quasikonformal

Quasiconformal is a term used in mathematics, specifically in the field of geometric function theory. It describes a type of mapping or transformation between regions in a Euclidean space, typically the complex plane. A quasiconformal map is a generalization of a conformal map. Conformal maps preserve angles locally, meaning they distort shapes but not angles. Quasiconformal maps are less restrictive; they are allowed to distort angles, but only by a bounded amount.

More formally, a homeomorphism is quasiconformal if it satisfies a certain geometric condition related to the distortion of shapes. This condition can be expressed in terms of the derivative of the mapping. A key parameter associated with a quasiconformal map is its dilatation, which quantifies the maximum amount of angular distortion. If the dilatation is bounded by a constant K, the map is called K-quasiconformal. A conformal map can be thought of as a 1-quasiconformal map.

Quasiconformal mappings have important applications in various areas of mathematics and physics. They play a crucial role in complex analysis, particularly in the study of Riemann surfaces and the uniformization theorem. They are also used in the theory of partial differential equations, differential geometry, and even in some areas of theoretical physics, such as fluid dynamics and elasticity. The study of quasiconformal maps has revealed deep connections between different branches of mathematics.