Burning Ship implementation

[LegalizeAdulthood]

@PaulTheLionHeart I was adding the formula for the burning ship fractal to the documentation and I'm confused about how you've implemented it compared to the BurningShip formula you submitted in 1.0 and the "official" formula given in wikipedia. Let's compare the formulas for all three and maybe you can help me understand the implementation.

WikiPedia

WikiPedia has this formula:

$$ z_0 = 0 $$ $$ z_{n+1} = (|Re(z_n)| + i|Im(z_n)|)^2 + c $$

Formula File

The formula file has this entry which is similar, except the imaginary component of $z_n$ is negated after taking the absolute value:

Burning_Ship { ; Burning Ship - the original Mandelbrot Derivative
   z = c = pixel:
   x = abs(real(z))
   y = -abs(imag(z))
   z = x + flip(y)
   z = z*z + pixel + p1,
   |z| < 4
}

Source Code

The implementation in burning_ship.cpp has this code (comments added by me):

    DComplex q;
    const int degree = (int) g_params[2];
    q.x = g_init.x;
    q.y = g_init.y;
    if (degree == 2)
    {
        // (|x| + i|y|)^2 = (|x| + i|y|)*(|x| + i|y|)
        //                = (|x|*|x| - |y|*|y|) + i(|x|*|y| + |x|*|y|)
        //                = (|x|^2 - |y|^2) + i(2*|x|*|y|)
        g_temp_sqr_x = sqr(g_old_z.x);                            // real(z)^2 = |real(z)|^2
        g_temp_sqr_y = sqr(g_old_z.y);                            // imag(z)^2 = |imag(z)|^2
        const double real_imag = std::abs(g_old_z.x * g_old_z.y); // |real(z)*imag(z)| = |real(z)|*|imag(z)|
        g_new_z.x = g_temp_sqr_x - g_temp_sqr_y + q.x;            // real(z)^2 - imag(z)^2 + real(c)
        g_new_z.y = real_imag + real_imag - q.y;                  // |real(z)*imag(z)|*2 - imag(c)
    }

This version is different again, because the imaginary component of $z_{n+1}$ has the imaginary component of $c$ subtracted instead of added.

Am I missing something subtle in your implementations? Why don't they correspond to the WikiPedia formula?

[PaulTheLionHeart]

The best explanation I can give is that if one uses the Wiki version, the image ends up upside down. Try it by changing the sign in the par file.

The kalles reference that I used to program Burning Ship is:

The implementation in C++ code tries to reduce the number of multiplications for speed: 3 as opposed to 4 in the interpreted version.

I suspect that the issue with the negative sign is the way Windows addresses pixels. Pixel 0 is the top of the screen as opposed to the bottom. I'm surprised this issue hasn't come up before with other fractals that don't have x-axis symmetry.

[LegalizeAdulthood]

OK, yes, increasing Y pixels going down (instead of up) explains it; I guess I never considered before in Id how the code handles this.

[LegalizeAdulthood]

Since we identified the discrepancy, I'm going to close this discussion :)