Feature request: Specify seed for initial Z value

[stardust4ever]

Sorry if this has been discussed before, but concerning fractals with polynomial coeeficients, for instance the HPDZ buffalo:

Z₁ = |Z₀|^{^2} - |Z| + C

where |Z| is defined as abs(Z_r) + abs(Z_i)*i

It was brought up several times that polynomial formula can often have multiple critical points, defined by taking the derivative of the equation and solving for zero. For the HPDZ Buffalo, the critical points are (±0.5 + 0i) which should result in a solid fractal image with minis that do not disintegrate.

Laser Blaster brought this up some time ago here in this post in the Kalles Fraktaler 2.7 thread.
http://www.fractalforums.com/kalles-fraktaler/kalles-fraktaler-2-7/msg77526/#msg77526


On immediate observation of this fractal, it has similar shape to the zero reference starting point, but upon closer observation, one can see the presence of minibrots within the set, and an unbroken western needle that does not decay into cantor dust. So if possible, it would be nice to allow for setting the seed value of Z in software, or at least adding a "Fixed HPDZ Buffalo" formula with a correct seed value of Z₀ = (±0.5 + 0i)

Either option would allow fractal explorers to zoom into both variations of the HPDZ Buffalo fractal, since the fractal characteristics of the HPDZ Buffalo, and possibly others as well, are markedly changed by setting the seed value.

Thanks and take care...  

[Kalles Fraktaler]

I made some tests however I cannot find any centered pattern when using seed (0.5,0) or (-0.5,0)

[xenodreambuie]

I agree with stardust4ever. Using correct critical points is necessary to get a correct Mandelbrot. For the purposes of art or curiosity one isn't limited to what is correct, but I think it preferable to at least include the correct results in the available choices where possible.

[TheRedshiftRider]

I agree as well, if there isn't the correct shapestacking does not mean the formula can't be interesting is some way. I thought the same when seeing f(x)=z^2+z+c.

[stardust4ever]

I made some tests however I cannot find any centered pattern when using seed (0.5,0) or (-0.5,0)

So it did not look anything like the above render in the OP? Curious.

I agree as well, if there isn't the correct shapestacking does not mean the formula can't be interesting is some way. I thought the same when seeing f(x)=z^2+z+c.

I think the critical points are necessary when using polynomial equations that contain first order terms, but as is the case with the HPDZ buffalo, even with a seed of zero you have an equation with highly unique properties. The plus Z variant seems like it might be interesting as well.

The critical points are always going to be complex solutions to f'(Z)=0. For any formula without a first order term, one of these critical points will always be zero, though more may exist. Oftentimes you get a chaotic fractal as a result of not choosing critical points. The connected version of the HPDZ mini, if the screenshot provided by Lazer Blaster is accurate, the fractal has minibrots, a solid needle, and masts on the west mini with obvious fractal detail. Upon inspection, the west minibrot most closely resembles the perpendicular Buffalo fractal in shape.

[xenodreambuie]

... if the screenshot provided by Lazer Blaster is accurate...

FWIW, Jux automatically finds 0.5,0 as the critical point and gets the same shape as that. (-0.5,0 is only a solution for z^2+z+c.)

[Kalles Fraktaler]

I think this image is pretty close

[Kalles Fraktaler]

For comparison, here is the same with seed=(0,0)

[stardust4ever]

Yes, that's it!

Notice that while both fractals look similar at first glance, there is a notable absence of minis in the zero seed version. Hence your "where's the minibrot?" zoom movie. I think both implementations of this formula should be included in the next version of KF. I would assume the use of non-zero seed value does not adversely impact perturbation calculations?

[Kalles Fraktaler]

I've uploaded an update, version 2.10.1
You can now specify seed values for all fractals

[TheRedshiftRider]

Thanks for implementing this. It looks very interesting (just looking at the examples here). Ill have a look at it as soon as I'm home.

[Kalles Fraktaler]

Thanks for implementing this. It looks very interesting (just looking at the examples here). Ill have a look at it as soon as I'm home.

Thanks, unfortunately there is a precision issue when going beyond e30 already in the HPDZ formula with seed (0.5,0)

[TheRedshiftRider]

Ok, that probably won't be a big deal. What about the others?

[stardust4ever]

I've uploaded an update, version 2.10.1
You can now specify seed values for all fractals

Whoo-hooo!!!

Thanks, unfortunately there is a precision issue when going beyond e30 already in the HPDZ formula with seed (0.5,0)

Could this be the result of using a float value for the seed? Perhaps it should use integer like the coordinate window. Number of digits should be unlimited. For instance, a seed of 1/3 would need at least a thousand "3"s at e1000. Ditto for irrational or arbitrary values, for instance setting the seed to the location of a deep zoomed mini. Number of digits should't be an issue with a value like 0.5 though.

[Kalles Fraktaler]

Could this be the result of using a float value for the seed? Perhaps it should use integer like the coordinate window. Number of digits should be unlimited. For instance, a seed of 1/3 would need at least a thousand "3"s at e1000. Ditto for irrational or arbitrary values, for instance setting the seed to the location of a deep zoomed mini. Number of digits should't be an issue with a value like 0.5 though.

The seed float value is converted to arbitrary precision before the iterations begin.

Code:

CFixedFloat xr = 0, xi = 0;

was simply changed to

Code:

CFixedFloat xr = g_SeedR, xi = g_SeedI;

And then the same code is able to continue unchanged.
The problem is when zooming into a minibrot deeper than something like e30.

The problem lies in the piece of code that is calculating the perturbation.
Perhaps it is the double abs values that are causing this.
a=delta real
b=delta imag
a2 = a*a
b2 = b*b
x=reference real
y=reference imag

Code:

Dnr = (-2*b*y+2*a*x-b2+a2) - lb_abs_db(x,a) + a0;
Dni = lb_abs_db(x*y,x*b+a*y+a*b) * 2.0 - lb_abs_db(y,b) + b0;

It is not possible to see if this would affect other formulas with seed, since using a seed value makes the minibrots disappear in all the other formulas.