From Wikipedia, the free encyclopedia
In
mathematics , a dilation is a
function
f
{\displaystyle f}
from a
metric space
M
{\displaystyle M}
into itself that satisfies the identity
d
(
f
(
x
)
,
f
(
y
)
)
=
r
d
(
x
,
y
)
{\displaystyle d(f(x),f(y))=rd(x,y)}
for all points
x
,
y
∈
M
{\displaystyle x,y\in M}
, where
d
(
x
,
y
)
{\displaystyle d(x,y)}
is the distance from
x
{\displaystyle x}
to
y
{\displaystyle y}
and
r
{\displaystyle r}
is some positive
real number .
[1]
In
Euclidean space , such a dilation is a
similarity of the space.
[2] 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
[3] that is called the center of dilation .
[4] Some congruences have fixed points and others do not.
[5]
See also
References
^ Montgomery, Richard (2002),
A tour of subriemannian geometries, their geodesics and applications , Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, Providence, RI, p. 122,
ISBN
0-8218-1391-9 ,
MR
1867362 .
^ King, James R. (1997), "An eye for similarity transformations", in King, James R.;
Schattschneider, Doris (eds.),
Geometry Turned On: Dynamic Software in Learning, Teaching, and Research , Mathematical Association of America Notes, vol. 41, Cambridge University Press, pp.
109–120 ,
ISBN
9780883850992 . See in particular
p. 110 .
^ Audin, Michele (2003),
Geometry , Universitext, Springer, Proposition 3.5, pp. 80–81,
ISBN
9783540434986 .
^ Gorini, Catherine A. (2009),
The Facts on File Geometry Handbook , Infobase Publishing, p. 49,
ISBN
9781438109572 .
^ Carstensen, Celine; Fine, Benjamin; Rosenberger, Gerhard (2011),
Abstract Algebra: Applications to Galois Theory, Algebraic Geometry and Cryptography , Walter de Gruyter, p. 140,
ISBN
9783110250091 .