Quasiconformal mapping

Remarks

Example: for $0 \le k < 1$, the real-linear map $f(z) = z + k \bar z$ has $f_z = 1$ and $f_{\bar z} = k$. Its Jacobian determinant is $\abs{f_z}^2 - \abs{f_{\bar z}}^2 = 1 - k^2 > 0$, so it is an orientation-preserving homeomorphism of $\C$, and it is $K$-quasiconformal with $K = (1 + k)/(1 - k)$. It maps the unit circle onto an ellipse with axis ratio $(1 + k)/(1 - k)$. Non-example: complex conjugation $f(z) = \bar z$ is a homeomorphism, but $f_z = 0$ and $f_{\bar z} = 1$, so clause 3 fails for every $K$; it reverses orientation. The quotient $\mu_f = f_{\bar z} / f_z$ is the complex dilatation of $f$, and $K_f = (1 + \norm{\mu_f}_\infty)/(1 - \norm{\mu_f}_\infty)$ is its maximal dilatation. A map is $1$-quasiconformal if and only if it is conformal. An equivalent geometric definition uses the moduli of quadrilaterals or annuli: $f$ is $K$-quasiconformal when it distorts each conformal modulus by a factor of at most $K$.