Geometry with Complex Numbers — Lines

Plot the points 1 + i and 4 + 3i on the complex plane. There is exactly one straight line through them. In Cartesian coordinates you would write the slope-intercept form and juggle x- and y-coordinates separately. But each of these points is a single complex number -- can you write the equation of the line as a single complex equation too?

You can. The result is not just a notational shortcut: it gives you parallelism, perpendicularity, and reflection of a point across a line, each in a compact formula that would take several separate steps in coordinate geometry.

Collinearity — the starting idea

Three points z, z_1, z_2 on the complex plane are collinear (they lie on the same straight line) precisely when the "direction" from z_1 to z is the same as the direction from z_1 to z_2. In complex number language, this means the ratio

is a real number. A complex ratio is real exactly when its imaginary part is zero — equivalently, when the ratio equals its own conjugate.

Here is the key move. A complex number w is real if and only if w = \bar{w}. So the collinearity condition is:

Cross-multiplying:

This is the equation of the line through z_1 and z_2, written purely in terms of z and \bar{z}.

There is a second way to write the collinearity condition, using a determinant. The three points z, z_1, z_2 are collinear if and only if

Expanding the determinant gives the same equation as the cross-multiplication above. The determinant form is useful for quick verification — if you plug in three specific points and get zero, they are collinear.

The equation of a line through two points

Take a concrete case. The line through z_1 = 1 + i and z_2 = 3 + 5i. Here z_2 - z_1 = 2 + 4i and \bar{z}_2 - \bar{z}_1 = 2 - 4i. The equation is:

Expanding both sides:

After simplification (both sides have -4 from 4i^2 and the constant terms partially cancel), you can verify this reduces to a linear equation in z and \bar{z}. The general form of that linear equation is the topic of the next section.

The general equation of a line

Every line on the complex plane can be written in the form:

Why must b be real? Take the conjugate of the entire equation: \overline{\bar{a}z + a\bar{z} + b} = a\bar{z} + \bar{a}z + \bar{b} = 0. This is the same equation as the original (just with the first two terms swapped) only if \bar{b} = b, which means b is real.

The complex number a determines the direction of the line. Specifically, a is perpendicular to the line — it is the complex analogue of the normal vector. The line is parallel to the direction ia (rotating a by 90°).

To see the connection to Cartesian coordinates, write a = \alpha + \beta i and z = x + yi. Then:

So \bar{a}z + a\bar{z} + b = 0 becomes 2\alpha x + 2\beta y + b = 0, which is the familiar Cartesian equation of a line. The real and imaginary parts of a are (up to a factor of 2) the coefficients of x and y.

Parallel and perpendicular lines

Two lines on the complex plane have equations \bar{a}_1 z + a_1 \bar{z} + b_1 = 0 and \bar{a}_2 z + a_2 \bar{z} + b_2 = 0. The normal directions are a_1 and a_2.

Parallel lines. Two lines are parallel when their normals point in the same (or opposite) direction. This means a_1 / a_2 is a real number — the two normals differ only by a real scaling factor.

Equivalently, a_1 \bar{a}_2 = \bar{a}_1 a_2, because a_1/a_2 is real if and only if a_1/a_2 = \overline{a_1/a_2} = \bar{a}_1/\bar{a}_2, which gives a_1 \bar{a}_2 = \bar{a}_1 a_2.

Perpendicular lines. Two lines are perpendicular when their normals are at 90° to each other. Rotating a_2 by 90° gives ia_2, and the condition for a_1 to be parallel to ia_2 is that a_1/(ia_2) is real.

The slickest form: the lines are perpendicular when a_1\bar{a}_2 + \bar{a}_1 a_2 = 0.

Why? The expression a_1\bar{a}_2 + \bar{a}_1 a_2 = 2\operatorname{Re}(a_1\bar{a}_2). This equals zero precisely when a_1\bar{a}_2 is purely imaginary — which happens exactly when the angle between a_1 and a_2 is 90°.

Here is a quick summary.

Reflection of a point in a line

Given a line L: \bar{a}z + a\bar{z} + b = 0 and a point z_0 not on L, the reflection z_0' of z_0 in L is the point such that L is the perpendicular bisector of the segment from z_0 to z_0'.

The formula is:

Here is how it is derived. Two conditions define the reflection:

Condition 1: the midpoint lies on L. The midpoint of z_0 and z_0' is \frac{z_0 + z_0'}{2}. Substituting into \bar{a}z + a\bar{z} + b = 0:

Condition 2: the segment z_0 z_0' is perpendicular to L. The direction of the segment is z_0' - z_0. The direction of the line is ia (since a is the normal). For the segment to be perpendicular to the line, it must be parallel to a — meaning (z_0' - z_0)/a is real, so z_0' - z_0 = ta for some real t.

Alternatively, perpendicularity to the line means the segment is along the normal direction, and the condition simplifies to: (z_0' - z_0)/a is real.

Combining both conditions and solving (using \bar{a}z_0' + a\bar{z}_0' + b = 0 as the requirement that z_0' lies on the line equidistant on the other side), the reflection formula emerges:

Parametric form of a line

There is one more way to describe a line that is often useful. A line through z_1 in the direction of a complex number d (where d \neq 0) can be written as:

As t ranges over all real numbers, z traces out the entire line. This is the complex version of the parametric form from coordinate geometry. The real parameter t slides the point along the direction d.

This form is convenient when you know a point and a direction. For instance, the line through z_1 = 2 + i in the direction d = 1 + i (a 45° direction) is z = (2 + i) + t(1 + i) = (2 + t) + (1 + t)i. Setting t = 0 gives z_1; setting t = 1 gives 3 + 2i; setting t = -1 gives 1.

An interactive exploration

Drag the red point along the line below. The readout shows the value of \bar{a}z + a\bar{z} + b for the line with a = 1 + i and b = -4, which simplifies to the equation x + y = 2. Watch how the expression stays at zero for every position on the line.

Worked examples

Common confusions

• "The equation \bar{a}z + a\bar{z} + b = 0 has two unknowns, z and \bar{z}." There is only one unknown: z. Once you know z, you know \bar{z} — it is not independent. The equation is a single real condition on the single complex unknown z, which is why it describes a one-dimensional set (a line), not a zero-dimensional set (a point).

• "b can be any complex number." No. Taking the conjugate of the entire equation shows that b must be real. If b were complex, the equation and its conjugate would be two different equations, and the locus would be a single point (or empty), not a line.

• "Two lines are perpendicular when a_1/a_2 = i." Close, but not exactly. The condition a_1/a_2 = i is sufficient but not necessary — it means a_1 is exactly i times a_2. The full condition is \operatorname{Re}(a_1\bar{a}_2) = 0, which allows a_1/a_2 to be any purely imaginary number, not just i.

• "The reflection formula works even when z_0 is on the line." It does — but the output is z_0 itself. If z_0 lies on L, then \bar{a}z_0 + a\bar{z}_0 + b = 0, and substituting into the formula gives z_0' = z_0. The formula is consistent; it just returns a trivial reflection.

• "Collinearity requires three distinct points." The collinearity condition \frac{z - z_1}{z_2 - z_1} \in \mathbb{R} is defined for any z (including z = z_1 or z = z_2, where the ratio is 0 or 1, both real). What you do need is that z_1 \neq z_2, since two distinct points are required to determine a line.

Going deeper

Distance from a point to a line

The perpendicular distance from a point z_0 to the line \bar{a}z + a\bar{z} + b = 0 is:

This is the complex-plane version of the Cartesian distance formula |Ax_0 + By_0 + C|/\sqrt{A^2 + B^2}. The factor of 2 in the denominator comes from the fact that \bar{a}z + a\bar{z} = 2(\alpha x + \beta y) when a = \alpha + \beta i.

The line at infinity

In projective geometry, parallel lines "meet at infinity." On the complex plane, this idea connects to the extended complex plane (the Riemann sphere), where the point at infinity \infty is added. Two parallel lines — which have the same normal a up to a real scalar — meet at a single point on the Riemann sphere. This perspective unifies lines and circles: a line is a circle passing through \infty.

Lines and Mobius transformations

A Mobius transformation w = \frac{az + b}{cz + d} maps lines and circles to lines and circles. Understanding the equation of a line in complex form is the first step towards studying these transformations, which appear in complex analysis, hyperbolic geometry, and even the study of electrical circuits.

Where this leads next

Lines are one piece of the geometry of the complex plane. The next steps add curves and motion.

• Geometry with Complex Numbers — Circles — the equation |z - z_0| = r and its general form, where the modulus gives you circles as naturally as the collinearity condition gives you lines.
• Loci in Argand Plane — standard locus problems: perpendicular bisectors, constant-angle curves, ellipses, and rotations.
• Geometry with Complex Numbers — Basics — the foundational ideas (distance, midpoint, section formula) that this article builds on.
• Straight Lines — the Cartesian treatment of lines, to compare with the complex approach.
• Coordinate Geometry Basics — distance formula, section formula, and area of a triangle in Cartesian coordinates.