a quasiconformal map
A conformal map preserves angles perfectly: it sends infinitesimal circles to infinitesimal circles, possibly rotated and scaled but never distorted. A quasiconformal map relaxes this just enough to be flexible without becoming wild: it is allowed to send infinitesimal circles to infinitesimal ELLIPSES, provided the eccentricity of those ellipses stays uniformly bounded. So shapes can be sheared and stretched, but only by a controlled, finite amount everywhere — a 'bounded distortion' homeomorphism.
Precisely, an orientation-preserving homeomorphism f between planar domains is K-quasiconformal (K ≥ 1) if it has locally integrable distributional derivatives and its complex dilatation mu = (f_zbar) / (f_z) satisfies |mu| ≤ k < 1 almost everywhere, where K = (1 + k)/(1 - k). The quantity mu is the Beltrami coefficient; it records, at almost every point, the direction and magnitude of the maximal stretch. When mu ≡ 0 the map is conformal (holomorphic), so K = 1 is exactly the conformal case. The infinitesimal ellipse picture is literal: at a point, df maps the unit circle to an ellipse whose axis ratio is the local dilatation K(z) = (1 + |mu(z)|)/(1 - |mu(z)|).
Quasiconformal maps are the natural maps of Teichmüller theory because they connect different complex structures with measured distortion. The Teichmüller distance between two marked Riemann surfaces is (1/2) log of the smallest K for which a K-quasiconformal map respecting the markings exists; Teichmüller's theorem says the optimal (extremal) map is essentially unique and stretches along a holomorphic quadratic differential. They also power Mostow rigidity (the boundary map in dimension ≥3 is a priori quasiconformal, then forced conformal) and the deformation theory of Kleinian groups. The bridge from a prescribed Beltrami coefficient back to an actual map is the measurable Riemann mapping theorem.
The real-linear map f(x + iy) = K x + i y, with K > 1, stretches the plane by factor K in the x-direction and leaves y alone. It sends the unit circle to an ellipse of axis ratio K, so it is K-quasiconformal with constant Beltrami coefficient mu = (K - 1)/(K + 1) (a real number, since the stretch direction is fixed). It is NOT conformal (it does not preserve angles — a 45-degree line tilts), yet it is a perfectly good homeomorphism with bounded distortion, the simplest non-conformal quasiconformal map.
The simplest qc map: stretch by K in one direction. Circles become ellipses of axis ratio K; angles are not preserved.
K = 1 means conformal, but K-quasiconformal does NOT mean 'almost conformal' for large K — a 100-quasiconformal map can be drastically distorted. Quasiconformal maps need only be differentiable almost everywhere (they can fail to be smooth on a measure-zero set), which is exactly why the measurable Riemann mapping theorem is needed to handle merely measurable Beltrami data.