Pathfinding in the Mandelbrot set

[MathJason]

Ok, I see now how to symbolically generate a path that can then be converted into a numerical path.  By symbolic, I mean internal angled addresses, rational exterior angles, and/or binary exterior angles.  By numerical, I mean, e.g., floating point complex numbers.

- Input the endpoints A and B of the path in symbolic form.
- Find the external angles associated with A and B. (Choose the pair theta_A, theta_B  with minimal angular distance).
- Enumerate the “largest” rational external angles between A and B (probably using lowest common denominator, but maybe there is a better measure of largeness).  Also enumerate the equivalences between these angles.
- Find the shortest path through these external angles (where one can “jump” to an equivalent external angle if possible).  I think this is quite simple: just go to the next rational angle in the ordered list and jump to the furthest equivalent angle if possible.  (If say, A is the 0 angle, then we are basically constructing a finite approximation of the long internal address, except that we are tracing along the boundary instead of jumping between interior components.)
- Finally, take this symbolic path and convert it to a numerical path.  (Claude said this is doable using a combination of ray tracing and Newton’s method.)

If the input is given in numerical form then first find a close symbolic approximation.  If the gaps in the final path are too big (say, more than a pixel apart), then redo the symbolic part with more angles to fill in the gaps.

Steps 1 through 4 are symbolic and likely quite fast.  Step 5 might be really computationally intensive; I am not sure.  If step 5 does take a long time, one could combine this method with my previous method.  Namely, start by drawing the M-set using DE in a high resolution, use the above method to find a few important points along the path, and then use pathfinding on the DE image data to fill in the rest of the path.

If one would prefer the path to follow the internal angles through the interior of the M-set, that is fairly easy to do as well.  Just symbolically associate points on the boundary of an interior component with a symbolic representation of that component.  Then follow the path through the component given by the internal angles.

To be clear, I don’t know if I am going to try an implement this.  It seems like all the ingredients are there in, say, Claude’s code base.  However, I don’t really know how to program in C or Haskell, and I feel like thinking about this project has taken up a bit too much of my time already.  (But I admit, it is fun to think about.)