Skip to content

Conformal Map Explorer

Learning goal

See what analyticity does to the plane: an analytic map bends the coordinate grid, but wherever \(f'(z) \neq 0\) the red and blue grid lines still cross at exactly 90° — angle preservation is conformality, and it is the geometric face of the Cauchy–Riemann equations.

Things to try

  1. \(w = z^2\) — hover along the real axis toward the origin: angles double at \(z = 0\), the one point where \(f' = 0\).
  2. \(w = e^z\) — the Cartesian grid becomes a polar grid: horizontal lines → rays, vertical lines → circles. This is why \(e^z\) solves strip-shaped boundary-value problems.
  3. Joukowski \(w = z + 1/z\) — the map behind airfoil theory; watch the region near \(z = \pm 1\) fold into sharp edges.
  4. Möbius \((z-1)/(z+1)\) — the right half-plane becomes the unit disk; circles map to circles, always.

Conformal mapping · Analytic functions · Potential-flow application (PC316)