by Roger L. Bagula 18 Sept 2000©
Abstract: A new nonlinear surface definition that appears to be very like a Steiner surface It is based on the differential twisting of the angular triangle minimal coordinate surface.
Partial angular differentiation in self-similar and affine fractal IFS's has appeared in previous TFTN articles. In this is meant:
1) cos(t)=sin(t-Pi/2)2) dcos(t)= -sin(t)*dt
3) f(t,s)=cos(t+Pi*s/2): domain: {s: 0,1}
Applying this kind of partial angular differential to the surface:
4) x=cos(2*Pi*t-s*Pi/2)*cos(2*Pi*(-t+p)+s*Pi/2)
5) y=cos(2*Pi*p-s*Pi/2)*cos(2*Pi*(-t+p)+s*Pi/2)
6) z=cos(2*Pi*p-s*Pi/2)*cos(2*Pi*t-s*Pi/2)
Unless s is a function of the two angular plane variables it appears as a new variable. Experiments were done with different dependent versions for s and a mistake was made to get:
7) s' = s + p - s * pwhich is an entropic fuzzy logic type of function in a nonlinear form for s. The result in terms of the surface was to remove one lobe of the original surface.
That the surface has an appearance like a Steiner surface or may even be another definition of a Steiner surface makes the original Angular Triangle as Minimal (ATM) surface more important and interesting as well.
It was a result of work in trying to get a Fano finite projective plane to express as a three dimensional surface using angular plane lines in a partition function. The suggestion is that the ATM surface is twisting of the topological projective plane that the Steiner surface represents and thus, is a new class of such surfaces. It may in fact be a more fundamental version of such fundamental topological surfaces, since it uses a more minimal definition in its coordinates.
This new procedure also represents a new kind of nonlinear coordinate procedure to produce surfaces which can be generalized even further to:
8) s' = f(s,t,p)This specific transform in 7) seems to leave both the cylinder and sphere invariant in experiments in my True Basic projections. More experiments would be necessary to get a better view of such nonlinear functions in s, but it seems to be a phase angle twisting that has to be just right for the symmetry of the surface.
The angular plane approach to surface geometry has resulted in a whole new variety of surfaces and the ATM surfaces seem to be the most fundamental results so far.