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In [[mathematics]], the '''distortion''' is a measure of the amount by which a [[function (mathematics)|function]] from the [[Euclidean plane]] to itself distorts circles to ellipses.  If the distortion of a function is equal to one, then it is [[conformal map|conformal]]; if the distortion is bounded and the function is a [[homeomorphism]], then it is [[quasiconformal mapping|quasiconformal]]. The distortion of a function ƒ of the plane is given by
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:<math>H(z,f)  = \limsup_{r\to 0}\frac{\max_{|h|=r}|f(z+h)-f(z)|}{\min_{|h|=r}|f(z+h)-f(z)|}</math>
 
which is the limiting eccentricity of the ellipse produced by applying ƒ to small circles centered at&nbsp;''z''. This geometrical definition is often very difficult to work with, and the necessary analytical features can be extrapolated to the following definition.  A mapping ''ƒ''&nbsp;:&nbsp;Ω&nbsp;→&nbsp;'''R'''<sup>2</sup> from an open domain in the plane to the plane has finite distortion at a point ''x''&nbsp;∈&nbsp;Ω if ''ƒ'' is in the [[Sobolev space]] W{{su|p=1,1|b=loc}}(Ω,&nbsp;'''R'''<sup>2</sup>), the [[Jacobian matrix and determinant|Jacobian determinant]] J(''x'',ƒ) is locally integrable and does not change sign in Ω, and there is a measurable function ''K''(''x'')&nbsp;≥&nbsp;1 such that
 
:<math>|Df(x)|^2 \le K(x)|J(x,f)| \, </math>
 
almost everywhere. Here ''Df'' is the [[weak derivative]] of ƒ, and |''Df''| is the [[Hilbert–Schmidt norm]].
 
For functions on a higher-dimensional [[Euclidean space]] '''R'''<sup>''n''</sup>, there are more measures of distortion because there are more than two [[principal axes]] of a symmetric tensor. The pointwise information is contained in the ''distortion tensor''
 
:<math>G(x,f) =
\begin{cases}
|J(x,f)^{-2/n}D^Tf(x)Df(x)&\text{if }J(x,f)\not=0\\
I &\text{if }J(x,f)=0.
\end{cases}</math>
 
The outer distortion ''K''<sub>O</sub> and inner distortion ''K''<sub>I</sub> are defined via the [[Raleigh quotient]]s
 
:<math>K_O(x) = \sup_{\xi\not=0}\frac{\langle G(x)\xi,\xi\rangle^{n/2}}{|\xi|^n},\quad K_O(x) = \sup_{\xi\not=0}\frac{\langle G^{-1}(x)\xi,\xi\rangle^{n/2}}{|\xi|^n}.</math>
 
The outer distortion can also be characterized by means of an inequality similar to that given in the two-dimensional case. If Ω is an open set in '''R'''<sup>''n''</sup>, then a function {{nowrap|&fnof; &isin; W{{su|p=1,1|b=loc}}(&Omega;,'''R'''<sup>''n''</sup>)}} has finite distortion if its Jacobian is locally integrable and does not change sign, and there is a measurable function ''K''<sub>''O''</sub> (the outer distortion) such that
 
:<math>|Df(x)|^n \le K_O(x)|J(x,f)| \, </math>
 
almost everywhere.
 
==See also==
* [[Deformation (mechanics)]]
 
==References==
* {{Citation | last1=Iwaniec | first1=Tadeusz | last2=Martin | first2=Gaven | title=Geometric function theory and non-linear analysis | publisher=The Clarendon Press Oxford University Press | series=Oxford Mathematical Monographs | isbn=978-0-19-850929-5 | mr=1859913 | year=2001}}.
*{{Citation | last1=Reshetnyak | first1=Yu. G. | title=Space mappings with bounded distortion | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Translations of Mathematical Monographs | isbn=978-0-8218-4526-4 | mr=994644 | year=1989 | volume=73}}.
 
[[Category:Conformal mapping]]
[[Category:Real analysis]]
[[Category:Complex analysis]]
[[Category:Topology]]
[[Category:Measure theory]]
[[Category:Euclidean geometry]]

Latest revision as of 18:26, 9 October 2014

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