Loci in Argand Plane

Take the condition |z - 2| = 3. You know from Geometry with Complex Numbers — Circles that this is a circle with centre 2 and radius 3. But what about |z - 1| = |z + 1|? Or \arg(z - i) = \pi/4? Or |z - 1| + |z + 1| = 4?

Each of these is a constraint on z. The set of all z that satisfies the constraint is a locus — a curve (or sometimes a region) on the Argand plane. The word comes from Latin: locus means "place." The locus is the place where z is allowed to live.

The power of complex numbers is that many loci that take several lines to describe in Cartesian coordinates collapse into a single equation involving |z|, \arg(z), or e^{i\theta}. This article catalogues the standard loci and shows how to decode each one.

Locus 1: |z - z_1| = |z - z_2| — the perpendicular bisector

The condition says: the distance from z to z_1 equals the distance from z to z_2. The set of all points equidistant from two fixed points is the perpendicular bisector of the segment joining them.

To verify algebraically: square both sides of |z - z_1| = |z - z_2|.

Expanding both sides and cancelling the z\bar{z} term:

Rearranging:

This is a linear equation in z and \bar{z} — the equation of a line, as expected. The line passes through the midpoint \frac{z_1 + z_2}{2} and is perpendicular to the direction z_2 - z_1.

For the specific case z_1 = -2, z_2 = 2: the equation becomes (2 - (-2))z + (2 - (-2))\bar{z} = 4 - 4, so 4z + 4\bar{z} = 0, giving z + \bar{z} = 0, which means 2x = 0, so x = 0 — the imaginary axis.

Locus 2: \arg(z - z_1) = \theta — a ray

The condition \arg(z - z_1) = \theta says: the argument (angle) of the vector from z_1 to z is fixed at \theta. Geometrically, this means z lies on a ray starting from z_1, making angle \theta with the positive real direction.

The locus is a half-line (ray), not a full line. The point z_1 itself is excluded (the argument is undefined at the origin of the vector). Points on the opposite ray (which would have argument \theta + \pi or \theta - \pi) do not satisfy the condition.

A related condition: \arg(z - z_1) - \arg(z - z_2) = \theta, or equivalently \arg\left(\frac{z - z_1}{z - z_2}\right) = \theta. This is the constant angle condition. The locus is an arc of a circle passing through z_1 and z_2, because the inscribed angle theorem says that all points on an arc subtend the same angle at the chord's endpoints. When \theta = \pi/2, the locus is a semicircle with z_1 z_2 as diameter.

Locus 3: |z - z_1| + |z - z_2| = k — an ellipse

The sum of the distances from z to two fixed points z_1 and z_2 is constant. This is the definition of an ellipse with foci z_1 and z_2.

The constant k must satisfy k > |z_1 - z_2| — the sum of distances must exceed the distance between the foci, otherwise no point can satisfy the condition (the triangle inequality would be violated).

When k = |z_1 - z_2|, the "ellipse" degenerates to the line segment from z_1 to z_2.

Similarly, \bigl||z - z_1| - |z - z_2|\bigr| = k (with 0 < k < |z_1 - z_2|) gives a hyperbola with foci z_1 and z_2. Without the absolute value, |z - z_1| - |z - z_2| = k gives one branch of the hyperbola (the branch closer to z_2).

Locus 4: \left|\dfrac{z - z_1}{z - z_2}\right| = k — the circle of Apollonius

When k \neq 1, this is a circle. The locus is the set of all points whose distances to z_1 and z_2 are in a fixed ratio k. The circle is called the circle of Apollonius.

When k = 1, you get |z - z_1| = |z - z_2|, which is the perpendicular bisector — a line, the degenerate case of a circle with infinite radius.

To find the centre and radius, square both sides:

Expand:

When k \neq 1, the z\bar{z} terms do not cancel, and rearranging gives a circle in general form.

Take a concrete case: z_1 = 0, z_2 = 4, k = 2. The condition is |z| = 2|z - 4|.

A circle with centre \frac{16}{3} and radius \frac{8}{3}.

Rotation: z' = ze^{i\theta}

Multiplying a complex number by e^{i\theta} = \cos\theta + i\sin\theta rotates it by angle \theta about the origin. The modulus stays the same: |z'| = |z||e^{i\theta}| = |z|. Only the argument changes: \arg(z') = \arg(z) + \theta.

This is one of the deepest facts about complex multiplication. Multiplying by a unit complex number e^{i\theta} is rotation. Multiplying by a general complex number w is rotation by \arg(w) followed by scaling by |w|.

Rotation about a general point

To rotate z by angle \theta about a point z_0 (not the origin):

1. Translate so that z_0 goes to the origin: w = z - z_0.
2. Rotate: w' = we^{i\theta}.
3. Translate back: z' = w' + z_0.

Combining: z' - z_0 = (z - z_0)e^{i\theta}, or equivalently:

This is the general rotation formula on the complex plane. It is far more compact than the Cartesian rotation formula, which involves separate expressions for the new x' and y'.

An interactive exploration

Drag the red point below. It is constrained to move on the locus |z - 2| = |z + 2| — the perpendicular bisector of the segment from -2 to 2. The readouts show the distances from the point to both fixed points, confirming they are always equal.

Standard loci summary

Here is a reference table of the loci you are most likely to encounter.

Worked examples

Common confusions

• "|z - z_1| = |z - z_2| gives a circle." No — it gives a line (the perpendicular bisector). The z\bar{z} terms cancel when you square both sides because the coefficient is 1 on both sides. You need unequal coefficients (like |z - z_1| = k|z - z_2| with k \neq 1) to get a circle.

• "\arg(z - z_1) = \theta gives a full line through z_1." It gives only a ray (half-line). The full line would include the opposite direction, where the argument is \theta \pm \pi. The argument function picks out a single direction.

• "The ellipse |z - z_1| + |z - z_2| = k exists for all k > 0." It exists only when k > |z_1 - z_2| (the triangle inequality). If k = |z_1 - z_2|, the ellipse collapses to the segment z_1 z_2. If k < |z_1 - z_2|, no point can satisfy the condition.

• "Rotation about the origin is z' = z + e^{i\theta}." That is translation, not rotation. Rotation is z' = ze^{i\theta} — multiplication, not addition. Multiplication by a unit complex number rotates; addition of a complex number translates.

• "The circle of Apollonius passes through both z_1 and z_2." Generally, it does not. At z = z_1, the ratio |z - z_1|/|z - z_2| is 0, not k (unless k = 0, which is degenerate). At z = z_2, the ratio is undefined (division by zero). The two fixed points are not on the Apollonius circle.

• "\arg\left(\frac{z - z_1}{z - z_2}\right) = \pi means z is on a full line through z_1 and z_2." Not quite. The condition \arg\left(\frac{z - z_1}{z - z_2}\right) = \pi means the vectors from z_2 to z and from z_1 to z point in "opposite directions" in a specific sense — z lies on the segment between z_1 and z_2 (the arc degenerates to the chord).

Going deeper

The inscribed angle theorem via complex numbers

The locus \arg\left(\frac{z - z_1}{z - z_2}\right) = \theta being an arc is a restatement of the inscribed angle theorem: all points on an arc of a circle subtend the same angle at the chord. The complex number formulation makes the proof a one-liner:

If \arg\left(\frac{z - z_1}{z - z_2}\right) = \theta, then the ratio (z - z_1)/(z - z_2) has fixed argument \theta, meaning it lies on a ray from the origin in the complex w-plane. The inverse map z = (z_1 - wz_2)/(1 - w) sends this ray to an arc of a circle in the z-plane. The inscribed angle theorem drops out as an algebraic consequence of the Mobius transformation.

Loci and transformations

Every locus equation can be viewed as the preimage of a simple set under a complex function. For example:

• |z - z_0| = r is the preimage of the circle |w| = r under w = z - z_0 (translation).
• |z - z_1|/|z - z_2| = k is the preimage of |w| = k under w = (z - z_1)/(z - z_2) (Mobius transformation).
• \arg(z - z_1) = \theta is the preimage of the ray \arg(w) = \theta under w = z - z_1.

This transformation perspective unifies all loci: find the right map, and the locus becomes obvious.

The spiral similarity

A map of the form z' = \alpha z + \beta (with \alpha \neq 0) is called a spiral similarity — it combines rotation by \arg(\alpha), scaling by |\alpha|, and translation by \beta. When |\alpha| = 1, it is a pure rotation plus translation (a rigid motion). These transformations preserve angles and map circles/lines to circles/lines. Understanding them is key to competition-level geometry problems that use complex coordinates.

Where this leads next

The loci in this article are the geometric consequences of modulus and argument conditions. The next steps explore the algebraic and analytical sides.

• Geometry with Complex Numbers — Lines — the equation \bar{a}z + a\bar{z} + b = 0 and the tools for parallel, perpendicular, and reflection.
• Geometry with Complex Numbers — Circles — the equation |z - z_0| = r and its general form, the building blocks for circular loci.
• Argument of Complex Number — the angle that determines the ray locus and the constant-angle arc.
• Polar Form of Complex Numbers — writing z = re^{i\theta}, which is the natural language for rotation.
• Conic Sections — Introduction — the Cartesian treatment of ellipses and hyperbolas that appear here as complex loci.