In mathematics, a dilation is a function

f

from a

metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...

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 measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

. In

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

, such a dilation is a

similarity Similarity may refer to: In mathematics and computing * Similarity (geometry), the property of sharing the same shape * Matrix similarity, a relation between matrices * Similarity measure, a function that quantifies the similarity of two objects * ...

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 Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...

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