float polygon_sides_length(float sides_count)
{
  return 2.0 * sin(M_PI / sides_count);
}

/* Returns intersection ratio between the radius edge at theta and the polygon edge.
 * Start first corners at theta == 0. */
float circle_to_polygon_radius(float sides_count, float theta)
{
  /* From Graphics Gems from CryENGINE 3 (Siggraph 2013) by Tiago Sousa (slide 36). */
  float side_angle = M_2PI / sides_count;
  float halfside_angle = side_angle * 0.5;
  theta += halfside_angle;
  return cos(side_angle * 0.5) /
         cos(theta - side_angle * floor((sides_count * theta + M_PI) / M_2PI));
}

/* Remap input angle to have homogenous spacing of points along a polygon edge.
 * Expect theta to be in [0..2pi] range. */
float circle_to_polygon_angle(float sides_count, float theta)
{
  float side_angle = M_2PI / sides_count;
  float side = floor(theta / side_angle);
  float halfside_angle = side_angle * 0.5;
  /* Length of segment from center to the middle of polygon side. */
  float adjacent = circle_to_polygon_radius(sides_count, halfside_angle);

  /* This is the relative position of the sample on the polygon half side. */
  float local_theta = theta - side * side_angle;
  float ratio = (local_theta - halfside_angle) / halfside_angle;

  float halfside_len = polygon_sides_length(sides_count) * 0.5;
  float oposite = ratio * halfside_len;

  /* NOTE: atan(y_over_x) has output range [-M_PI_2..M_PI_2]. */
  float final_local_theta = halfside_angle + atan(oposite / adjacent);

  return side * side_angle + final_local_theta;
}