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Re: Conway on Trilinear vs Barycentric
Posted:
Jul 2, 1998 1:19 PM
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> 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.
Yes, the example I gave is for mapping a surface. Three
vectors fan out from the origin with coordinates (1,0,0)
(0,1,0)(0,0,1). In volume, I would use 4 vectors and
4-tuples {1,0,0,0} where {} means "any permutation of
the enclosed numbers".
> Whenever you describe a two dimensional idea (a point) in terms
> of 3 numbers, you are giving up something important, namely
> that your basis vectors are orthogonal with an operation (the
> dot product) representing that orthogonality. With barycentrics
> and trilinears (both homogeneous) we gain something by doing
> this. Many of the properties of a triangle do not depend on
> scale, for which homogeneous systems are nice. Also the dual
> nature of points and lines comes out nicely with this. What
> advantages does your system have?
Quadrays have some useful properties. All centers of close
packed unit radius spheres in the fcc are linear combinations
of vectors {2,1,1,0} i.e. have positive integer coordinates.
Other primitive polys have integer coordinates also. The
'home base' tetrahedron defined by the 4 basis vectors has
edges of length 2 and a volume of 1 (by definition). The
unit volume tetrahedron divides evenly into the octahedron,
cube (face diag = 2) and rhombic dodecahedron (long face
diag = 2). Any tetrahedron defined by 4 fcc centers has a
whole number volume.
Mostly I use the quadrays apparatus as a philosophers' toy to
investigate concepts of "dimensionality" (I proffer this system
as "4D" even though it maps conceptual volume) and "linear
independence" (not necessarily the same as orthogonality).
Whereas in XYZ lingo we would say one of the basis vectors
is a linear combination of 3 others, this depends on a
change in orientation (vector reversal) being lumped in
with grow/shrink scaling, as another instance of "scalar
multiplication". In the quadrays system (and the flat
surface analog), no negative scalars are needed to span
vector space, although the "-" operator is still defined
i.e. -(1,0,0,0) is a vector pointing 180 degrees from
(1,0,0,0).
In XYZ (left handed), the +++ octant is not symmetrical
about the origin and signage is permuted to define 7
other quadrants. The apparent greater simplicity of
3-tuples is in trade-off with the all-positive economy
of this alternative 4-tuples game.
I say "game" and "toy" because I'm not trying to give
the misleading impression that I think quadrays are
anything but -- useful in their own context, but not
a replacement or substitute for anything that's already
well-established. The more games the better, is what
math is all about, no?
Kirby
References:
http://www.teleport.com/~pdx4d/quadrays.html
http://www.teleport.com/~pdx4d/quadphil.html
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