Dilation (metric space)

For dilation in operator theory, see Dilation (operator theory).

In mathematics, a dilation is a function ${\displaystyle f}$ from a metric space ${\displaystyle M}$ into itself that satisfies the identity

${\displaystyle d(f(x),f(y))=rd(x,y)}$

for all points ${\displaystyle x,y\in M}$, where ${\displaystyle d(x,y)}$ is the distance from ${\displaystyle x}$ to ${\displaystyle y}$ and ${\displaystyle r}$ 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

• Homothety
• Dilation (operator theory)

References

1. 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 .
2. 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.
3. Audin, Michele (2003), Geometry, Universitext, Springer, Proposition 3.5, pp. 80–81, ISBN   9783540434986 .
4. Gorini, Catherine A. (2009), The Facts on File Geometry Handbook, Infobase Publishing, p. 49, ISBN   9781438109572 .
5. Carstensen, Celine; Fine, Benjamin; Rosenberger, Gerhard (2011), Abstract Algebra: Applications to Galois Theory, Algebraic Geometry and Cryptography, Walter de Gruyter, p. 140, ISBN   9783110250091 .