External rays of Julia and Mandelbrot sets

Question

I am familiar with the JuliaSetPlot function of Mathematica, but I was wondering if it is possible to get Mathematica to plot external rays of some specified angles. I can see in the JuliaSetBoettcher there is an example but this isn't very good and does not seem immediately adaptable. Ideally I would like something of the form externalray[c_,j_] which would plot the external ray of angle j for the Julia set of $f_c(z)=z^2+c$. Is there some simple implementation of this?

Edit: A common program that can do this is Wolf Jung's Mandel. There is some unofficial documentation here. Perhaps this would be useful to someone with more programming experience.

Answer 1

Here's a start you can perhaps work on to make better. Sorry, kinda crude. Not very familiar with the background.

theC = -1.2564 + 0.3803 I;
jsp = JuliaSetPlot[theC, PlotRange -> {{-2, 2}, {-1,    
1}}];

jb = Flatten[
   Table[{x, y, 
   JuliaSetBoettcher[theC, x + I y, MaxIterations -> 
30]}, {y, -1, 
 1, .01}, {x, -2, 2, .01}], 1];
theRays = Array[# &, 20, {-Pi, Pi}];
rayTable = Table[
theRayPoints = 
Select[jb, (theta - 0.001) < Arg[#[[3]]] < (theta + 
0.001) &];
Graphics@Line@({#[[1]], #[[2]]} & /@ theRayPoints),
{theta, theRays}];
Show[{jsp, rayTable}]

Here's a link for background: external rays of Mandelbrot