mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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

y

and

r

is some positive

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

. In

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

, 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..

References

{{DEFAULTSORT:Dilation (Metric Space) Metric geometry

See also

References