
Julibrot Fractal Type

formulas: julibrot


The Julibrot fractal type uses a general-purpose renderer for visualizing
three dimensional solid fractals. Originally Mark Peterson developed
this rendering mechanism to view a 3-D sections of a 4-D structure he
called a "Julibrot". This structure, also called "layered Julia set" in
the fractal literature, hinges on the relationship between the Mandelbrot
and Julia sets. Each Julia set is created using a fixed value c in the
iterated formula z^2 + c. The Julibrot is created by layering Julia sets
in the x-y plane and continuously varying c, creating new Julia sets as z is
incremented. The solid shape thus created is rendered by shading the surface
using a brightness inversely proportional to the virtual viewer's eye.

Starting with xmfract version 1.2, the Julibrot engine can be used
with other Julia formulas besides the classic z^2 + c. The first field on
the Julibrot parameter screen lets you select which orbit formula to use.

You can also use the Julibrot renderer to visualize 3D cross sections of
TRUE four dimensional Quaternion and Hypercomplex fractals.

The Julibrot Parameter Screens

Orbit Algorithm - select the orbit algorithm to use. The available
   possibilities include 2-D Julia and both mandelbrot and Julia variants
   of the 4-D Quaternion and Hypercomplex fractals.

Orbit parameters - the next screen lets you fill in any parameters
   belonging to the orbit algorithm. This list of parameters is not
   necessarily the same as the list normally presented for the orbit
   algorithm, because some of these parameters are used in the Julibrot
   layering process.

From/To Parameters
   These parameters allow you to specify the "Mandelbrot" values used to
   generate the layered Julias. The parameter c in the Julia formulas will
   be incremented in steps ranging from the "from" x and y values to the
   "to" x and y values. If the orbit formula is one of the "TRUE" four
   dimensional fractal types quat, quatj, hypercomplex, or hypercomplexj,
   then these numbers are used with the 3rd and 4th dimensional values.

   The "from/to" variables are different for the different kinds of orbit
   algorithm.

      2D Julia sets - complex number formula z' = f(z) + c\
         The "from/to" parameters change the values of c.\
      4D Julia sets - Quaternion or Hypercomplex formula z' = f(z) + c\
         The four dimensions of c are set by the orbit parameters.\
         The first two dimensions of z are determined by the corners values.\
         The third and fourth dimensions of z are the "to/from" variables.\
      4D Mandelbrot sets - Quaternion or Hypercomplex formula z' = f(z) + c\
         The first two dimensions of c are determined by the corners values.\
         The third and fourth dimensions of c are the "to/from" variables.\

Distance between the eyes - set this to 2.5 if you want a red/blue
   anaglyph image, 0 for a normal greyscale image.

Number of z pixels - this sets how many layers are rendered in the screen
   z-axis. Use a higher value with higher resolution displays.

The remainder of the parameters are needed to construct the red/blue
picture so that the fractal appears with the desired depth and proper 'z'
location. With the origin set to 8 inches beyond the screen plane and the
depth of the fractal at 8 inches the default fractal will appear to start
at 4 inches beyond the screen and extend to 12 inches if your eyeballs are
2.5 inches apart and located at a distance of 24 inches from the screen.
The screen dimensions provide the reference frame.

