A $C^1$-map $f:U\to \mathbb{C}$ of an open set $U\subseteq \mathbb{C}\cong {\mathbb{R}}^2$ is called:

1. Locally conformal, if 1;;$\mathrm{\forall }{z}_0\in U$, the differential $df(z_0):\mathbb{C}\to \mathbb{C}$ is angle preserving.
2. conformal, if it is 1;;locally conformal and acts bijectively ($U\to f(U)$ is injective)