Dilation (metric space)

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

d(f(x),f(y))=rd(x,y)

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

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.

[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.

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See also

References