Evaluating position on Mandelbrot set boundary

[Furan]

I'm not a mathematician but I was wondering, is there a way to assign a value of parameter t (from 0 to 1 with 0 = 1 on some important point of the M-set boundary) to any point z (or its close outside projection) of the M-set boundary at iteration n so that its inverse z_bound(t,n) is a continuous function?
I'm really stretching my English/Math skills here. Let me know whether you understand.

We could use that function to animate a movement alongside the boundary and possibly get some intriguing results at very deep zoom.

[hobold]

This is sort of possible ... :
http://en.wikipedia.org/wiki/External_ray

But the rays don't seem to sample the border evenly (and I am unsure how to even define mathematically what I mean by that). I have yet to see practical algorithms that put external rays to good use.

[David Makin]

The problem is that calculation of the external rays - also known as field lines - breaks down at higher iterations.
You can use it in combination with either distance estimation or smooth iteration colouring to give x,y coords for mapping into the fractal - either calculated shapes/objects or indeed images used as source texture maps.
When done reasonably accurately this means you can get an "x" value or field-line "angle" from 0 to 360 around the "inside".

Examples:

http://makinmagic.deviantart.com/art/Potential-and-Field-Lines-103356230

And see UF parameters below.

Using the values in polar form:

http://makinmagic.deviantart.com/art/Fractals-are-hard-to-swallow-102574524

For more esoteric formulas than polynomials it's very difficult to get field lines anything like "correct", however the same technique can be achieved by using distance estimation values combined with an angle returned from slight additions to the DE calciulation - these angles do not match the field lines in the same way that DE doesn't match smooth iteration but the combination used for mapping can be effective:

http://makinmagic.deviantart.com/art/Magnetic-Blooms-104516876
http://makinmagic.deviantart.com/art/Newton-Makin-104248676

I've considered trying a similar technique to get colouring values for the surfaces of 3D+ fractals but always wimped out due to the extra complexity and the lack of definition of a 3D equivalent to trig - though I've been wondering if there's something similar using area ratios for irregular tetrahedrons...


UF example:

Code:

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[lkmitch]

I'm not a mathematician but I was wondering, is there a way to assign a value of parameter t (from 0 to 1 with 0 = 1 on some important point of the M-set boundary) to any point z (or its close outside projection) of the M-set boundary at iteration n so that its inverse z_bound(t,n) is a continuous function?
I'm really stretching my English/Math skills here. Let me know whether you understand.

We could use that function to animate a movement alongside the boundary and possibly get some intriguing results at very deep zoom.

My meager understand is that the boundary of the M-set has been mapped to a circle, so it should conceptually be possible to do what you're describing.  However, the implementation of such, particularly at deep zooms, would be tricky to say the least.  One could approximate it by mapping the boundaries of iteration bands to circles, then ratcheting up the numbers of iterations.