Does anyone know how to parametrize the boundary of the Mandelbrot set? I am not a fractal-geometer or a dynamical systems person. I just have some idle curiosity about this question.

The Mandelbrot set is customarily defined as the set $M$ of all points $c\in\mathbb{C}$ such that the iterates of the function $z\mapsto z^2+c$, starting at $z=0$, remain bounded forever. Most very pretty depictions of the Mandelbrot set show $M$ as an intersection of an infinite sequence of sets $M_1\supset M_2\supset M_3\supset\cdots$, where the boundary of $M_i$ is the curve $|z_i(c)|=K$. Here $z_i(c)$ is the $i$th iterate of $z\mapsto z^2+c$, starting at $z=0$, and $K$ is some constant which guarantees that future iterates will escape. These curves $\partial (M_i)$ guide the viewer to see the increasingly intricate parts of the Mandelbrot set.

Each of these curves $\partial(M_i)$ is analytic and closed. They can thus be parametrized nicely with a trigonometric series. To be more specific, each boundary has a parametrization of the form $$z(t)=\sum_{k=0}^\infty a_k\cos(kt)+i\sum_{k=0}^\infty b_k\sin(kt).$$ (In fact, since each boundary $\partial(M_i)$ is determined by a polynomial equation in the real and imaginary parts of $c$, I think each of these series should terminate. Correct me if I am wrong.) I would think that the limiting path should also have some nice parametrization with a trigonometric series. Is this limit the same for all $K$? If the limit is not the same for all $K$, then is there a limit as $K\rightarrow\infty$? What are the Fourier coefficients?