Conformal Mappings

Beautiful angle-preserving transformations and the Joukowsky airfoil

🎨 Conformal Mappings

Some maps bend and stretch the plane but never change an angle. Every analytic function with a nonzero derivative is one of them, and one of them turns a circle into a wing.

1. Warp a Picture

Choose a map and watch the picture warp. Drag the probe anywhere in either plane: its tiny right angle stays a right angle, except where f′(z) = 0.

z-plane (domain)[−2, 2]²
f′ = 0
w-plane (image of w = z²)[−4, 4]²
image of the imaginary axis
0.90 + 0.60i
probe z₀
90.0°
angle between the image arms
× 2.16
local stretch |f′(z₀)|
33.7°
local rotation arg f′(z₀)
Map
Picture

z and −z land on the same point, so only the bright right half-plane is painted on the right; the dim left half would cover the w-plane a second time. The imaginary axis folds onto the negative reals, and at the critical point z = 0 angles double.

Each pixel of the w-plane is colored by pulling it back through the inverse map to the z-plane, so the picture is warped exactly, pixel by pixel. The angle readout measures the image arms with f itself, not with a formula.

Go deeper:Conformal Mappings

A conformal mapping is a function that preserves angles locally. These transformations are not only mathematically beautiful but also incredibly useful in physics and engineering for solving problems in electrostatics, fluid dynamics, and aerodynamics.

Key Property: If two curves intersect at angle θ in the z-plane, their images intersect at the same angle θ in the w-plane!
Conformal ⟺ Analytic with f'(z) ≠ 0

Every analytic function with non-zero derivative is conformal. This deep connection between complex differentiability and geometry makes conformal mappings a powerful tool for transforming difficult problems into simple ones.

Go deeper:The Maps in the Demo, and Their Branches
z² and √z
z² doubles angles at 0, so the right half-plane already covers the whole w-plane. Its inverse √z must pick one of two roots; the principal root has Re √z ≥ 0 and jumps across the negative real axis, its branch cut.
eᶻ and log z
eᶻ sends vertical lines to circles and horizontal lines to rays, and repeats every 2πi. The principal log z = ln|z| + i Arg z, with −π < Arg z ≤ π, undoes it on the strip −π < Im z ≤ π and is cut along the negative reals.
1/z and Möbius maps
w = (az + b)/(cz + d) with ad − bc ≠ 0 sends circles and lines to circles and lines. These are exactly the one-to-one conformal maps of the Riemann sphere onto itself, and any three distinct points can be sent to any other three.
Joukowsky
w = z + 1/z takes the same value at z and 1/z, so it is two-to-one. The outside of the unit circle covers the plane minus the slit [−2, 2] once; f′ = 1 − 1/z² vanishes at z = ±1, the ends of the slit.

How the picture is drawn: each pixel w of the right panel is pulled back through the chosen inverse branch to a point z, and painted with the color the picture has at z. Pixels with no preimage on that branch stay dark, which is why the √z image is a half-plane and the log z image is a horizontal strip.

2. Grid Transformer

Choose a transformation and watch the grid deform. Perpendicular lines stay perpendicular everywhere except at critical points, where f'(z) = 0.

Identity (no transformation): w = z
No transformation - grid stays rectangular

Transformed Grid (w-plane):

Notice how perpendicular grid lines remain perpendicular (angles preserved!)

📐 Angle Preservation in Action:

In the original z-plane, the grid has perpendicular horizontal and vertical lines (intersecting at 90°).

After applying the conformal mapping, the grid may stretch, compress, or curve, but at every intersection point, the transformed curves still meet at 90°!

Why? Because conformal mappings preserve angles. This is the defining property that makes them so useful in physics and engineering.

Technical note: Conformality requires f'(z) ≠ 0. At critical points (where f'(z) = 0), the angle-preserving property breaks down. For example, w = z² has a critical point at z = 0.
Go deeper:What to Look For

Watch how a conformal mapping deforms a coordinate grid while preserving angles at every intersection:

  • Perpendicular grid lines in the z-plane remain perpendicular in the w-plane
  • Small circles map to small ellipses (but locally look circular)
  • The grid may stretch, compress, or rotate, but angles are always preserved
  • Where f'(z) = 0, conformality breaks down (critical points)

3. Riemann Mapping Theorem

Choose a domain, such as a half-plane, a strip, or a wedge, and watch it map onto the unit disk with every angle kept intact.

Upper Half-Plane
Original Domain: The set H = {z : Im(z) > 0}
Mapping: w = (z-i)/(z+i)
This Möbius transformation maps the upper half-plane (Im z > 0) to the unit disk |w| < 1. The real axis maps to the unit circle, and the point at infinity maps to w = 1.

Mapped to Unit Disk (w-plane)

Conformally mapped to the unit disk |w| < 1

Riemann Mapping Theorem:

Every simply connected domain D ⊂ ℂ (except ℂ itself) can be conformally mapped onto the unit disk D = {w : |w| < 1}.

Simply connected means the domain has no holes - you can shrink any closed loop to a point.

Conformal means angles are preserved by the mapping.

Onto means every point in the unit disk is the image of some point in D.

Existence Result
The theorem guarantees a conformal map exists, but doesn't tell us what it is! Finding explicit formulas is often very difficult.
Uniqueness
If you fix a point z₀ ∈ D and its image w₀ ∈ disk, plus an angle condition, the mapping is unique!

🎯 Why This Matters:

  • Solve PDEs: Transform Laplace's equation from complicated domain to simple disk, solve there, then transform back
  • Universal property: All simply connected domains are "the same" from a conformal viewpoint
  • Numerical methods: Modern algorithms (Schwarz-Christoffel, CRDT) can approximate these mappings
  • Proof technique: The proof uses normal families and the maximum modulus principle - deep complex analysis!
Go deeper:Riemann Mapping Theorem

One of the most remarkable theorems in complex analysis:

Riemann Mapping Theorem:
Every simply connected domain (except ℂ itself) can be conformally mapped to the unit disk D = {z : |z| < 1}.
This means any "blob" without holes can be smoothly transformed into a circle!

While the theorem guarantees existence, finding the explicit mapping is often challenging. Numerical methods can approximate these mappings effectively.

4. Joukowsky Airfoil Transformation

Move the sliders to shift and resize the circle, and watch w = z + 1/z turn it into an airfoil with more or less thickness and camber.

Joukowsky Airfoil Transformation: w = z + 1/z
Adjust the circle parameters to change the airfoil shape. The circle must pass through or near z = 1 to create a realistic airfoil!
Controls airfoil thickness
Controls airfoil camber (curvature)
Controls airfoil thickness

Original Circle (z-plane):

Blue: adjustable circle, Orange: critical points z = ±1

Joukowsky Airfoil (w-plane):

Green: transformed airfoil shape

How the Transform Works:

Step 1: Start with a circle in the z-plane. The circle must have radius > 1 and should pass through or near the point z = 1.

Step 2: Apply w = z + 1/z. This maps:

  • The circle to a closed curve (the airfoil)
  • z = 1 to w = 2 (the trailing edge, often a cusp)
  • z = -1 to w = -2 (the leading edge)

Step 3: The resulting shape looks like an aircraft wing! By adjusting the circle's center and radius, we control the airfoil's camber (curvature) and thickness.

Historical Impact:

Nikolai Joukowsky (1847-1921) - Russian scientist who developed this transformation in the early 1900s.

Before computational fluid dynamics, engineers needed analytical solutions. The Joukowsky transform allowed them to:

  1. Calculate flow around a circle (easy!)
  2. Transform to flow around an airfoil (via conformal mapping)
  3. Compute lift force on wings analytically

This mathematical technique literally helped design the first aircraft wings and establish the field of aerodynamics!

🔍 Key Observations:

  • Camber: Moving the circle center up/down changes the airfoil curvature
  • Thickness: Moving the center left and enlarging the radius makes the airfoil thicker
  • Cusp: If the circle passes exactly through z = 1, you get a sharp trailing edge
  • Conformal: Angles are preserved everywhere except at z = ±1 (critical points where w' = 0)
  • Flow: Potential flow around the circle maps to potential flow around the airfoil, allowing lift calculation!
Go deeper:The Joukowsky Transform: w = z + 1/z

One of the most spectacular applications of conformal mappings: designing aircraft wings!

This simple formula maps circles in the z-plane to airfoil-like shapes in the w-plane!

History: Developed by Russian scientist Nikolai Joukowsky in the early 1900s, this transform helped establish the mathematical foundation for aerodynamics.

How it works:

  • Start with a circle passing through z = 1
  • Apply w = z + 1/z
  • The circle transforms into an airfoil shape
  • Adjust the circle's position/size to change the airfoil's camber and thickness

By studying fluid flow around the circle (easy!), we can understand flow around the airfoil (hard!) via the conformal mapping. This is how lift on wings was first calculated!

Why Conformal Mappings Matter

  • Analytic with f'(z) ≠ 0 means angles are preserved
  • Map a hard domain to a simple one, solve there, and map back
  • Every simply connected domain except ℂ maps conformally onto the unit disk
  • The Joukowsky map w = z + 1/z turns circles into wing shapes

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