Dilation (mathematics)
Encyclopedia
In mathematics
, a dilation is a function
from a metric space
into itself that satisfies the identity
for all points
where 
is the distance from 
to 
and 
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a dilation is a function
from a metric spaceMetric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
into itself that satisfies the identity
for all points
where is the distance from to and is some positive real numberReal number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
.
In Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, such a dilation is a similarity
Similarity (geometry)
Two geometrical objects are called similar if they both have the same shape. More precisely, either one is congruent to the result of a uniform scaling of the other...
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 (geometry)
In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...
has a unique fixed point that is called the center of dilation. Some congruences have fixed points and others do not.
See also
- homothety
- Dilation (operator theory)