扩张 (度量空间)

数学中,扩张(英语:dilation)是从度量空间映射到本身的函数,使得恒等式

对任意 都成立,其中的距离,是正实数[1]

对于欧氏空间,这样的扩张相当于空间的相似[2]扩张只改变对象或者说图形的大小,而不改变其形状。

欧氏空间的每个不是全等的扩张都有唯一的不动点[3] ,称为扩张中心。 [4]而在全等关系中,有些全等有固定点,有些则没有。 [5]

参见

参考文献

  1. ^ Montgomery, Richard, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs 91, American Mathematical Society, Providence, RI: 122, 2002 [2021-09-29], ISBN 0-8218-1391-9, MR 1867362, (原始内容存档于2021-09-29) .
  2. ^ King, James R., An eye for similarity transformations, King, James R.; Schattschneider, Doris (编), Geometry Turned On: Dynamic Software in Learning, Teaching, and Research, Mathematical Association of America Notes 41, Cambridge University Press: 109–120, 1997, ISBN 9780883850992 . See in particular p. 110页面存档备份,存于互联网档案馆).
  3. ^ Audin, Michele, Geometry, Universitext, Springer, Proposition 3.5, pp. 80–81, 2003 [2021-09-29], ISBN 9783540434986, (原始内容存档于2021-09-29) .
  4. ^ Gorini, Catherine A., The Facts on File Geometry Handbook, Infobase Publishing: 49, 2009 [2021-09-29], ISBN 9781438109572, (原始内容存档于2021-09-29) .
  5. ^ Carstensen, Celine; Fine, Benjamin; Rosenberger, Gerhard, Abstract Algebra: Applications to Galois Theory, Algebraic Geometry and Cryptography, Walter de Gruyter: 140, 2011 [2021-09-29], ISBN 9783110250091, (原始内容存档于2021-09-29) .