Dilation (metric space)

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

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

for all points ${\displaystyle (x,y)}$, 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 an 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, 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, 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.