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<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">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.</p>
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<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">The attached images:</p>
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<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> solid6: Interpenetrating tetrahedra with arc of intersection</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> solid7: Boolean Union</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> solid8: Boolean Intersection</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> solid9: Open manifold</p>
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<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">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.</p>
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<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">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.</p>
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<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">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.</p>
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<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">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.</p>
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<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">Regards,</p>
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<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">Daniel</p>
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