|
|
| Line 1: |
Line 1: |
| In mathematical [[complex analysis]], a '''quasiconformal mapping''', introduced by {{harvtxt|Grötzsch|1928}} and named by {{harvtxt|Ahlfors|1935}}, is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded [[ellipse#Eccentricity|eccentricity]].
| | Alyson Meagher is the name her parents gave her but she doesn't like when people use her full title. To climb is something I really enjoy doing. For a while I've been in Alaska but I will have to move in a yr or two. Office supervising is what she does for a residing.<br><br>Stop by my blog: [http://kjhkkb.net/xe/notice/374835 free online tarot card readings] |
| | |
| Intuitively, let ''f'' : ''D'' → ''D''′ be an [[orientation (mathematics)|orientation]]-preserving [[homeomorphism]] between [[open set]]s in the plane. If ''f'' is [[continuously differentiable]], then it is ''K''-quasiconformal if the derivative of ''f'' at every point maps circles to ellipses with eccentricity bounded by ''K''.
| |
| | |
| ==Definition==
| |
| Suppose ''f'' : ''D'' → ''D''′ where ''D'' and ''D''′ are two domains in '''C'''. There are a variety of equivalent definitions, depending on the required smoothness of ''f''. If ''f'' is assumed to have [[continuous function|continuous]] [[partial derivative]]s, then ''f'' is quasiconformal provided it satisfies the [[Beltrami equation]]
| |
| | |
| {{NumBlk|:|<math>\frac{\partial f}{\partial\bar{z}} = \mu(z)\frac{\partial f}{\partial z},</math>|{{EquationRef|1}}}}
| |
| | |
| for some complex valued [[Lebesgue measurable]] μ satisfying sup |μ| < 1 {{harv|Bers|1977}}. This equation admits a geometrical interpretation. Equip ''D'' with the [[metric tensor]]
| |
| | |
| :<math>ds^2 = \Omega(z)^2\left| \, dz + \mu(z) \, d\bar{z}\right|^2,</math>
| |
| | |
| where Ω(''z'') > 0. Then ''f'' satisfies ({{EquationNote|1}}) precisely when it is a conformal transformation from ''D'' equipped with this metric to the domain ''D''′ equipped with the standard Euclidean metric. The function ''f'' is then called '''μ-conformal'''. More generally, the continuous differentiability of ''f'' can be replaced by the weaker condition that ''f'' be in the [[Sobolev space]] ''W''<sup>1,2</sup>(''D'') of functions whose first-order [[distributional derivative]]s are in [[Lp space|L<sup>2</sup>(''D'')]]. In this case, ''f'' is required to be a [[weak solution]] of ({{EquationNote|1}}). When μ is zero almost everywhere, any homeomorphism in ''W''<sup>1,2</sup>(''D'') that is a weak solution of ({{EquationNote|1}}) is conformal.
| |
| | |
| Without appeal to an auxiliary metric, consider the effect of the [[pullback (differential geometry)|pullback]] under ''f'' of the usual Euclidean metric. The resulting metric is then given by
| |
| | |
| :<math>\left|\frac{\partial f}{\partial z}\right|^2\left|\,dz+\mu(z)\,d\bar{z}\right|^2</math>
| |
| | |
| which, relative to the background Euclidean metric <math>dz d\bar{z}</math>, has [[eigenvalues]]
| |
| | |
| :<math>(1+|\mu|)^2\textstyle{\left|\frac{\partial f}{\partial z}\right|^2},\qquad (1-|\mu|)^2\textstyle{\left|\frac{\partial f}{\partial z}\right|^2}.</math>
| |
| | |
| The eigenvalues represent, respectively, the squared length of the major and minor axis of the ellipse obtained by pulling back along ''f'' the unit circle in the tangent plane.
| |
| | |
| Accordingly, the ''dilatation'' of ''f'' at a point ''z'' is defined by
| |
| | |
| :<math>K(z) = \frac{1+|\mu(z)|}{1-|\mu(z)|}.</math>
| |
| | |
| The (essential) [[supremum]] of ''K''(''z'') is given by
| |
| | |
| :<math>K = \sup_{z\in D} |K(z)| = \frac{1+\|\mu\|_\infty}{1-\|\mu\|_\infty}</math>
| |
| | |
| and is called the dilatation of ''f''.
| |
| | |
| A definition based on the notion of [[extremal length]] is as follows. If there is a finite ''K'' such that for every collection '''Γ''' of curves in ''D'' the extremal length of '''Γ''' is at most ''K'' times the extremal length of {''f'' o γ : γ ∈ '''Γ'''}. Then ''f'' is ''K''-quasiconformal.
| |
| | |
| If ''f'' is ''K''-quasiconformal for some finite ''K'', then ''f'' is quasiconformal.
| |
| | |
| ==A few facts about quasiconformal mappings==
| |
| | |
| If ''K'' > 1 then the maps ''x'' + ''iy'' ↦ ''Kx'' + ''iy'' and ''x'' + ''iy'' ↦ ''x'' + ''iKy'' are both quasiconformal and have constant dilatation ''K''.
| |
| | |
| If ''s'' > −1 then the map <math>z\mapsto z\,|z|^{s}</math> is quasiconformal (here ''z'' is a complex number) and has constant dilatation <math>\max(1+s, \frac{1}{1+s})</math>. When ''s'' ≠ 0, this is an example of a quasiconformal homeomorphism that is not smooth. If ''s'' = 0, this is simply the identity map.
| |
| | |
| A homeomophism is 1-quasiconformal if and only if it is conformal. Hence the identity map is always 1-quasiconformal. If ''f'' : ''D'' → ''D''′ is ''K''-quasiconformal and ''g'' : ''D''′ → ''D''′′ is ''K''′-quasiconformal, then ''g'' o ''f'' is ''KK''′-quasiconformal. The inverse of a ''K''-quasiconformal homeomorphism is ''K''-quasiconformal. The set of 1-quasiconformal maps forms a group under composition.
| |
| | |
| The space of K-quasiconformal mappings from the complex plane to itself mapping three distinct points to three given points is compact.
| |
| | |
| {{Expand section|date=May 2012}}
| |
| | |
| ==Measurable Riemann mapping theorem==
| |
| Of central importance in the theory of quasiconformal mappings in two dimensions is the [[measurable Riemann mapping theorem]], proved by {{harvtxt|Morrey|1938}}. The theorem generalizes the [[Riemann mapping theorem]] from conformal to quasiconformal homeomorphisms, and is stated as follows. Suppose that ''D'' is a simply connected domain in '''C''' that is not equal to '''C''', and suppose that μ : ''D'' → '''C''' is [[Lebesgue measurable]] and satisfies <math>\|\mu\|_\infty<1</math>. Then there is a quasiconformal homeomorphism ''f'' from ''D'' to the unit disk which is in the Sobolev space ''W''<sup>1,2</sup>(''D'') and satisfies the corresponding Beltrami equation ({{EquationNote|1}}) in the [[weak solution|distributional sense]]. As with Riemann's mapping theorem, this ''f'' is unique up to 3 real parameters.
| |
| | |
| ==''n''-dimensional generalization==
| |
| {{Empty section|date=August 2008}}
| |
| | |
| ==Computational quasi-conformal geometry==
| |
| Recently, quasi-conformal geometry has attracted attention from different fields, such as applied mathematics, computer vision and medical imaging. Computational quasi-conformal geometry has been developed, which extends the quasi-conformal theory into a discrete setting. It has found various important applications in medical image analysis, computer vision and graphics.
| |
| | |
| ==See also==
| |
| *[[Isothermal coordinates]]
| |
| *[[Pseudoanalytic function]]
| |
| *[[Teichmüller space]]
| |
| | |
| ==References==
| |
| | |
| *{{Citation | last1=Ahlfors | first1=Lars | author1-link=Lars Ahlfors | title=Zur Theorie der Überlagerungsflächen | publisher=Springer Netherlands | language=German | doi=10.1007/BF02420945 | zbl=0012.17204 | year=1935 | journal=[[Acta Mathematica]] | issn=0001-5962 | volume=65 | issue=1 | pages=157–194}}
| |
| *{{Citation | last1=Ahlfors | first1=Lars V. | title=Lectures on quasiconformal mappings | origyear=1966 | url=http://books.google.com/books?id=4oWFH7FPb50C | publisher=[[American Mathematical Society]] | location=Providence, R.I. | edition=2nd | series=University Lecture Series | isbn=978-0-8218-3644-6 |id = {{MR|0200442}}, {{MR|2241787}} | year=2006 | volume=38}}
| |
| *{{citation|title=Quasiconformal mappings, with applications to differential equations, function theory and topology |first=Lipman|last=Bers |authorlink=Lipman Bers|journal=Bull. Amer. Math. Soc.|volume=83|issue=6|year=1977|pages=1083–1100|mr=463433|doi=10.1090/S0002-9904-1977-14390-5 }}
| |
| *{{Citation | last1=Grötzsch | first1=Herbert | authorlink = Herbert Grötzsch | title=Über einige Extremalprobleme der konformen Abbildung. I, II. | language=German | jfm=54.0378.01 | year=1928 | journal=Berichte Leipzig | volume=80 | pages=367–376}}
| |
| * {{citation| last=Heinonen|first=Juha | url=http://www.ams.org/notices/200611/whatis-heinonen.pdf | format=PDF | title=What Is ... a Quasiconformal Mapping? | journal=Notices of the American Mathematical Society | volume=53 | issue=11|date=December 2006}}
| |
| *{{citation | first1=O.|last1=Lehto | first2=K.I.|last2=Virtanen | title=Quasiconformal mappings in the plane | publisher=Springer-Verlag | location=Berlin, New York | edition=2nd | year=1973}}
| |
| * Papadopoulos, Athanase, ed. (2007), Handbook of Teichmüller theory. Vol. I, IRMA Lectures in Mathematics and Theoretical Physics, 11, European Mathematical Society (EMS), Zürich, {{doi|10.4171/029}}, ISBN 978-3-03719-029-6, MR2284826
| |
| * Papadopoulos, Athanase, ed. (2009), Handbook of Teichmüller theory. Vol. II, IRMA Lectures in Mathematics and Theoretical Physics, 13, European Mathematical Society (EMS), Zürich, {{doi|10.4171/055}}, ISBN 978-3-03719-055-5, MR2524085
| |
| *{{citation|title=On the Solutions of Quasi-Linear Elliptic Partial Differential Equations|first=Charles B. Jr.|last=Morrey|authorlink=Charles B. Morrey, Jr.|journal=Transactions of the American Mathematical Society|volume=43|year=1938|pages=126–166|doi=10.2307/1989904|issue=1|jstor=1989904|publisher=American Mathematical Society}}.
| |
| *{{eom|id=Q/q076430|first=V.A.|last= Zorich}}
| |
| | |
| {{DEFAULTSORT:Quasiconformal Mapping}}
| |
| [[Category:Conformal mapping]]
| |
| [[Category:Homeomorphisms]]
| |
| [[Category:Complex analysis]]
| |
Alyson Meagher is the name her parents gave her but she doesn't like when people use her full title. To climb is something I really enjoy doing. For a while I've been in Alaska but I will have to move in a yr or two. Office supervising is what she does for a residing.
Stop by my blog: free online tarot card readings