Distortion (mathematics)
Introduction 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'''2 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'''2), 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'''''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'':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 '''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:almost everywhere.
See also * Deformation (mechanics)
References * . *.