DescriptionJulia-set z2+c -0.742 0.1 5000.jpg	English: Julia set (in white) and Fatou set (in green and dark violet tones) for ${\displaystyle z\mapsto z^{2}+c}$ with ${\displaystyle c=-0.742+0.1i}$. This value of ${\displaystyle c}$ lies inside the period 1 component (the main cardioid) of ${\displaystyle M}$, the Mandelbrot set, but near its boundary. The Julia set is connected because ${\displaystyle c\in M}$. The violet tinted points are attracted by infinity unter the iteration, and for the green points the attractor is the single complex number ${\displaystyle z}$ that satisfies ${\displaystyle z=z^{2}+c}$ and lies in the main cardioid of ${\displaystyle M}$.
Date	2 April 2008
Source	Own work
Author	Georg-Johann Lay
Permission (Reusing this file)	Public domainPublic domainfalsefalse I, the copyright holder of this work, release this work into the public domain. This applies worldwide. In some countries this may not be legally possible; if so: I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

Coloring

For definition of ${\displaystyle E_{n},\!\,\sigma _{n},\Sigma _{n}(z),h_{a}(x),w_{a}(x)}$ see Visualising Julia sets and the helper functions.

${\displaystyle {\begin{aligned}n&=5000\\K&=E_{n}+2.2\sigma _{n}\\J&=\ln(\delta +\Sigma _{n}(z))-K\\V_{1}&={\begin{cases}h_{0.6}(J)&,{\text{ if }}z\notin A(\infty ){\text{ or }}J>0\\h_{0.09}(J)&,{\text{ else }}\end{cases}}\\V_{2}&=1-(1-V_{1})^{2}\\V_{3}&=0.7(1-w_{7.5}(\Sigma _{n}(z)))\\V&={\begin{cases}1-(1-V_{2})(1-V_{3})&,{\text{ if }}z\notin A(\infty )\\V_{2}&,{\text{ else }}\end{cases}}\\H&={\begin{cases}0.3&,{\text{ if }}z\notin A(\infty )\\0.75&,{\text{ else }}\end{cases}}\\\mathrm {color} &=\operatorname {HSV} (H,1-V,V)\end{aligned}}}$

${\displaystyle V_{1}}$ is sufficient to show the Julia set. ${\displaystyle V_{2}}$ stresses the bright parts. ${\displaystyle V_{3}}$ serves to shade the Fatou set (green) and is "merged" with ${\displaystyle V_{2}}$ to ${\displaystyle V}$, the brightness. The hue ${\displaystyle H}$ is set according to the basin of attraction. There are two basins for this ƒ: one belonging to the attractor ∞.

        int basin = getBasin (z);

        K = EX+2.3*sigma; 
        X = log(ss) - K;

        if (basin == 0 || X > 0)
            v = hfunc(X, .6);
        else
            v = hfunc(X, .09);
        
        v = 1-pow(1-v,2);

        double x;
        if (abstand2 (o, _0) < 10)
        {
            x = (bfunc (15, ss));
            
            x = 0.7*(1-x);
            v = 1-(1-v)*(1-x);
            return hsv2rgb (0.3, 1-v, v);
        }    
        else
        {
            return hsv2rgb (0.75, 1-v, v);
            
            iv = (-.1+JCOLORS) * v;
        }

        color.index = iv;

        return color;
    }
}