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Concept
Quasiconformal mapping
Summary
In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity.
Intuitively, let f : D → D′ be an orientation-preserving homeomorphism between open sets in the plane. If f is continuously differentiable, then it is K-quasiconformal if the derivative of f at every point maps circles to ellipses with eccentricity bounded by K.
Suppose f : D → D′ where D and D′ are two domains in C. There are a variety of equivalent definitions, depending on the required smoothness of f. If f is assumed to have continuous partial derivatives, then f is quasiconformal provided it satisfies the Beltrami equation
for some complex valued Lebesgue measurable μ satisfying sup |μ| < 1 . This equation admits a geometrical interpretation. Equip D with the metric tensor
where Ω(z) > 0. Then f satisfies () precisely when it is a conformal transformation from D equipped with this metric to the domain D′ equipped with the standard Euclidean metric. The function f is then called μ-conformal. More generally, the continuous differentiability of f can be replaced by the weaker condition that f be in the Sobolev space W1,2(D) of functions whose first-order distributional derivatives are in L2(D). In this case, f is required to be a weak solution of (). When μ is zero almost everywhere, any homeomorphism in W1,2(D) that is a weak solution of () is conformal.
Without appeal to an auxiliary metric, consider the effect of the pullback under f of the usual Euclidean metric. The resulting metric is then given by
which, relative to the background Euclidean metric , has eigenvalues
The eigenvalues represent, respectively, the squared length of the major and minor axis of the ellipse obtained by pulling back along f the unit circle in the tangent plane.
Accordingly, the dilatation of f at a point z is defined by
The (essential) supremum of K(z) is given by
and is called the dilatation of f.
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