Dilation (metric space)
In mathematics, a dilation is a function [math]\displaystyle{ f }[/math] from a metric space [math]\displaystyle{ M }[/math] into itself that satisfies the identity
- [math]\displaystyle{ d(f(x),f(y))=rd(x,y) }[/math]
for all points [math]\displaystyle{ x, y \in M }[/math], where [math]\displaystyle{ d(x, y) }[/math] is the distance from [math]\displaystyle{ x }[/math] to [math]\displaystyle{ y }[/math] and [math]\displaystyle{ r }[/math] 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, 91, American Mathematical Society, Providence, RI, p. 122, ISBN 0-8218-1391-9, https://books.google.com/books?id=DYAt3gVB7Q4C&pg=PA122.
- ↑ King, James R. (1997), "An eye for similarity transformations", in King, James R.; Schattschneider, Doris, Geometry Turned On: Dynamic Software in Learning, Teaching, and Research, Mathematical Association of America Notes, 41, Cambridge University Press, pp. 109–120, ISBN 9780883850992, https://archive.org/details/geometryturnedon0000unse/page/109. See in particular p. 110.
- ↑ Audin, Michele (2003), Geometry, Universitext, Springer, Proposition 3.5, pp. 80–81, ISBN 9783540434986, https://books.google.com/books?id=U_cTJMCIzdUC&pg=PA80.
- ↑ Gorini, Catherine A. (2009), The Facts on File Geometry Handbook, Infobase Publishing, p. 49, ISBN 9781438109572, https://books.google.com/books?id=PlYCcvgLJxYC&pg=PA49.
- ↑ Carstensen, Celine; Fine, Benjamin; Rosenberger, Gerhard (2011), Abstract Algebra: Applications to Galois Theory, Algebraic Geometry and Cryptography, Walter de Gruyter, p. 140, ISBN 9783110250091, https://books.google.com/books?id=X1SJ_ywbgy8C&pg=PA140.
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