In mathematical complex analysis, a quasiconformal mapping, introduced by Grötzsch (1928) and named by Ahlfors (1935), is a (weakly differentiable) 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
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).
When μ is zero almost everywhere, any homeomorphism in W1,2(D) that is a weak solution of (1) is conformal.
, 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.
is quasiconformal (here z is a complex number) and has constant dilatation
Of central importance in the theory of quasiconformal mappings in two dimensions is the measurable Riemann mapping theorem, proved by Lars Ahlfors and Lipman Bers.
Then there is a quasiconformal homeomorphism f from D to the unit disk which is in the Sobolev space W1,2(D) and satisfies the corresponding Beltrami equation (1) in the distributional sense.
As with Riemann's mapping theorem, this f is unique up to 3 real parameters.
Recently, quasi-conformal geometry has attracted attention from different fields, such as applied mathematics, computer vision and medical imaging.
It has found various important applications in medical image analysis, computer vision and graphics.