
Re: Volume of geodesic "sphere" (project)
Posted:
May 1, 1998 7:51 PM


In a previous article, pdx4d@teleport.com (Kirby Urner) says:
i.e.trilinear coordinates, which are "homogenous" in the plane, just as quadriplanar or quadrilinear coords.are in space. now, using "linespace", I'm not sure whether Pluecker's 4D line coordinates are homogenous, also,or even sure that homogenous ones are generally *projective*, but I assume, so. anyway, since you have "negative moves" along any of the quadray directions, as I recall, it is hard to jsutify lack of minus rays; except pedagogically, of course  the best reason!
>To be more precise, 3 vectors might "splay" symmetrically in >a plane at 120 degrees (MercedesBenz logo) to divide volume >into two halves, but these spokes won't vectorsum to span >volume, i.e. all grow/shrink scaling, cloning and sliding of >vectors into tiptotail zigzags will keep us stuck on the >flatlander plane. > >With 4 vectors splayed to the vertices of a regular tetrahedron >(or through face centers  same diff = to the vertices of the >dual), we have a minimal spacespanning set. Although we might >define a vector reversing 'negatation' operation  and in fact >do so, to give meaning to (1,0,0,0)  'vector reversal' is >not needed to span volume. Positive scaling (which includes >shrinking, but without an orientation switch) will do fine.
anyway, one can coordinate *points* in space, entirely positively, using tripolar coordinates, although it may not be the most efficient method.
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