mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the distortion is a measure of the amount by which a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...

from the

Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...

to itself distorts circles to ellipses. If the distortion of a function is equal to one, then it is conformal; if the distortion is bounded and the function is a

homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...

, then it is

quasiconformal In mathematical complex analysis, a quasiconformal mapping is a (weakly differentiable) homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity. Quasiconformal mappings are a generaliz ...

. The distortion of a function ƒ of the plane is given by :

H(z,f)  = \limsup_\frac

which is the limiting eccentricity of the ellipse produced by applying ƒ to small circles centered at ''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'' : Ω → R² from an open domain in the plane to the plane has finite distortion at a point ''x'' ∈ Ω if ''ƒ'' is in the

Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...

W(Ω, R²), the

Jacobian determinant In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of components ...

J(''x'',ƒ) is locally integrable and does not change sign in Ω, and there is a measurable function ''K''(''x'') ≥ 1 such that :

, Df(x), ^2 \le K(x), J(x,f),

almost everywhere. Here ''Df'' is the

weak derivative In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (''strong derivative'') for functions not assumed differentiable, but only integrable, i.e., to lie in the L''p'' space L^1( ,b. The method o ...

of ƒ, and , ''Df'', is the

Hilbert–Schmidt norm In mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. Th ...

. For functions on a higher-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

R^''n'', 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'' :

G(x,f) =
\begin
, J(x,f), ^D^Tf(x)Df(x)&\textJ(x,f)\not=0\\
I &\textJ(x,f)=0.
\end

The outer distortion ''K''_O and inner distortion ''K''_I are defined via the

Rayleigh quotient In mathematics, the Rayleigh quotient () for a given complex Hermitian matrix M and nonzero vector ''x'' is defined as:R(M,x) = .For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugat ...

s :

K_O(x) = \sup_\frac,\quad K_I(x) = \sup_\frac.

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^''n'', then a function has finite distortion if its Jacobian is locally integrable and does not change sign, and there is a measurable function ''K''_''O'' (the outer distortion) such that :

, Df(x), ^n \le K_O(x), J(x,f),

almost everywhere.

References

* . *{{Citation , last1=Reshetnyak , first1=Yu. G., authorlink=Yurii Reshetnyak , title=Space mappings with bounded distortion , publisher=

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

, location=Providence, R.I. , series=Translations of Mathematical Monographs , isbn=978-0-8218-4526-4 , mr=994644 , year=1989 , volume=73. Conformal mappings Real analysis Complex analysis Topology Measure theory Euclidean geometry

See also

References