In a previous article, email@example.com (Kirby Urner) says:
i.e.trilinear co-ordinates, which are "homogenous" in the plane, just as quadriplanar or quadrilinear co-ords.are in space. now, using "line-space", I'm not sure whether Pluecker's 4D line- co-ordinates 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 (Mercedes-Benz logo) to divide volume >into two halves, but these spokes won't vector-sum to span >volume, i.e. all grow/shrink scaling, cloning and sliding of >vectors into tip-to-tail 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 space-spanning 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 co-ordinate *points* in space, entirely positively, using tripolar co-ordinates, although it may not be the most efficient method.