Location Dependent Critical Points

[element90]

The current version of Neptune/Triton does not include any formulae that have location dependent critical points, Saturn/Titan doesn't include support for critical points at all. I'm adding support for location dependent critical points to Neptune/Triton and where appropriate to some formulae in Saturn/Titan.

Saturn/Titan allowed parameters to be substituted with the +ve or -ve location in the complex plane (c), this will change for the next release instead the parameter will be modified by either multiplying it by c or dividing it by c allowing greater flexibility to each formula. Neptune/Triton only allowed parameters to be a number, it too will have the same parameter modification scheme as Saturn/Titan, as Neptune/Triton only deals with multiple critical point fractals the handling of parameters will be restricted so that a formula is not turned in a Julia (no critical points).

An example:





more to come.

[element90]

An other example, this time a quartic:





The fractal exhibits both M2 and M3 Mandelbrot features, the M3 features can be seen more easily in the picture below:



The preceding pictures painted black areas where the orbits did not escape for at least one critical point. The following picture averages inner and outer colouring where those areas overlap:



There are three critical points: each produces a component picture that can be used to produce a composite picture using two or more critical points. The component pictures are as follows:

[cKleinhuis]

nice, but can you explain what you mean by critical points and how they influence a formula!?

[element90]

Fractals generated using the Mandelbrot algorithm use a critical point as the initial value of z. If a value other than the critical value is used you get a "perturbed" fractal, just try using a value other than zero for the Mandelbrot set.

Using the Mandelbrot formula as an example, the critical point can found as follows:

z = z^2 + c

f(z)  = z^2 + c

The critical point(s) are the solutions of f'(z) = 0 which for the Mandelbrot set is:

f'(z) = 2z = 0

so the critical point is zero.

For higher power Mandelbrots i.e.

z = z^n + 1 where n > 2

f'(z) = nz^(n-1) = 0

so their critical points are all zero.

It gets much more interesting when extra terms are added e.g.

z = z^5 + z + c

f'(z) = 5z^4 + 1 = 0

so the critical points are the four 4th roots of -1/5, changing the z term to -5z results in the four 4th roots of 1 which are 1, -1, i and -i. Each critical point produces a picture, some times more than one critical point produces the same picture.

When the solutions of f'(z) = 0 include c (the location in the complex plane) you get location dependent critical points which have to be determined for each location. e.g.

z = z^2 + cz + c

f'(z) = 2z + c = 0

so the critical point is -c/2 which, at first glance, just produces a Mandelbrot of a different size in a different location. Zooming out, however, shows another Mandelbrot pointing the other way

I initially thought that the Quadratic equation z = az^2 + bz + c just produced a Mandelbrot but that was before adding support for location dependent critical points to Saturn.

For more information on the Quadratic formula see https://element90.wordpress.com/2014/11/20/quadratic-observations/.

[element90]

The current version of Neptune/Triton includes a fractal called MC SIN that has no parameters and two critical points.

The formula is:

z = sin(z) + c

The critical points are the solutions to f'(z) = 0:

f'(z) = cos(z)

so the solutions are arccos(0) which are pi/2 + n*pi where n is an integer which implies an infinite number of critical points. There are only two distinct component pictures so only the critical points -pi/2 and pi/2 are required.



showing the overlaps betwen the two component pictures:



The triton summary is:



For the next version of Neptune/Triton three parameters and been added, the Triton summary for the equivalent formula is:



It may seem odd that there are two standalone parameters but there is a reason.

An MC COS formula was not included as it produced the same fractal shifted horizontally:



Adding parameters to the formula makes the following formula possible:

z = sin(cz)

its critical points:

f'(z) = c*cos(cz) = 0

so the critical points are arccos(0)/c or (pi/2 + n*pi)/c which is dependent on location.

There is only one distinct component picture:



Setting the formula to:

z = sin(cz) + c

leaves the critical points as they are and there are two distinct component pictures:



Using the extra parameter:

z = sin(cz) + c - 2

again leaves the critical points as they are but the two component pictures are different and is reason it exists:



The cos version of this formula produces markedly different pictures to the sin version once the formula diverges from cos(z) + c and sin(z) +c.

The critical points:

z = cos(alpha*z) + beta + gamma
f'(z) = -alpha*sin(alpha*z) = 0

so the critical points are arcsin(0)/alpha or (n*pi)/alpha

For

z = cos(cz)

there is one component picture which is clearly different to the sin version consequently MC COS has been added to Neptune/Triton.



For

z = cos(cz) + c



and for

z = cos(cz) + c - 2



There will be additional variations on the formula such as

z = (sin(alpha*z))^2 + beta + gamma
z = sin(alpha*z^2) + beta + gamma

and higher powers and ones that use cos. There will be example pictures using those formulae soon.

[element90]

Now for some examples using MC COS2.

z = (cos(alpha*z)^2 + beta + gamma

So its critical points are found using:

f(z) = (cos(alpha*z))^ + beta + gamma
f'(z) = -2*sin(alpha*z)*cos(alpha*z) = 0

so there are critical points that at arcsin(0)/alpha and at arccos(0)/alpha i.e. (pi/2 + n*pi)/alpha and n*pi/alpha. The component pictures for the arcsin critical points are identical are as those for the arccos critical points.










Note the spot in the final picture that is the result of a lack of precision and can be reduced in size by using higher precision or increasing the bailout limit.

[element90]

The Quadratic formula is much more interesting than I had supposed.

z = alpha*z^2 + beta*z + gamma

finding its critical point is simple:

f(z) = alpha*z^2 + beta*z + gamma
f'(z) = 2*alpha*z + beta = 0

so it has one critical point at -beta/(2*alpha).




closer in ...