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.
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.
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. 0-8218-1391-9 ↩
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. 9780883850992 ↩
Audin, Michele (2003), Geometry, Universitext, Springer, Proposition 3.5, pp. 80–81, ISBN 9783540434986. 9783540434986 ↩
Gorini, Catherine A. (2009), The Facts on File Geometry Handbook, Infobase Publishing, p. 49, ISBN 9781438109572. 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. 9783110250091 ↩