Archive for the ‘multiple critical points’ Tag
So far adding critical point support to some of the formulae in Saturn and Titan, for the next release, has been straight forward. The next formula “Mandelbrot Division” is somewhat trickier. The formula features two parts that match a standard Mandelbrot with powers that are not necessarily 2, one part is divided into the other, in addition there are two parameters added to the result. All parameters in the new version of Saturn can be a number or a number modified by multiplying or dividing the location in the complex plane, strictly speaking the “Mandelbrot parts” aren’t Mandelbrot unless the parameter is modified, so Mandelbrot Division is only a loose description but it is convenient as a name.
The current formula is as follows:
z = (zα + β)/(zγ + δ) + ε + ζ
To ensure that the critical points can be found a new version of the formula “Mandelbrot Division IP” will be added where IP indicates Integer Power.
z = (zA + α)/(zB + β) + γ + δ
The critical points are found as usual by solving f'(z) = 0, γ and δ simply disappear leaving the Mandelbrot parts to be differentiated using the quotient rule:
(f(z)/g(z))’ = (f'(z)g(z) – g'(z)f(z))/g(z)2 = 0
Multiplying both side by g(z)2 simplifies things so critical points are the solutions of the following:
f'(z)g(z) – g'(z)f(z) = 0
The Mandelbrot parts form the definitions of f(z) and g(z):
f(z) = zA + α
g(z) = zB + β
Their differentials are:
f'(z) = AzA-1
g'(z) = BzB-1
So the formula to solve is:
(AzA-1)(zB + β) – (BzB-1)(zA + α) = 0
which simplifies to:
(A – B)zA+B-1 + βAzA-1 – αBzB-1 = 0
When the values of A and B are known the smallest power can be factored out and the other powers reduced accordingly, as a power of z has been factored out it follows that Mandelbrot Division IP always has a critical point of zero. The other critical points are found by solving the remaining polynomial which will be handled by Saturn and Titan.
This formula hasn’t yet been implemented so I can only show some pictures using the current Mandelbrot Division formula where the initial value is set to zero (a critical point).
The two Titan summaries show changes from the currently released software the most noticeable of which is the display of colour maps at the bottom.