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Topic: Volume of geodesic "sphere" (project)
Replies: 21   Last Post: Aug 15, 2001 3:09 AM

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Brian Hutchings

Posts: 635
Registered: 12/6/04
Re: Volume of geodesic "sphere" (project)
Posted: May 1, 1998 7:51 PM
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In a previous article, pdx4d@teleport.com (Kirby Urner) says:

i.e.trilinear co-ordinates,
which are "homogenous" in the plane, just as
quadriplanar or quadrilinear co-ords.are in space. now,
using "line-space", I'm not sure whether Pluecker's 4D line-
co-ordinates are homogenous, also,or even sure that
homogenous ones are generally *projective*, but I assume, so.
anyway, since you have "negative moves"
along any of the quadray directions, as I recall,
it is hard to jsutify lack of minus rays;
except pedagogically, of course -- the best reason!

>To be more precise, 3 vectors might "splay" symmetrically in
>a plane at 120 degrees (Mercedes-Benz logo) to divide volume
>into two halves, but these spokes won't vector-sum to span
>volume, i.e. all grow/shrink scaling, cloning and sliding of
>vectors into tip-to-tail zigzags will keep us stuck on the
>flatlander plane.
>
>With 4 vectors splayed to the vertices of a regular tetrahedron
>(or through face centers -- same diff = to the vertices of the
>dual), we have a minimal space-spanning set. Although we might
>define a vector reversing 'negatation' operation -- and in fact
>do so, to give meaning to -(1,0,0,0) -- 'vector reversal' is
>not needed to span volume. Positive scaling (which includes
>shrinking, but without an orientation switch) will do fine.


anyway, one can co-ordinate *points* in space,
entirely positively, using tripolar co-ordinates, although
it may not be the most efficient method.

--
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*---<Brian Hutchings, Living Space Programs, Santa Monica College>---*







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