Imagine we have a parametrization of a particle in 2D space like this
http://i.minus.com/iXL64EfdJe6w5.gif
How do we go about finding an explicit way to express these functions ($f(x)$ and $g(x)$) based on the angle $a$ (see the figure below)? (In the same way we get to express $\sin$ and $\cos$ in the unit circle using many ways)
https://i.stack.imgur.com/3UH0F.png
And what field of maths can answer those questions? (I guess it's diff. geometry) Can we always find explicit ways to express these functions?
To solve this it'll be extremely useful to know how to add and subtract vectors, and how to multiply them by scalars. If you don't know how to do this yet, it's not hard, and now is a great time to learn!
Let's order the circles in descending order of size with radii $R_0>R_1>R_2>R_3$. Suppose circle $i$ makes one rotation around its containing circle every $2\pi\omega_i$ time units.
Let $\vec{c}_i(t)$ be the location of circle $i$'s center at time $t$.
It's not hard to see that $\vec{c}_1(t)=(R_0-R_1)(\cos(\omega_1 t),\sin(\omega_1 t))$, since the center of the larger circle just traces out another circle of radius $R_0-R_1$.
It's also not terribly hard to see that $$\vec{c}_2(t)=\vec{c}_1(t) + (R_2-R_1)(\cos(\omega_2 t), \sin(\omega_2 t))$$ and
$$\vec{c}_3(t)=\vec{c}_2(t) + (R_3-R_2)(\cos(\omega_3 t), \sin(\omega_3 t))$$.
Back-substituting gives us a nice parametrization of $\vec{c}_3(t)$, which is what we were after.