The Mathematics
A function w = f(z) of a complex variable z = x + iy is conformal at every point where f'(z) ≠ 0. The derivative f'(z) simultaneously encodes:
- Scaling:
|f'(z)|— how much the map stretches lengths. - Rotation:
arg(f'(z))— how much the map rotates.
Because multiplication by a complex number is a rotation + scale (never a shear), infinitesimal circles map to infinitesimal circles — angles are preserved.
The Joukowski map: z → z + 1/z
w = z + 1/z
Pick Joukowski in the playground and a circle offset from the origin maps to a smooth airfoil — the classic teardrop wing profile that made early aerodynamics tractable. Lines and circles in z-space bend into the wing's contour while every crossing stays perpendicular.
Inversion: z → 1/z
w = 1/z
Pick Inversion and the grid turns inside-out: points near the origin fly outward, the far field collapses inward, and straight lines that miss the origin become circles through it. It's the simplest map that swaps "near" and "far."
Inversion is the simplest Möbius transformation (az + b)/(cz + d) — the family that maps circles to circles (straight lines counting as circles through infinity). Joukowski is built from it: z + 1/z is an inversion bolted onto the identity.
Where angles break down
Conformality fails at critical points where f'(z) = 0. For z², the origin doubles all angles — two crossing curves that meet at 90° in z-space meet at 180° in w-space. This is the same mechanism as branch points in multivalued functions.
Head back up to the playground and switch the Map dropdown between Power, Inversion, Joukowski, and Exponential. Watch how wildly the same rectangular grid bends from one map to the next — yet every intersection stays a perfect right angle. That invariance is conformality.
Further reading
- Needham, Visual Complex Analysis — the most beautiful mathematics textbook ever written. Chapter 3 is all about conformal maps.
- Conformal map (Wikipedia)
- Ahlfors, Complex Analysis