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.

In Euclidean space, such a dilation is a similarity 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 congruence has a unique fixed point that is called the center of dilation. Some congruences have fixed points and others do not.

• Homothety
• Dilation (operator theory)