
Re: Conway on Trilinear vs Barycentric
Posted:
Jul 2, 1998 1:19 PM


> 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 4tuples {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 3tuples is in tradeoff with the allpositive economy of this alternative 4tuples 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 wellestablished. 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

