In
mathematics, a dilation is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
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 set ...
into itself that satisfies the identity
:
for all points
, where
is the distance from
to
and
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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
.
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 ...
, 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.
[.]
See also
*
Homothety
In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point ''S'' called its ''center'' and a nonzero number ''k'' called its ''ratio'', which sends point X to a point X' by th ...
*
Dilation (operator theory) In operator theory, a dilation of an operator ''T'' on a Hilbert space ''H'' is an operator on a larger Hilbert space ''K'', whose restriction to ''H'' composed with the orthogonal projection onto ''H'' is ''T''.
More formally, let ''T'' be a boun ...
References
{{DEFAULTSORT:Dilation (Metric Space)
Metric geometry