quasiconformal mapping

quasiconformal mapping

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

f

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

.

Define the

dilatation

of the mapping

f

at the point

z

as

D

f

⁢

(

z

)

:=

|

f

z

|

+

|

f

z

¯

|

|

f

z

|

-

|

f

z

¯

|

≥

1

,

and define
the

maximal dilatation

of the mapping as

K

f

:=

sup

z

⁡

D

f

⁢

(

z

)

.

Now we are ready to define what it means for

f

to be quasiconformal.

Definition

.

For

f

as above, we will call

f

quasiconformal

if the maximal
dilatation of

f

is finite. We will say that

f

is

K

-quasiconformal mapping if the maximal dilatation of this mapping is

K

.

Note that sometimes the

K

-quasiconformal is used to that the dilatation is

K

or lower.

It is easy to see that a conformal sense-preserving mapping has a dilatation of

1

since

|

f

z

¯

|

=

0

. We can further define several
other related quantities

Definition

.

For

f

as above, define the

small dilatation

as

d

f

⁢

(

z

)

:=

|

f

z

¯

|

|

f

z

|

.

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

k

-quasiconformal if the small dilatation is

bounded

by

k

. 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

d

f

:=

D

f

-

1

D

f

+

1

.

Definition

.

For

f

as above, define the

complex dilatation

as

μ

f

⁢

(

z

)

:=

f

z

¯

f

z

.

The complex dilatation now appears in the

Beltrami differential equation

f

z

¯

⁢

(

z

)

=

μ

f

⁢

(

z

)

⁢

f

z

⁢

(

z

)

.

This means that a quasiconformal mapping is a solution to the Beltrami equation where a non-negative

measurable

μ

f

is uniformly bounded by some

k

<

1

.

The above results are stated for

f

:

ℂ

→

ℂ

, but
the statements are exactly the same if you take

f

:

G

⊂

ℂ

→

ℂ

for an open set

G

.

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 (

http://planetmath.org/BeurlingAhlforsQuasiconformalExtension

) that an of a mapping of the real line to itself is quasiconformal if and only if the mapping is quasisymmetric.

References

1

L. V. Ahlfors.

. Van Nostrand-Reinhold, Princeton, New Jersey, 1966

2

J. Lebl.

. . Also available at

http://www.jirka.org/thesis.pdf

http://www.jirka.org/thesis.pdf

Title

quasiconformal mapping

Canonical

name

QuasiconformalMapping

Date of creation

2013-03-22 14:06:43

Last modified on

2013-03-22 14:06:43

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

13

Author

jirka (4157)

Entry type

Definition

Classification

msc 30C65

Classification

msc 30C62

Synonym

K-quasiconformal mapping

Related topic

QuasisymmetricMapping

Related topic

BeurlingAhlforsQuasiconformalExtension

Related topic

ConformalMapping

Related topic

BeltramiDifferentialEquation

Defines

dilatation

Defines

small dilatation

Defines

maximal dilatation

Defines

complex dilatation

Defines

K

-quasiconformal

Defines

K-quasiconformal

Defines

quasiconformal

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