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Sample point z
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Image w = f(z)
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|w|
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arg(w)
Conformal map w = z² (morphing identity ⇄ z²)
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Simultaneously visualize |f(z)| and argument via domain coloring. Interactively explore conformal mapping, Joukowski transform, and residue calculation in real time.
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The fundamental condition for a complex function $f(z) = u(x,y) + i v(x,y)$ to be holomorphic (and thus generate a conformal map where $f'(z) \neq 0$) is given by the Cauchy-Riemann equations. These couple the real and imaginary parts.
$$ \frac{\partial u}{\partial x}= \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x}$$$u(x,y)$: Real part of the complex function (scalar potential).
$v(x,y)$: Imaginary part of the complex function (stream function or orthogonal potential).
$x, y$: Real and imaginary parts of the input $z = x + iy$.
For applications in potential flow, we define the Complex Potential. Its derivative gives the fluid velocity, and its real and imaginary parts define equipotential lines and streamlines, which are always orthogonal—a direct result of conformality.
$$ \Phi(z) = \phi(x,y) + i \psi(x,y), \quad \text{with }\frac{d\Phi}{dz} = u_x - i u_y $$$\phi$: Velocity potential (equipotential lines).
$\psi$: Stream function (streamlines).
$u_x, u_y$: $x$- and $y$-components of the fluid velocity vector. The orthogonality of the grid in the simulator visually represents this perpendicular relationship between $\phi$ and $\psi$.
Aerodynamic Airfoil Design: As shown with the Joukowski transform in the simulator, conformal mapping is a classical method for designing airfoil profiles. Engineers analyze the lift and flow around a simple circle and then map the solution to a wing shape, allowing for rapid preliminary design before more complex CFD simulations.
Control System Stability (Nyquist Plot): In control theory, the Nyquist stability criterion uses a conformal mapping. The plot of a system's open-loop transfer function is analyzed, and the number of times it "winds around" a critical point (its winding number, related to the argument principle) tells engineers if the closed-loop system will be unstable.
Electrostatic & Heat Conduction Field Solving: The electric potential in a 2D electrostatic field or the temperature in steady-state heat conduction obeys Laplace's equation. Conformal maps can transform a complicated geometry (like a capacitor with irregular plates) into a simpler one (parallel plates), making the solution for potential and field lines straightforward.
Cartography & Geographic Mapping: While not perfectly conformal at large scales, map projections like the Mercator projection aim to preserve local angles (rhumb lines). This is crucial for navigation, as a constant compass bearing appears as a straight line on the map, analogous to the angle-preserving property you see in the simulator's grid.
When you start using this tool, there are a few common pitfalls. First, you might think "the domain coloring represents everything," but the color change represents the function's "argument" (phase), while the magnitude (absolute value) of the function itself is shown by brightness. So, a completely black area simply means the absolute value is near zero (the function value is small), not necessarily a singularity. Singularities often appear as a messy "vortex" where all colors of the hue circle converge at a single point.
Next, tips for parameter settings. Increasing the "Grid Line Count" too much can make the transformed shape overly complex, making it harder to grasp the overall picture. It's more efficient to start with a smaller number (e.g., around 10) to understand the general flow, then zoom in on areas of interest and increase the grid lines. Also, adjusting "Zoom" and "Contour Circle Radius" simultaneously can make it unclear what's happening. When you want to observe the behavior of the circular integral, keep the zoom fixed and slowly change only the radius.
A practical pitfall is understanding that "the mapping is not a panacea." For example, when creating an airfoil shape with the Joukowsky transform, shifting the center of the original circle just slightly can drastically change the airfoil's thickness or camber. Even if you create a beautiful airfoil shape in the tool, it's not necessarily optimal for real aerodynamic characteristics. Use it strictly as a theoretical starting point.
For the Joukowski airfoil transform f(z) = z + 1/z with zoomR = 1.5 and gridN = 80: A circle of radius r = 1.2 in the z-plane maps to a Joukowski airfoil in the w-plane. Poles occur at z = ±1 (simple poles with residue ±1 each). Setting contourR = 0.2 displays level curves in 0.2-unit increments. The winding number around the airfoil trailing edge is +1, confirming one pole encirclement. Domain coloring reveals the characteristic cusp singularity mapping.