# 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 \(|F^p(0, c)|\) 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;
}