The map z → z² squares the complex plane. A rectangular grid becomes two families of parabolas — but every intersection remains perpendicular.
The Core Idea
A conformal map is a function that preserves angles. Take a grid of perpendicular lines, apply the map, and — no matter how wildly the grid bends — every intersection is still a right angle. This seemingly abstract property makes conformal maps indispensable in aerodynamics (Joukowski airfoils), electrostatics (solving Laplace's equation by mapping to simpler geometries), and complex analysis (the Riemann mapping theorem).
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.
z → z³ triples all angles. The origin is a branch point; three copies of the plane wrap around it.
The power map: z → zⁿ
w = zⁿ → rⁿ eⁱⁿᶿ
The power map takes the ray at angle θ to the ray at angle nθ, and stretches the radius from r to rⁿ. A full circle in z-space covers n circles in w-space.
The exponential map. Horizontal lines become rays; vertical lines become circles. Strips of height 2π tile the entire plane.
The exponential map: z → eᶻ
w = eˣ⁺ⁱʸ = eˣ(cos y + i sin y)
The exponential map turns horizontal lines (constant y) into rays from the origin and vertical lines (constant x) into concentric circles. A rectangular grid in z-space maps to the polar grid in w-space — still orthogonal everywhere.
The Möbius transformation
w = (az + b) / (cz + d)
Möbius transformations map circles to circles (including lines as circles through infinity). They are the "rigid motions" of the Riemann sphere — the most general conformal automorphisms of the extended complex plane. Every Möbius map is a composition of translations, rotations, dilations, and inversions.
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.
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