Conformal Mappings

Beautiful angle-preserving transformations and the Joukowsky airfoil

🎨 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.

1. Conformal Mapping Gallery

Explore classic conformal mappings and see how they transform the complex plane:

Möbius Transform
w = (az+b)/(cz+d) - Most general conformal map. Circles → circles or lines.
Exponential
w = e^z - Maps vertical strips to sectors. Horizontal lines → circles.
Sine/Cosine
w = sin(z) - Maps vertical strips to plane. Used in potential theory.
Joukowsky
w = z + 1/z - Circle → Airfoil! Used in aerodynamics for wing design.
Möbius Transformation: w = (z-i)/(z+i)
The most general conformal map. w = (az+b)/(cz+d) where ad-bc ≠ 0. Maps circles to circles or lines.

Image of the mapping (w-plane):

Domain coloring shows how the function maps the z-plane to the w-plane

Key Properties:

  • •Preserves circles and lines (generalized circles)
  • •Forms a group under composition
  • •Three points determine a Möbius transformation
  • •Maps upper half-plane ↔ unit disk

Example Application:

This example maps upper half-plane to unit disk

🎯 What Makes It Conformal?

A function f is conformal at a point z₀ if:

  1. f is analytic at z₀
  2. f'(z₀) ≠ 0 (non-zero derivative)

This ensures that angles between curves are preserved at z₀. The mapping may rotate and scale, but the angle between any two intersecting curves remains the same!

Note: At critical points where f'(z) = 0, conformality breaks down. For example, z² has a critical point at z = 0 where angles are doubled rather than preserved.

2. Interactive Grid Transformer

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

What to Look For:
  • 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)
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.

3. 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.

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!

4. Joukowsky Airfoil Transformation

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

The Joukowsky Transform: w = z + 1/z

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!

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 camber (curvature)
Controls angle of attack
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 left/right changes the airfoil curvature
  • Thickness: Larger radius = thicker airfoil
  • 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!

Why Conformal Mappings Matter

1. Angle Preservation: Every analytic function with f'(z) ≠ 0 is conformal. This deep connection between analysis and geometry is unique to complex analysis!

2. Problem Simplification: Transform hard domains into simple ones. Solve Laplace's equation on a weird shape? Map it to a disk first!

3. Physical Applications: Electrostatic potentials, heat flow, fluid dynamics, and aerodynamics all use conformal mappings to solve real-world problems.

4. Möbius Magic: Möbius transformations map circles to circles (or lines). They form a group and are the automorphisms of the Riemann sphere!

5. Riemann's Insight: The Riemann Mapping Theorem is an existence result - every simply connected domain is "the same" as the disk, conformally speaking.

6. Engineering Impact: The Joukowsky transform literally helped design the first aircraft wings. Complex analysis enabled flight!

Conformal mappings are where the beauty of mathematics meets the power of engineering!