Conformal Maps of the Complex Plane

An analytic function $w = f(z)$ is a map of the plane onto itself, and it is a remarkably gentle one: wherever its derivative is nonzero it bends and stretches the plane but never tears an angle. Two curves that cross at some angle in the $z$-plane have images that cross at the very same angle in the $w$-plane. This is conformality, and it follows from the derivative: near a point, $f$ acts as multiplication by the complex number $f'(z)$, which is just a rotation by $\arg f'(z)$ together with a uniform scaling by $|f'(z)|$, and a rotation-plus-scale leaves every angle alone. The left panel draws a square grid in the $z$-plane; the right panel draws its image, the same grid bent into curves but still meeting at right angles wherever the original did. Drag the probe and watch its little perpendicular cross map to another perpendicular cross, rotated and resized but square. The spell breaks only at a critical point, where $f'(z) = 0$ and the cross collapses: there angles are not preserved but multiplied, so $z^2$ opens a right angle into a straight line at the origin. The bottom panel plots the local magnification $|f'(z)|$ along the probe's row, dropping to zero at critical points and blowing up at poles. Cycle through squaring, inversion, the exponential, a Mobius map, and the Joukowski map that turns a circle into an aerofoil.