Coordinate Systems

Subject:      RE: Coordinate systems
Author:       Kirby Urner <pdx4d@teleport.com>
Date:         Sun, 08 Feb 1998 12:13:20 -0800


Re: Coordinate Systems

Lets not forget "latitude and longitude" as an important
coordinate system.  I know some would argue these are a 
subclass of "polar coordinate" and they'd have a case, 
but lots of the specifics are unique to the actual 
practice of navigation, including the real time use of
global positioning devices (not just the paper and 
pencil nomenclature, but the gizmos applied, has a
bearing on what we mean by "coordinate system" I would
argue).

Cliff has some simplex coordinates he associates with
Synergetics.  In keeping with my philosophy that what 
Fuller meant by '4D' was quite simply Platonic-Euclidean 
space of the ordinary kind (volumetric conceptuality), 
but minus any 0,1,2,3- D 'dimensional ladder', I've 
been offering quadray coordinates as yet another 
coordinate system for service as a pedagogical tool
for math teachers, aimed at keeping our minds flexible 
and open to "new gizmos" generally (the future will 
doubtless have many new games for us to learn).

In quadrays, we spoke out in 4 directions from the center
of a tetrahedron to its vertices, and label these basis
rays (definitional move):

   a (1,0,0,0)
   b (0,1,0,0)
   c (0,0,1,0)
   d (0,0,0,1)

You can then clone and translate these basis vectors, scaled 
by floating point numbers, and add them tip-to-tail, as per 
usual, such that any address (fp, fp, fp, fp) will signify a 
point in volume surrounding the origin.  However, because any 
given point only needs to make use of at most 3 basis vectors, 
and because shrink/grow scaling can take care of spanning any 
quadrant without making use of the 'vector reversal' operator 
(i.e. negation or - ), we will always have a 'lowest terms' 
expression of a coordinate address of the form {fp, fp, fp, 0} 
where all fp are positive floating point numbers at least one 
of which will be zero -- the curly braces indicate that we're 
not tacking down which.

I've derived an alternative distance formula for dealing with 
quadrays, thereby giving myself a metric, and written object 
oriented computer code for translating to/from xyz.  This gives 
me what I need to bring polyhedra to the screen using a database 
of 4D coordinates, with the xyz conversion happening 'on the fly' 
as I write out to my ray tracer engine, which of course expects 
input using the time-tested Cartesian protocol.  I also have 
an alternative volume expression, which syncs with Synergetics.
Plug in the four coordinates of any tetrahedron, and get back
its volume in terms of the unit-volume tet defined by the 
centers of 4 adjacent IVM spheres.  Turns out any tetrahedron
with IVM vertices has a whole-number volume by this method
of reckoning, no matter how skew.

What I can use quadrays for is to challenge the idea that
the "linear independence" necessarily gets us to "3-D" as 
the only logical result.  I claim that my system in many 
ways streamlines, by making negative scalars unnecessary, 
and by getting by with 4 spokes, omnisymmetrically or 
spherically arranged, instead of the Cartesian 6 (which
includes 3 positive and 3 negative).  Plus I can derive 
the Cartesian apparatus by adding all pairs of my basis 
rays, to get vectors poking the the 6 mid-edges of my home 
base tetrahedron.  These vector sums have the form {1,0,1,0}
and I can tell kids to "paint them black and relabel with 
positive and negative numbers" to play the standard textbook 
xyz games, which of course they still need to learn and will 
for the foreseeable future.

Kirby

Cite:
http://www.teleport.com/~pdx4d/quadrays.html