Multiple Critical Points

[kram1032]

I'm sure somebody else already thought of this, since it's such a simple thought, but here is a method to "invert" the problem of finding critical points to a polynomial and to, instead, craft polynomials with precisely the critical points you want (here for degree 6) :

http://www.wolframalpha.com/input/?i=Integrate+%28x-c0%29%28x-c1%29%28x-c2%29%28x-c3%29%28x-c4%29+dx

as long as you don't put anything in all those constants (including the additional constant from integrating), which depends on x, you can use arbitrary terms, including arbitrary functions on the iteration constant c. The critical points will be solely determined by the constants (not including the integration constant).

[jdebord]

here is a method to "invert" the problem of finding critical points to a polynomial and to, instead, craft polynomials with precisely the critical points you want

The method is also discussed in Ingvar Kullberg's site:

http://klippan.seths.se/fractals/articles/28.pdf 

[jdebord]

The formula z = z^2 + c/z^2 has four critical points which are the roots of z^4 = c

Yes. In the general case the critical points are

Here is another example: a four-leaf clover from p=5 and q=1



When p > 2 and q = 1 the Mandelbrot set displays (p-1) symmetrical lobes.

[element90]

Yes. In the general case z^p+c/z^q the critical points are (qc/p)^{1/(p+q)}

Here is another example: a four-leaf clover from p=5 and q=1



When p > 2 and q = 1 the Mandelbrot set displays (p-1) symmetrical lobes.

I originally had my doubts about the value of p used for that picture as I had a go at reproducing it with Gnofract4d but I haven't got the initial value of z right as it lacks a power function, however reducing p to 3 so that I could set the initial value of z two lobes are produced which would be 3 when p is 4 and 4 when it is 5. A similar picture can be obtained using p = 3 if both z terms have c as part of their coefficients and the critical points are simply numbers and not expressions involving c.

z = c(z^3 + 3/z)


https://copy.com/Js4eqBMVvpMm

The fractal has four critical points (1, -1, i & -i), as it is well formed, i.e. has normal Mandelbrot type buds and no areas lacking in fractal detail it produces the same image for all four critical points.

I have a sequence of posts on my blog on the formula

z = c(alpha*z^beta + gamma*z^delta)

http://element90.wordpress.com/2012/08/08/guide-to-cczcpaczcp-part-1/
http://element90.wordpress.com/2012/08/10/guide-to-cczcpaczcp-part-2/
http://element90.wordpress.com/2012/08/15/guide-to-cczcpaczcp-part-3/
http://element90.wordpress.com/2012/08/16/guide-to-cczpaczcp-part-4/
http://element90.wordpress.com/2012/08/17/guide-to-cczcpaczp-part-5/ - Note: the central dots are the result of a bug that has since been fixed.
http://element90.wordpress.com/2012/08/27/guide-to-cczcpaczcp-part-6/ - Note: the odd fine loops within the black areas are the result of a bug that has since been fixed.


The posts were published long before I considered combining the pictures of versions of the same fractal using multiple critical points.

I've briefly looked at

z = alpha*z^3 + beta*z^-1 + c

and it produces four pictures one for each critical point, pictures later.

[element90]

I've briefly looked at

z = alpha*z^3 + beta*z^-1 + c

and it produces four pictures one for each critical point, pictures later.

I forgot about this, there won't be any pictures for that formula (sorry can't be bothered).

Meanwhile I've been making progress with my new programs Neptune and Triton. The programs are based on Saturn and Titan, so Neptune is the explorer and Triton is the expander. I expect to release them towards the end of April.

The construction of the pictures boils down to how the pictures for each critical point are combined, for areas that are outer only the colour data is either averaged or the data for the highest iteration is used, for areas that are inner only the colour data is averaged, where inner and outer areas overlap there are three options: averaging the inner colour with outer colour, ignoring outer colour and using only the inner colour, and finally, ignoring inner colour data and using only outer colouring.

Some fractal formulae produce pictures that are identical for more than one critical point so Neptune provides an option for ignoring critical points when calculating the fractal so that only the necessary calculations are performed. A side effect is to allow for "incomplete" multiple critical fractals to be produced, e.g. the final image may be made up of two out of three possible images.

The critical points are automatically determined from the fractal's parameters.

Now for some example pictures (average outer, average inner/outer, average inner):


https://copy.com/M3SiSGSvIirr


https://copy.com/Ni1abEbnqxCw


https://copy.com/M2nGsC1Pbf1M


https://copy.com/CV1OcO4HQuJV

average outer, ignore outer average inner, average inner:


https://copy.com/lJzjqIKWeGFs

average outer, ignore inner average outer, average inner:


https://copy.com/pTB4SXovQOZI

Note: all the above pictures use quartic formulae and have three critical points.