An orientation-preserving homeomorphism with locally weak derivatives and essentially bounded infinitesimal distortion. In holomorphic coordinates, its Beltrami coefficient satisfies , and the maximal dilatation is .
The essential supremum of the ratio of the largest to the smallest infinitesimal singular value of a quasiconformal map. It equals in terms of the Beltrami coefficient.
For an orientation-preserving quasiconformal map, the almost-everywhere coefficient . It is a coordinate-dependent representative of a tensor; the modulus and the resulting maximal dilatation are coordinate-independent.
The equation prescribing the Beltrami coefficient of a quasiconformal map. Two homeomorphic solutions on the same source differ by a biholomorphism between their targets.
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Quasiconformal mapping is a type of mapping between different spaces that generalizes the concept of conformal mappings. While conformal mappings preserve angles and are holomorphic (complex differentiable) in a neighborhood, quasiconformal mappings allow for some distortion but still maintain a controlled relationship between the shapes of the mapped objects. ### Key Concepts of Quasiconformal Mapping: 1. **Distortion Control**: In a quasiconformal mapping, the angle distortion is bounded.