that is not a <a href="/facts/Congruence_(geometry)/2tKJ6AhB">congruence</a> has a unique <a href="/facts/Fixed_point_(mathematics)/vVVddTKc">fixed point</a> that is called the center of dilation. Some congruences have fixed points and others do not.

Dilation (metric space) open-in-new

Dilation (metric space)