- See also
- References
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 W{{su|p=1,1|b=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 KO and inner distortion KI 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 {{nowrap|ƒ ∈ W{{su|p=1,1|b=loc}}(Ω,Rn)}} 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
almost everywhere.
See also
- Deformation (mechanics)
References
- {{Citation | last1=Iwaniec | first1=Tadeusz |author1-link=Tadeusz Iwaniec| last2=Martin | first2=Gaven|author2-link= Gaven Martin | title=Geometric function theory and non-linear analysis | publisher=The Clarendon Press Oxford University Press | series=Oxford Mathematical Monographs | isbn=978-0-19-850929-5 | mr=1859913 | year=2001}}.
- {{Citation | last1=Reshetnyak | first1=Yu. G.|authorlink=Yurii Reshetnyak | title=Space mappings with bounded distortion | publisher=American Mathematical Society | location=Providence, R.I. | series=Translations of Mathematical Monographs | isbn=978-0-8218-4526-4 | mr=994644 | year=1989 | volume=73}}.
- U.S. Route 278 Business (Hilton Head Island, South Carolina)
- U.S. Route 278 Business (Hilton Head, South Carolina)
- U.S. Route 278 Business (Hope, Arkansas)
- U.S. Route 278 Business (Piedmont, Alabama)
- U.S. Route 278 Business (Rockmart, Georgia)
- U.S. Route 278 Business (Warren, Arkansas)
- U.S. Route 278B (Warren, Arkansas)
- U.S. Route 278 Bypass (Warrenton, Georgia)
- U.S. Route 278 Truck (Warrenton, Georgia)
- U.S. Route 27 Alternate (Avon Park-Sebring, Florida)
- U.S. Route 27 Alternate (Belleview-Lady Lake, Florida)
- U.S. Route 27 Alternate (Carrollton-Columbus, Georgia)
- U.S. Route 27 Alternate (Frostproof-Haines City, Florida)
- U.S. Route 27 Alternate (Georgia)
- U.S. Route 27 Alternate (Haines City)
- U.S. Route 27 Alternate (Lady Lake-Belleview, Florida)
- U.S. Route 27 Alternate (Lake Weir)
- U.S. Route 27 Alternate (Lake Weir, Florida)
- U.S. Route 27 Alternate (Perry-Williston, Florida)
- U.S. Route 27 Alternate (Sebring)
- U.S. Route 27 Alternate (Sebring, Florida)
- U.S. Route 27 Alternate (Williston-Perry, Florida)
- U.S. Route 27 Business (Alma, Michigan)
- U.S. Route 27 Business (Attapulgus, Georgia)
- U.S. Route 27 Business (Bainbridge, Georgia)