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 $$x, y \in M$$, where $$d(x, y)$$ is the distance from $$x$$ to $$y$$ and $$r$$ 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.