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dist = sqrt(dx * dx + dy * dy)
ratio = dy / dx
angle = atan(ratio)

cos(angle) = adjacent / hypotenuse = dx / sqrt(dx * dx + dy * dy)
sin(angle) = opposite / hypotenuse = dy / sqrt(dx * dx + dy * dy)

cos(angle) * dist = dx
sin(angle) * dist = dy

double _Complex inflectionmapping(int n, double _Complex *cs, double _Complex c)
{
  for (int i = 0; i < n; ++i)
  {
    double _Complex d = c - cs[i];
    c = cs[i] + d * d;
  }
  return c;
}

// Map the screen pixels to points in the fractal
r = r_start+x*(r_stop-r_start)/nX;
i = i_start+y*(i_stop-i_start)/nY;
int inf;
std::complex<double> c(r,i);
for(inf=g_nInflections-1;inf>=0;inf--){
	std::complex<double> cs(g_pInflections[inf].r,g_pInflections[inf].i);
	std::complex<double> d = c-cs;
	c=cs+d*d;				
}
r=c.real();
i=c.imag();
// Proceed with the fractal sequence

#include "Progressive2D.frag"

vec2 cMul(vec2 a, vec2 b)
{
  return vec2(a.x * b.x - a.y * b.y, a.x * b.y + a.y * b.x);
}

vec2 cSqr(vec2 a)
{
  return cMul(a, a);
}

vec4 cMul(vec4 a, vec4 b)
{
  return vec4(cMul(a.xy, b.xy), cMul(a.xy, b.zw) + cMul(a.zw, b.xy));
}

vec4 cSqr(vec4 a)
{
  return cMul(a, a);
}

#define N 6
const vec2 cs[N] = vec2[N]
  ( vec2(-0.75, 0.1)
  , vec2(-0.7780062, 0.9591821)
  , vec2(-0.7186571, 1.0251421)
  , vec2(-0.5475953, 0.6401844)
  , vec2(-0.5129835, 0.6237739)
  , vec2(-0.4531227, 0.8943247)
  );

vec3 color(vec2 c0)
{
  float px = length(vec4(dFdx(c0), dFdy(c0)));
  vec4 c = vec4(c0, px, 0.0);
  for (int i = N-1; i >= 0; --i)
  {
    vec4 f = vec4(cs[i], 0.0, 0.0);
    vec4 d = c - f;
    c = cSqr(d) + f;
  }
  int n = 0;
  vec4 z = c;
  for (n = 0; n < 1000; ++n)
  {
    if (dot(z.xy, z.xy) >= 1.0e10)
      break;
    z = cSqr(z) + vec4(cs[0], 0.0, 0.0);
  }
  if (dot(z.xy, z.xy) < 1.0e10)
    return vec3(1.0, 0.0, 0.0);
  float r = length(z.xy);
  float dr = length(z.zw);
  float de = 2.0 * r * log(r) / dr;
  float g = tanh(clamp(de, 0.0, 4.0));
  if (isnan(de) || isinf(de) || isnan(g) || isinf(g))
    g = 1.0;
  return vec3(g);
}