[World] Sliced and diced

[Daniel Phillips]

Theory turns into practice. As expected, the surface slicing turns out to be 
rugged and consistent. And importantly, does indeed produce the desired 
result: boolean solid combinations.

The attached images:

	solid6:  Interpenetrating tetrahedra with arc of intersection
	solid7:  Boolean Union
	solid8:  Boolean Intersection
	solid9:  Open manifold

The last image shows something that is not a solid at all but an open 
manifold. Actually, the others are all open manifolds too, but they are just 
one glue step away from being solid. In many cases, an open manifold may be 
exactly the result we want. It gives us a way of drilling holes in surfaces.

What we have here is not an algebra on solids, but an algebra on open 
manifolds. I think that may be something new, at least I have not seen it 
discussed in the computer graphics literature. But it seems obvious at this 
point that this is the Right Way To Do It, and neatly sidesteps some awkward 
difficulties of traditional methods. While also showing promise in terms of 
robustness and efficiency, though we can't be sure of that until it matures a 
little.

Though the principle is now satisfactorily demonstrated, a considerable amount 
of finishing work remains. The next big item will be open arcs, that is, 
cutting arcs that run from boundary to boundary instead of joining back to 
themselves. After that I think we need some fancier images, which most 
probably will entail some debugging. And then I suppose it is time to think in 
detail about the coplanar face problem.

Inconsistencies due to numerical accuracy also need to be addressed but I am 
pretty sure the plan I have in mind for that will work, and I will wait until 
some bugs actually show up before wading in. And there is also a nasty N**2 
step in the algorithm that needs to go away. Until that is fixed, images with 
hundreds of thousands of triangles will be impractical.

Regards,

Daniel
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