Conformal Maps

Definition

A conformal map $w = f(z)$ is a transformation that preserves the angles between curves in both magnitude and orientation. Any analytic function $f$ is conformal wherever $f'(z) \neq 0$.

• How to read: “The function w equals f of z.”
• Meaning: A conformal map preserves angles between curves; any analytic $f$ is conformal wherever $f'(z) \neq 0$—locally it rotates and uniformly scales.

Why It Matters

They allow us to solve complex physics problems—like airflow over a wing—by mapping them to simpler shapes without losing geometric relationships.

Core Concepts

• Angle Preservation: If two lines intersect at $30^\circ$ in the $z$-plane, their mapped images intersect at $30^\circ$ in the $w$-plane.
• Local Isometry: Although the map distorts large regions, at an infinitesimal scale, it acts as a rotation and uniform scaling (similarity transform).
• Orthogonality: Conformal maps preserve the orthogonality of coordinate grids, making them ideal for transforming physical domains while keeping flux and potential lines perpendicular.
• Critical Points: Points where $f'(z) = 0$ are not conformal; angles are typically multiplied at these points.
  • How to read: “The derivative f prime of z equals zero.”
  • Meaning: Where the derivative vanishes, the map folds or stretches angles—conformality breaks down.

Connected Concepts

• Analytic Geometry
• Physics
• Differentials
• Limits of Complex Functions, Continuity of Complex Functions