Optimizations for bulbs other than main cardioid and period-1 bulb?

[Levi]

I know that there are equations to determine if a point is in the main cardioid or period-1 bulb that can be used to optimize code, but has anybody tried to find equations for the period-2 bulbs, period-3 bulbs, etc.? It seems like this could make Mandelbrot programs much more efficient.

Just for reference, the bailout conditions for the cardioid and period-1 are (according to wikipedia) :

(x + 1)^2 + y^2 < 1/16
q*(q + x - 1/4) < 1/4*y^2

where  q = (x - 1/4)^2 + y^2

[M Benesi]

  The other buds aren't perfect circles... at least with the standard mapping.  Wonder if altering the pixels mapping would make the buds perfect circles at some point?

  I mapped the pixel from   pixel^1.25  to pixel ^3  in the following video.  Can map higher z^n as well- just map pixels to X / (n-1), with X being the type of mandy you want to force map.

<a href="http://www.youtube.com/v/28Yb8IS0CxQ&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/28Yb8IS0CxQ&rel=1&fs=1&hd=1</a>


  Wondering about the eccentricity of the period 3 bulb when X=2?  (It's shaped like a z^3 Mandelbrot).  Doubt it's worth pursuing. 

[Levi]

They're not perfect circles, but are they some kind of ellipsoid or other well-defined shape?

Even equations that don't remove the entire bulb, but just a large percent of their area, could be good.

I know there are equations for the points at which every single bulb connects to the main cardioid, which is a good start...http://en.wikipedia.org/wiki/Mandelbrot_set#Main_cardioid_and_period_bulbs

Additionally, the center of each bulb can be found by solving  Q(n+1)(c) = (Q(n)(c))^2 + c  for Q(n)(c) = 0 and n is any natural number (in the wiki article under hyperbolic components).