Distortion (mathematics)

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

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 W1,1
loc(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

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

The outer distortion K_O and inner distortion K_I are defined via the Rayleigh quotients

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 ƒ ∈ W1,1
loc(Ω,Rⁿ) 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

almost everywhere.