Title: Help about mandelbrot negative exponents
Post by: trenmost on April 30, 2015, 08:28:00 AM
Hello!
What is the formula used for negative (z --> z^-n + c) rendering?
i applied the following formula:
but it only gives me a circle...
What is the formula used for negative (z --> z^-n + c) rendering?
i applied the following formula:
Code:
RealNumber l = cLength(c1); //sqrt(c1.r*c1.r+c1.i*c1.i)
c1.i = sqrt(0.5*(l - c1.r));
c1.r = sqrt(0.5*(l + c1.r));
but it only gives me a circle...
Title: Re: Help about mandelbrot negative exponents
Post by: element90 on April 30, 2015, 02:07:24 PM
To find the square root of a complex number
if z = a + bi
r = sqrt(a*a + b*b)
theta = atan2(b/a)
z.r = sqrt(r)*cos(theta/2)
z.i = sqrt(r)*sin(theta/2)
if z = a + bi
r = sqrt(a*a + b*b)
theta = atan2(b/a)
z.r = sqrt(r)*cos(theta/2)
z.i = sqrt(r)*sin(theta/2)
Title: Re: Help about mandelbrot negative exponents
Post by: lycium on April 30, 2015, 02:16:58 PM
Getting the inverse of z^n gives n roots (http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra), so element90's code is a little incomplete for the case n=2.
You can solve it quite easily for rational n = p/q (where p and q integers), however dealing with negative p/q is probably also a bit trickier and there might be infinitely many roots...
You can solve it quite easily for rational n = p/q (where p and q integers), however dealing with negative p/q is probably also a bit trickier and there might be infinitely many roots...
Title: Re: Help about mandelbrot negative exponents
Post by: claude on April 30, 2015, 03:16:09 PM
Negative powers are just reciprocals:
Complex reciprocal is defined by:
This is derived by multiplying both the top and bottom of the left hand side by
(aka the complex conjugate of 
) which leaves the fraction the same.
The escape time algorithm doesn't work for negative powers (there is no possible escape radius, because large values can get small again), see here:
http://math.stackexchange.com/questions/1257555/how-to-compute-a-negative-multibrot-set
Roots are fractional powers,
is the n-valued n'th root - the multiple values cause all kinds of problems, not to mention not being analytic at 0 - I wasted some hours trying to find an interior distance estimate for 
, but the pre-conditions of the Koebe quarter theorem aren't met (, c))
wraps around the unit circle 3 times for that function, for the usual 
it wraps around just once, which I think is probably crucial).
Complex reciprocal is defined by:
This is derived by multiplying both the top and bottom of the left hand side by
The escape time algorithm doesn't work for negative powers (there is no possible escape radius, because large values can get small again), see here:
http://math.stackexchange.com/questions/1257555/how-to-compute-a-negative-multibrot-set
Roots are fractional powers,
Title: Re: Help about mandelbrot negative exponents
Post by: DarkBeam on April 30, 2015, 04:29:22 PM
As far as I can tell it's easy to find the inverse and it's an unique solution.
1/(a+ib) = (a-ib)/((a+ib)(a-ib)) ... etc ;)
1/(a+ib) = (a-ib)/((a+ib)(a-ib)) ... etc ;)
Title: Re: Help about mandelbrot negative exponents
Post by: youhn on April 30, 2015, 05:22:21 PM
See also http://www.fractalforums.com/new-theories-and-research/negative-multibrots/
Title: Re: Help about mandelbrot negative exponents
Post by: xenodreambuie on May 03, 2015, 12:28:28 AM
The escape time algorithm doesn't work for negative powers (there is no possible escape radius, because large values can get small again), see here:
http://math.stackexchange.com/questions/1257555/how-to-compute-a-negative-multibrot-set
Thanks for that article. Using a more general method for convergent Mandelbrot and Julia formulas, finding fixed critical points by Newton-Raphson fails for z-n+c. Using c for the critical point solves the problem (although I don't have a smooth transition between these methods).