The locus of points of the form $ae^{i\theta}+be^{i\phi}$

Question

Let $a$ and $b\in\mathbb{R}^{>0}$ be two positive real numbers. What is the locus of points of the form $ae^{i\theta}+be^{i\phi}$ where $\theta$ and $\phi\in[-\pi,\pi)$? Does it have any specific name?

Answer 1

WLOG, $a\le b$. $$ae^{i\theta}$$ is a circle centered at the origin and of radius $a$.

Now the sum

$$ae^{i\theta}+be^{i\phi}$$ corresponds to a circle of radius $b$, the center of which is swept along the first circle. Hence the locus is a filled ring of inner diameter $a-b$ and outer diameter $a+b$. (See @achille's comment.)