Geometry and Locus of Complex Numbers

This section studies the geometric interpretation of a complex number $z$ in the complex plane, and how to convert a condition on $z$ (given in terms of $z,\overline z,|z|$ or $\arg z$) into an ordinary Cartesian equation in $x,y$ — the equation of the locus traced out by $z$.

Definition (circle). A circle is the locus of a point that moves in a plane such that its distance from a fixed point in that plane is always a constant. The fixed point is the centre, and the constant distance is the radius.

Complex form of the equation of a circle. Since $|z-z_0|$ is the distance from $z$ to the fixed point $z_0$, the locus of $z$ satisfying

consists of all points at distance $r$ from $z_0$ — exactly a circle with centre $z_0$ and radius $r$. Correspondingly:

• $|z-z_0|<r$ represents the points interior to the circle;
• $|z-z_0|>r$ represents the points exterior to the circle.

For instance, $|z|=r$ (i.e. $z_0=0$) gives $\sqrt{x^2+y^2}=r$, i.e. $x^2+y^2=r^2$: a circle centred at the origin, radius $r$. An equation such as $|\alpha z-\beta|=\gamma$ ($\alpha\ne0$) is first divided through by $|\alpha|$ and rewritten as $\left|z-\dfrac\beta\alpha\right|=\dfrac\gamma{|\alpha|}$, from which the centre $\dfrac\beta\alpha$ and radius $\dfrac\gamma{|\alpha|}$ can be read off directly.

General loci. More general conditions on $z$ also trace out recognisable curves once translated to Cartesian form:

• $|z-a|=|z-b|$ (equidistant from two fixed points $a,b$) always gives the perpendicular bisector of the segment joining $a$ and $b$ — a straight line, not a circle.
• Conditions phrased with $\operatorname{Re}(\cdots)$, $\operatorname{Im}(\cdots)$ or $\overline z$ (e.g. $[\operatorname{Re}(iz)]^2=3$, $\operatorname{Im}[(1-i)z+1]=0$, $\overline z=z^{-1}$) become ordinary Cartesian equations once $z=x+iy$ is substituted and the real/imaginary parts are separated — the result may be a line, a circle, or another simple curve.
• An argument condition, e.g. $\arg\!\left(\dfrac{z-i}{z+2}\right)=\dfrac\pi4$, fixes the angle subtended by the segment joining two fixed points, and typically traces an arc of a circle. …