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Topic: Conway on Trilinear vs Barycentric coordinates
Replies: 13   Last Post: Feb 16, 1999 8:05 AM

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Kirby Urner

Posts: 803
Registered: 12/4/04
Re: Conway on Trilinear vs Barycentric
Posted: Aug 3, 1998 12:57 AM
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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 4-tuple 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, sum-total weight).

But 4-tuple 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 not-basis, carve volume into 8 regions, with 8 permutations
of + and - (+++, +-+ +-- ... ---) giving a 3-tuple 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 non-zero (no negative
signs in the normalized form for any point address).

But there's no stipulation of a "balanced constant weight"
(constant n-tuple 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 out-rank 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
wire-frame polyhedra in object-oriented xBase.

Kirby






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