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quasiconformal mapping
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(Definition)
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Quasiconformal mappings are mappings of the complex plane to itself that are “almost” conformal. That is, they do not distort angles arbitrarily and this “distortion” is uniformly bounded throughout their domain of definition. Alternatively one can think of quasiconformal mappings as mappings which take infinitesimal circles to infinitesimal ellipses. For example invertible linear maps are quasiconformal.
More rigorously, suppose  is a mapping of the complex plane to itself, and here we will only consider sense-preserving mappings, that is mappings with a positive jacobian.
Definition 1 Define the dilatation of the mapping  at the point  as
and define the maximal dilatation of the mapping as
Now we are ready to define what it means for  to be quasiconformal.
Definition 2 For  as above, we will call  quasiconformal if the maximal dilatation of  is finite. We will say that  is  -quasiconformal mapping if the maximal dilatation of this mapping is  .
Note that sometimes the term  -quasiconformal is used to mean that the dilatation is  or lower.
It is easy to see that a conformal sense-preserving mapping has a dilatation of  since
 . We can further define several other related quantities
Definition 3 For  as above, define the small dilatation as
Again for sense-preserving maps this quantity is less then 1 and it is equal to 0 if the mapping is conformal. Some authors call a map  -quasiconformal if the small dilatation is bounded by  . It is however not ambiguous as the large dilatation is always greater then or equal to 1. Furthermore this is related to the large dilatation by
Definition 4 For  as above, define the complex dilatation as
The complex dilatation now appears in the Beltrami differential equation
This means that a quasiconformal mapping is a solution to the Beltrami equation where a non-negative measurable  is uniformly bounded by some  .
The above results are stated for
 , but the statements are exactly the same if you take
 for an open set  .
The theory generalizes to other dimensions as well. For example in one real dimension, the analogous mappings are called quasisymmetric. It is a well-known theorem of Beurling and Ahlfors that an extension of a mapping of the real line to itself is quasiconformal if and only if the mapping is quasisymmetric.
- 1
- L. V. Ahlfors. Lectures on Quasiconformal Mappings. Van Nostrand-Reinhold, Princeton, New Jersey, 1966
- 2
- J. Lebl. Quasiconformal Extensions of Quasisymmetric Mappings. Masters thesis, San Diego State University, San Diego, CA, May 2003. Also available at http://www.jirka.org/thesis.pdf
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"quasiconformal mapping" is owned by jirka.
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See Also: quasisymmetric mapping, Beurling-Ahlfors quasiconformal extension, conformal mapping, Beltrami differential equation
| Other names: |
K-quasiconformal mapping |
| Also defines: |
dilatation, small dilatation, maximal dilatation, complex dilatation,  -quasiconformal, K-quasiconformal, quasiconformal |
| Keywords: |
quasiconformal, quasisymmetric, dilatation |
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Cross-references: line, quasisymmetric, real, dimensions, theory, open set, measurable, equation, solution, Beltrami differential equation, maps, sense-preserving, easy to see, finite, point, Jacobian, positive, sense-preserving mappings, invertible linear maps, ellipses, circles, infinitesimal, domain, bounded, angles, conformal, complex plane, mappings
There are 7 references to this entry.
This is version 10 of quasiconformal mapping, born on 2004-01-15, modified 2005-03-07.
Object id is 5515, canonical name is QuasiconformalMapping.
Accessed 12282 times total.
Classification:
| AMS MSC: | 30C62 (Functions of a complex variable :: Geometric function theory :: Quasiconformal mappings in the plane) | | | 30C65 (Functions of a complex variable :: Geometric function theory :: Quasiconformal mappings in $R^n$, other generalizations) |
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Pending Errata and Addenda
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