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
By Geometrical Interpretation of Complex Subtraction:
\(\ds \arg \dfrac {z - b} {z - a}\) | \(=\) | \(\ds \lambda\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \arg {z - b} - \map \arg {z - a}\) | \(=\) | \(\ds \lambda\) | 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
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations: Example $3$.