Consider the four curves $A=\{t\}$ $B=\{t+it\}$ $C=\{it\}$ and $D=\{-t+it\}$ $t\in[-10,10]$ Suppose there is a mapping $f:\mathbb{C}\mapsto\mathbb{C}$ which maps $A$ to $D$ and $B$ to $C$ Is $f$ conformal at $z_0=0$ The size of the angles between $A$ and $B$ at the point of intersection $z_0=0$ is preserved, however the orientation is not. Therefore $f$ is not conformal at $z_0=0$ Now suppose there is a function $g:\mathbb{C}\mapsto\mathbb{C}$ which maps $A$ to $C$ and $B$ to $D$ In this case we see not only that the size of the angles is preserved, but also the orientation. Therefore $g$ is conformal at $z_0=0$