Dilation (metric space)

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

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

for all points (x, y), 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]

See also

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, 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.