In mathematics, the distortion is a measure of the amount by which a 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; if the distortion is bounded and the function is a homeomorphism, then it is quasiconformal. The distortion of a function ƒ of the plane is given by
H(z,f)=\limsupr\to
| max|h|=r|f(z+h)-f(z)| | |
| min|h|=r|f(z+h)-f(z)| |
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 W(Ω, R2), the Jacobian determinant 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\leK(x)|J(x,f)|
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 Rn, 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{cases} |J(x,f)|-2/nDTf(x)Df(x)&ifJ(x,f)\not=0\\ I&ifJ(x,f)=0. \end{cases}
The outer distortion KO and inner distortion KI are defined via the Rayleigh quotients
KO(x)=\sup\xi\not=0
| \langleG(x)\xi,\xi\ranglen/2 | |
| |\xi|n |
, KI(x)=\sup\xi\not=0
| \langleG-1(x)\xi,\xi\ranglen/2 | |
| |\xi|n |
.
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 Rn, then a function has finite distortion if its Jacobian is locally integrable and does not change sign, and there is a measurable function KO (the outer distortion) such that
|Df(x)|n\leKO(x)|J(x,f)|
almost everywhere.