First of all the 3d fractal is bounded by a bounding-box so that no ray-fractal test is made if the a ray does not intersect the boundig box. Then, if the ray hits the box, the segment between the two intersection points have to be tested against the fractal. For the Julia sets the "interior" of the fractal is considered the Fatou set, ie, the points that do not escape to infinity under iteration. So each point of the segment should be tested under iteration to see if the orbit diverges or not. Of course this is provided a fixed maximal number of iterations (N). That point of the segment separating the interior from the exterior that is nearer to the ray origin is considered the intersection point.
The easier but slower way to find this point is to travel thru the segment in steps of some fixed width until the first non-scaping point is detected. This is really slow, and a modification of this "basic" algorithm is used in the generation of the images of this page.
Instead of using constant-width jumps, increments depending of the fractal potential are used. This makes the algorithm advance much faster in distant points than in the nearest ones. With this modification the algorithms catchs the intersection point in just some iterations.
Once the intersection point rutine is made, the next step is the normal calculation rutine. It is much easier than the intersection rutine. The only thing to do is to iterate the fractal in the intersection point together with the partial derivatives. Then the normal is the gradient of the escalar function under the fractal test. This function is the modulo of the Nth point of the orbit minus the bailout value:
so that the interior of the fractal are those points for whitch
The normal is therefore
for the Julia sets and
for the Mandelbrot set. Diferenciating, for the julia set, we get
so the normal (gradient) can be writen as
Don't forget to normalize this vetor to unit length if you plan to use the normal to do shading calculations on your rendering system. Anyway, I gess a better normal might be obtained thru the gradient of the Douady-Hubbard potential.
