📂Complex Anaylsis

Conformal Mappings Preserve the Magnitude of Interior Angles

Theorem ¹

Let function $f$ be a conformal mapping on a complex domain $\mathscr{R}$, and suppose curves $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ meet at a point $\alpha$ with interior angle $\psi$.

If $\mathscr{C}_{1} ' $ and $\mathscr{C}_{2} ' $ denote the images of $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ under $f$, then the two curves meet at $\beta = f ( \alpha )$ and their interior angle is likewise $\psi$.

Explanation

As is typical of analysis, the statement is worded in a difficult way, but the point is that a conformal mapping preserves the interior angles formed by figures. Indeed, the very name "conformal mapping" originates from this property.

Meanwhile, a mapping that preserves the magnitude of angles but reverses their sign is called an isogonal mapping.

Proof

$f$ is a conformal mapping that sends $z = x + iy$ to $w = u + iv$.

Let the magnitude of the interior angle formed by $\mathscr{C}_{1}$ and the $x$-axis be $\psi_{1}$, and let a point on $\mathscr{C}_{1}$ be $z_{1}$. Similarly, let the magnitude of the interior angle formed by $\mathscr{C}_{2}$ and the $x$-axis be $\psi_{2}$, and let a point on $\mathscr{C}_{2}$ be $z_{2}$. Then the interior angle formed by $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ will be $\psi_{2} - \psi_{1} = \psi$.

$$ z - \alpha := r e^{i \theta_{1}} \\ z_{2} - \alpha = r e^{i \theta_{2}} $$ Setting it this way, when $r \to 0$ we have $$ \theta_{1} \to \psi_{1} \\ \theta_{2} \to \psi_{2} $$ Meanwhile, letting $w_{k}: = f(z_{k})$, $$ w_{1} - \beta = R_{1} e^{i \phi _{1}} \\ w_{2} - \beta = R_{2} e^{i \phi _{2}} $$ Since by assumption $f ' (\alpha) \ne 0$ exists, for $\rho > 0$ we may set $f ' (\alpha) = \rho e^{ i \lambda }$.

$$ f ’ ( \alpha) = \lim_{z_{1} \to \alpha } {{w_{1} - \beta } \over {z_{1} - \alpha }} = \lim_{z_{1} \to \alpha} {{R_{1}} \over {r}} e^{ i ( \phi_{1} - \theta_{1} )} = \rho e^{ i \lambda } $$ Thus $$ \lim_{z_{1} \to \alpha } (\phi_{1} - \theta_{1}) = \lambda $$ and accordingly we obtain $$ \lim_{w_{1} \to \beta } \phi_{1} = \psi_{1} + \lambda \\ \lim_{w_{2} \to \beta } \phi_{2} = \psi_{2} + \lambda $$ Therefore the magnitude of the interior angle formed by $\mathscr{C}_{1} ' $ and the $u$-axis is $\psi_{1} + \lambda$, and the magnitude of the interior angle formed by $\mathscr{C}_{2} ' $ and the $u$-axis is $\psi_{2} + \lambda$. Finally, the interior angle formed by $\mathscr{C}_{1} ' $ and $\mathscr{C}_{2} ' $ has the $\lambda$ terms cancel out, giving $$ (\psi_{2} + \lambda) - (\psi_{1} + \lambda) = \psi_{2} - \psi_{1} = \psi $$

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