Title: Optimizations for bulbs other than main cardioid and period-1 bulb? Post by: Levi on May 23, 2013, 11:36:24 PM 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 Title: Re: Optimizations for bulbs other than main cardioid and period-1 bulb? Post by: M Benesi on May 24, 2013, 08:37:18 AM 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. http://www.youtube.com/watch?v=28Yb8IS0CxQ&feature=youtu.be (http://www.youtube.com/watch?v=28Yb8IS0CxQ&feature=youtu.be) 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. Title: Re: Optimizations for bulbs other than main cardioid and period-1 bulb? Post by: Levi on May 24, 2013, 06:37:57 PM 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). |