查看“︁扩张 (度量空间)”︁的源代码

在[[数学|数学]]中，'''扩张'''（{{lang-en|dilation}}）是从[[度量空间]]<math>M</math>映射到<math>M</math>本身的[[函数]]<math>f</math>，使得恒等式

: <math>d(f(x),f(y))=rd(x,y)</math>

对任意<math>x, y \in M</math> 都成立，其中<math>d(x, y)</math>是<math>x</math>到<math>y</math>的距离，<math>r</math>是正[[实数]]。 <ref>{{Citation|last=Montgomery|first=Richard|isbn=0-8218-1391-9|mr=1867362|page=122|publisher=American Mathematical Society, Providence, RI|series=Mathematical Surveys and Monographs|title=A tour of subriemannian geometries, their geodesics and applications|url=https://books.google.com/books?id=DYAt3gVB7Q4C&pg=PA122|volume=91|year=2002|accessdate=2021-09-29|archive-date=2021-09-29|archive-url=https://web.archive.org/web/20210929131244/https://books.google.com/books?id=DYAt3gVB7Q4C&pg=PA122|dead-url=no}}.</ref>

对于[[欧几里得空间|欧氏空间]]，这样的扩张相当于空间的[[相似 (幾何)|相似]]。 <ref>{{Citation|last=King|first=James R.|editor1-last=King|editor1-first=James R.|editor2-last=Schattschneider|editor2-first=Doris|editor2-link=Doris Schattschneider|contribution=An eye for similarity transformations|isbn=9780883850992|pages=[https://archive.org/details/geometryturnedon0000unse/page/109 109–120]|publisher=Cambridge University Press|series=Mathematical Association of America Notes|title=Geometry Turned On: Dynamic Software in Learning, Teaching, and Research|volume=41|year=1997|url=https://archive.org/details/geometryturnedon0000unse/page/109}}. See in particular [https://books.google.com/books?id=lR0SDnl2bPwC&pg=PA110 p.&nbsp;110] {{Wayback|url=https://books.google.com/books?id=lR0SDnl2bPwC&pg=PA110 |date=20210929131242 }}.</ref>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]]<ref>{{Citation|title=Geometry|series=Universitext|first=Michele|last=Audin|publisher=Springer|year=2003|isbn=9783540434986|at=Proposition&nbsp;3.5, pp.&nbsp;80–81|url=https://books.google.com/books?id=U_cTJMCIzdUC&pg=PA80|accessdate=2021-09-29|archive-date=2021-09-29|archive-url=https://web.archive.org/web/20210929131247/https://books.google.com/books?id=U_cTJMCIzdUC&pg=PA80|dead-url=no}}.</ref> ，称为扩张中心。 <ref>{{Citation|title=The Facts on File Geometry Handbook|first=Catherine A.|last=Gorini|publisher=Infobase Publishing|year=2009|isbn=9781438109572|page=49|url=https://books.google.com/books?id=PlYCcvgLJxYC&pg=PA49|accessdate=2021-09-29|archive-date=2021-09-29|archive-url=https://web.archive.org/web/20210929131245/https://books.google.com/books?id=PlYCcvgLJxYC&pg=PA49|dead-url=no}}.</ref>而在全等关系中，有些全等有固定点，有些则没有。 <ref>{{Citation|title=Abstract Algebra: Applications to Galois Theory, Algebraic Geometry and Cryptography|first=Celine|last=Carstensen|first2=Benjamin|last2=Fine|first3=Gerhard|last3=Rosenberger|publisher=Walter de Gruyter|year=2011|isbn=9783110250091|page=140|url=https://books.google.com/books?id=X1SJ_ywbgy8C&pg=PA140|accessdate=2021-09-29|archive-date=2021-09-29|archive-url=https://web.archive.org/web/20210929131243/https://books.google.com/books?id=X1SJ_ywbgy8C&pg=PA140|dead-url=no}}.</ref>

== 参见 ==

* [[位似变换]]
* [[扩张（算子理论）]]

== 参考文献 ==
{{Reflist}}
[[Category:度量几何]]