What is the shape in the complex plane generated by all possible points $z_1 + z_2$, where $z_1$ and $z_2$ can be any two points on the unit circle?

Question

What is the shape in the complex plane generated by all possible points $z_1 + z_2$, where $z_1$ and $z_2$ can be any two points on the unit circle centered at $0$

Answer 1

First take any point $z_1$ on the unit circle. Now adding all points from the unit circle to it is equivalent to drawing a new circle of radius 1 around that point (by viewing them as vectors).

Doing this for all points $z_1$ on the unit circle will "smear" new circles centered on these points, around the origin, thus filling the disk of radius 2 centered at the origin.

Answer 2

We start by noting that $|z_1+z_2|\leqslant|z_1|+|z_2|=2$. Therefore, the desired locus lies on or within the circle with radius $2$, centered at the origin.

We prove that every point $z$ with $|z|\leqslant2$ can be written as $z=z_1+z_2$, where $z_1$ and $z_2$ are points on the unit circle. Write $z=re^{i\theta}$, with $0\leqslant r\leqslant2$ and $0\leqslant \theta \lt 2\pi$. Then, we can write $r=2\cos\alpha$, where $0\leqslant\alpha\leqslant\pi$ is uniquely determined. Now, take $z_1=e^{i(\theta-\alpha)} $ and $z_2=e^{i(\theta+\alpha)}$; clearly these are points on the unit circle. Then, $z=z_1+z_2$.

It follows that the desired locus is the disk centered at the origin with radius $2$.