# Interior Coordinates

Interior coordinates rendering

Interior coordinates in the Mandelbrot set (2013) derives an algorithm for calculating interior coordinates $b$ , and it turned out that only periods that are “partials” (corresponding to atom domains) need be considered:

For each period p, starting from 1 and increasing:
- If |Fp(0,c)||F^p(0, c)| reaches a new minimum:
  - Find $z_0$ such that $F^p(z_0,c)=z_0$ using Newton’s method in one complex variable (“attractor”, with initial guess $F^p(0,c)$ );
  - Find $b =\frac{\partial}{\partial z} F^p(z_0,c)$ ;
  - If $|b| \le 1$ then return $b$ , otherwise continue with the next $p$ .

# 1 C99 Code

#include <complex.h>

double _Complex m_interior_coordinates
    (int N, int M, double _Complex c)
{
    double _Complex z = 0;
    double mz = 1.0 / 0.0;
    for (int n = 0; n < N; ++n)
    {
        z = z * z + c;
        double zp = cabs(z);
        if (zp < mz)
        {
            mz = zp;
            double _Complex w = m_attractor(z, n, c, M);
            double _Complex dw = 1;
            for (int m = 0; m < n; ++m)
            {
                dw = 2 * w * dw;
                w = w * w + c;
            }
            if (cabs(dw) <= 1)
                return dw;
        }
    }
    return 0;
}