`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`

http://forum.swarthmore.edu/epigone/geometry-research/sarswimpbli/3.0.3.32.19980208121320.0318c764@mail.teleport.com