[preliminary - > evolutionary]
The steiner surface is both similar to and quite different from the other equations we have examined thus far in gsd7303.2. It is parametric, and in that way shares certain properties with formula we have seen before. It does not require differential equations and the like to solve, merely simple algebra.
However, this equation, with its many singularities and irregularities, becomes quite tricky to visualize as a smooth surface in three dimensions. Instead, I have employed a sampling methodology (proposed by Bourke) which is to divide up the known space for which solutions are possible and test discrete points. If a given point in space satisfies the equation for the surface, we plot it.
We can see from the above equations that solutions exist for the surface within a sqhere of radius 0.5 centered at the origin. Therefore we only need sample points within this region. For each axis, we sample 1/delta number of discrete sections.
For purposes of further work, a chief consideration in solving the steiner surface was to gain control of the alias software package for the purposes of rendering geometric and dynamic systems. This led me to examine the .sdl file interface to the alias renderer. I had originally planned to output data points from mathcad and pull them into alias in a manner similar to that laid out previously in gsd7303.2. I soon found, however, that the .sdl file format provided for a wide variety of primitive mathematical and nesting functions. In this way, I was able to code the steiner surface directly in sdl.