Complex Locus - Mathematics.lu

In maths, we are often interested in understanding what rule governs a certain condition. Today’s lesson focuses on that very topic in the context of complex numbers.

Example: What can be said about all the complex numbers $z$ satisfying that: $\arg(z) = 0$ ?

Solution: The answer is that they must be positive real numbers! If they form 0 degrees with the real axis, then the imaginary part is zero ( $y = 0$ ), which means $z = x + i \cdot 0 = x$ . Since they point along the positive side of the real axis, $x$ must be greater than zero, so $z \in \mathbb{R}^+$ .

Definition (Complex Locus): A complex locus is a geometric path, curve, or region formed in the complex plane by a set of points whose coordinates all satisfy a specific algebraic condition or equation involving a complex variable $z$ .

Instead of mapping standard $x$ and $y$ variables, a complex locus uses geometric constraints based on lengths, distances, and angles from a fixed point. Common examples include:

Circles: Defined by equations like $|z - z_1| = r$ , representing all points $z$ whose distance from a fixed center point $z_1$ is exactly equal to a radius $r$ .
Lines and Rays: Defined by equations like $\arg(z - z_0) = \theta$ , representing a straight line ray starting at a point $z_0$ and shooting outward at a constant fixed angle $\theta$ .
Perpendicular Bisectors: Defined by equations like $|z - z_1| = |z - z_2|$ , representing the straight line of all points that are perfectly equidistant from two distinct complex anchor points $z_1$ and $z_2$ .

Example: What is the locus of all points $z$ that have the same fixed modulus?

Solution: We know that the modulus is the distance from the origin. If we fix this distance, then all the points with this modulus create a circle with its center at the origin and a radius equal to the given modulus. Algebraically, this is expressed as $|z| = r$ , where $r$ is the constant modulus value.

Let us now show some of the examples that we mentioned above and why that is the case.

Table of Contents

Circles

An easy way to conclude this is by noticing that $|z_1-z_2|$ represents the distance between two complex numbers $z_1$ and $z_2$ . However, let us show formally how the locus is understood.

Claim (Geometric Locus of a Circle): Given a fixed complex number $z_1$ , the geometric locus of all points $z$ satisfying the equation $|z - z_1| = r$ (where $r > 0$ ) is a circle centered at the point $z_1$ with a radius of $r$ .

Example: Identify the center and radius of the circular locus described by the complex equation $|z - 3 + 4i| = 5$ . Sketch the locus on the complex plane.

Rays and Lines

Claim (Geometric Ray in the Complex Plane): The locus of points satisfying the complex equation $\arg(z - z_0) = \theta$ represents a half-line or ray that starts at the point $z_0$ and extends infinitely in the direction making a fixed angle $\theta$ with the positive real axis.

$\arg(z - z_0) = \theta \iff z = z_0 + re^{i\theta} \quad \text{for } r > 0$

Example: Determine the geometric locus of points in the complex plane defined by the equation $\arg(z + 1 - i) = -\frac{\pi}{3}$ , and describe its path parameters.

Step 1: Rewrite the equation into standard shift form
Factor out a negative sign inside the argument parenthesis to expose the true complex source coordinate point $z_0$ :

$\arg(z - (-1 + i)) = -\frac{\pi}{3}$

Step 2: Extract the geometric origin and vector direction angle
Identify the coordinate location where the ray begins and find the fixed angle trajectory of the shooting ray path:

$z_0 = -1 + i, \quad \theta = -\frac{\pi}{3}$

Step 3: TikZ Diagram Visual
Here is the TikZ code to generate the geometric representation of this complex ray locus:

Rendered by QuickLaTeX.com

Conclusion:
The locus is a straight half-line ray originating at the coordinate point $(-1, 1)$ and shooting outward infinitely at a downward angle of $60^\circ$ relative to the positive real axis direction line.

Bisectors

We finish by showing how to conclude the locus of a perpendicular bisector. The idea is the following, since we know that $|z-z_1|$ is the distance between $z$ and $z_2$ , the locus of all complex numbers $z$ such that $|z-z_1|=|z-z_2|$ is the set of points $z$ whose distance from $z_1$ is the same as that for $z_2$ . This is the definition of perpendicular bisector.

Claim (Perpendicular Bisector Locus): The locus of points satisfying the complex equation $|z - z_1| = |z - z_2|$ represents the perpendicular bisector of the line segment joining the two fixed complex points $z_1$ and $z_2$ .

$|z - z_1| = |z - z_2| \iff \text{dist}(z, z_1) = \text{dist}(z, z_2)$

Example: Determine the geometric locus of points in the complex plane defined by the equation $|z - 3| = |z - i|$ , and describe its equation parameters.

Step 1: Identify the anchor coordinates
Extract the two fixed reference endpoints $z_1$ and $z_2$ from the standard shift form layout parameters:

$z_1 = 3 \implies (3,0), \quad z_2 = i \implies (0,1)$

Step 2: Calculate the geometric line properties
Find the midpoint of the segment connecting $(3,0)$ and $(0,1)$ , then find the negative reciprocal slope: