
Re: Conway on Trilinear vs Barycentric
Posted:
Aug 3, 1998 12:57 AM


In posting to geometry_research: > Subject: Re: Conway on Trilinear vs Barycentric > Author: steve sigur <ssigur@netdepot.com> > Date: Thu, 2 Jul 98 10:59:10 0400
Steve wrote: > I do not think your system is either. Neither trilinears > nor barycentric coordinates represent distances from a > common origin. Yours looks like the regular basis vectors > in 3D space. If yours is in 2D space then I think it is > different from the other two.
This made sense to me at the time. From what I read, a hallmark of the barycentrics is the coordinates add up to some constant. Also they seem to always be used for planar applications (which is why the link to "trilinear" coordinates)  although no doubt a 4tuple version inside a tetrahedron has been formalized (quadrilinear coordinates?). I recall seeing tetrahedral graphs in a chemistry book on the Gibbs Phase Rule... barycentrics?
With barycentrics, positive coordinates keep you inside the region defined by the 3 (or 4) vertices, while negatives get you outside. The idea is a "roving center of gravity" controlled by "weighted vertices" (redistributing their constant, sumtotal weight).
But 4tuple quadrays, with 4 basis rays from the common origin (0,0,0,0) seem more NeoCartesian to me, a completely analogous system operating in 4 quadrants instead of 8 octants. In other words, the 6 XYZ vectors, 3 basis + 3 notbasis, carve volume into 8 regions, with 8 permutations of + and  (+++, ++ + ... ) giving a 3tuple its "octant address". In quadrays, the 4 basis vectors carve volume into 4 regions, and you know which quadrant your point is in based on what coordinates are nonzero (no negative signs in the normalized form for any point address).
But there's no stipulation of a "balanced constant weight" (constant ntuple sum) or "bounding vertices" with an "inside" versus an "outside" related to signage in quadrays (at least not obviously).
Nevertheless, Brian Scott and Robin Chapman, both very knowledgable mathematicians active on sci.math, have handed down their decision that quadrays are a subclass of barycentric coordinates. I mentioned that maybe some on geometry_research might have a differing opinion, but Chapman especially was dismissive of geometry_research posters (me in particular):
I gave up reading geometry.research ages ago because of its dire signal/noise ratio, mostly due to Urner's logorrhoeic drivel. The group should be retitled geometry.remedial. Were I still reading it I would certainly assent to Brian's assertion that Urner's "quadrays" are basically barycentrics.
Robin Chapman (7/31/98)
These guys clearly outrank me when it comes to having strong academic credentials in math. Given how the game is played, I think I have to defer to their judgement. So now my plan is to write to webmasters who have material relating to barycentrics on the web and suggest they link to my 'Quadray Papers' (at http://www.teleport.com/~pdx4d/quadrays.html) as examplary of how barycentrics have been used to define a system for rendering a concentric hierarchy of nested wireframe polyhedra in objectoriented xBase.
Kirby

