Equation of Circular Arc in Complex Plane

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Theorem

Let $a, b \in \C$ be complex constants representing the points $A$ and $B$ respectively in the complex plane.

Let $z \in \C$ be a complex variable representing the point $Z$ in the complex plane.

Let $\lambda \in \R$ be a real constant such that $-\pi < \lambda < \pi$.


Then the equation:

$\arg \dfrac {z - b} {z - a} = \lambda$

represents the arc of a circle with $AB$ as a chord subtending an angle $\lambda$ at $Z$ on the circumference.


Proof

Circular-Arc-in-Complex-Plane.png


By Geometrical Interpretation of Complex Subtraction:

$z - a$ represents the line from $A$ to $Z$
$z - b$ represents the line from $B$ to $Z$


\(\ds \arg \dfrac {z - b} {z - a}\) \(=\) \(\ds \lambda\)
\(\ds \leadsto \ \ \) \(\ds \map \arg {z - b} - \map \arg {z - a}\) \(=\) \(\ds \lambda\) Argument of Quotient equals Difference of Arguments


Thus:

$\arg \dfrac {z - b} {z - a} = \lambda$

represents the statement that the angle between $AZ$ and $BZ$ is constant:

$\angle AZB = \lambda$

That is, the angle subtended by $AB$ at $Z$ is $\lambda$.

The result follows from the Inscribed Angle Theorem.

$\blacksquare$


Sources