Dilation (metric space)

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

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

for all points ${\displaystyle x,y\in M}$, where ${\displaystyle d(x,y)}$ is the distance from ${\displaystyle x}$ to ${\displaystyle y}$ and ${\displaystyle r}$ is some positive real number.[1]

In Euclidean space, such a dilation is a similarity of the space.[2] 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[3] that is called the center of dilation.[4] Some congruences have fixed points and others do not.[5]