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 → R 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, R), 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 R, 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" and inner distortion "K" 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 R, then a function has finite distortion if its Jacobian is locally integrable and does not change sign, and there is a measurable function "K" (the outer distortion) such that almost everywhere.