Orthogonal level curves of harmonic conjugates

For a holomorphic function $f=u+iv$, the Cauchy-Riemann equations imply $\nabla u\cdot \nabla v=0$ and $|\nabla u|=|\nabla v|=|f^{\prime }|$. At a point where $f^{\prime }\ne 0$, the two regular level curves therefore meet at right angles, since each tangent is perpendicular to its nonzero gradient. The noncritical condition is necessary: for $f(z)=z^2$, the zero level sets are the coordinate axes and the two diagonals, which meet at angles of $\pi /4$ at the critical point zero. Orthogonality at regular points is one geometric expression of conformality.