From: sshws@convx1.lerc.nasa.gov (Herb Schilling)
Subject: SUMMARY: Equispaced points on sphere?
Organization: NASA Lewis Research Center
Date: Mon, 28 Dec 1992 16:11:33 GMT


After a long delay ( sorry ) , here is a summary of replies to my question
on how do you place N points on a 3-D sphere such that the minimum 
distance between any two points is a maximum. 

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From: Jim Roth MLO1-2/U2 223-6562 Workstations Software  01-Dec-1992 0923
<roth@3d.enet.dec.com>
To: sshws@convx1.lerc.nasa.gov
Apparently-To: sshws@convx1.lerc.nasa.gov
Subject: Re: Equispaced points on sphere? ( Repeat?)

In article <1992Dec1.134328.19116@eagle.lerc.nasa.gov>, you write...
> 
>I apologize if this was just discussed ( I think it was ), but I
>am trying to figure out how to determine the locations of n equispaced 
>points on the surface of 3-D sphere. 

As far as I know, such equidistant points only exist for cases coming
from the classcial polyhedra (squares, cubes, octahedra, icosahedron...)

You can get a close approximation using geodesic dome techniques for some
values of N though.

Start with a parent polyhedron like an icosahedron and triangulate the
faces into more equilateral triangles.  Projecting the points on the sphere
will not have too bad a discrepancy.

For general N I don't know of a simple way to do this, or if it's even
possible without being related to geodesic domes.

- Jim

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Date: Tue, 1 Dec 92 10:52:48 EST
From: "Phil Hanna" <pehanna@vnet.ibm.com>
To: sshws@convx1.lerc.nasa.gov
Subject: Equispaced points on a sphere

Herb, this is an unsolved problem I have been interested in for years.
Are you sure you are describing exactly what you want?  Do you mean
equidistant or maximally distant?  For example, for n=5, you can
locate four points equidistant around the equator and one point at
the north pole, and if you think of these points as vertices of a
square pyramid, all of the edges are the same length.  But all five
points are not equidistant from each other (consider the diagonals
of the base).  If you are thinking of "equidistant" in terms of
some definition of "neighboring points", this is not the solution.
If you think of circular caps extending out from each point and
try to maximize the common radius, it turns out that a better solution
is north pole/south pole/three points at 120 degrees along the
equator.  I conjecture that for sufficiently large n, the solution
is not unique.

I have found some references in the literature.  Martin Gardner
devoted one of his columns in Scientific American (I think it was
in mid to late 1969) to this problem.  He cast it in terms of
how to locate n lunar bases as far apart from each other as possible.
There is a more rigorous treatment in the book "Regular Figures"
by Fejes-Toth.  It turns out that the problem occurs in nature, in
the optimum placement of dimples on a pollen grain.

As of 1969, there were solutions for specific values of n up to
about 13 but no general solution.  It is a rich problem; think
of solutions in terms of the polyhedra that they determine.  What
do these polyhedra have in common?  Of the five platonic solids,
you would think they all are solutions for n equal to their
number of vertices, but this is only true for four of them.  The
cube is not the solution for n=8; a 45 degree twisted cube is.

Please let me know what you find out on this problem.  In particular,
see if you can come up with a good definition, where "good" means
that it produces solutions that intuitively look right.  Thanks.

Phil Hanna
pehanna@vnet.ibm.com


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From: lhf@visgraf.impa.br (Luiz Henrique de Figueiredo)
Message-Id: <9212011828.AA10235@Newton.visgraf.impa.br>
To: sshws@convx1.lerc.nasa.gov
Subject: Re: Equispaced points on sphere?

If you can settle for *almost* equispaced points, then a simple algorithm is:
        1. start with a icosaheadron
        2. foreach triangle
                - divide triangle into 4, by connecting midpoints of sides
        3. repeat step 2 as much as neeeded.

I can send you code for that if you need.
--
Luiz Henrique de Figueiredo                        email: lhf@impa.br
IMPA-Instituto de Matematica Pura e Aplicada              lhfig@brlncc.bitnet
Estrada Dona Castorina 110                         voice: +55 21 294-9032 x226
22460 Rio de Janeiro, RJ, Brasil                     fax: +55 21 512-4115
--

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From: huuskone@cc.helsinki.fi (Taneli Huuskonen)
Message-Id: <9212021409.AA28515@klaava.Helsinki.FI>
To: sshws@convx1.lerc.nasa.gov
Subject: Re: Equispaced points on sphere? ( Repeat?)
Newsgroups: sci.math

In sci.math you write:


>I apologize if this was just discussed ( I think it was ), but I
>am trying to figure out how to determine the locations of n equispaced 
>points on the surface of 3-D sphere. 

Depends on how you define "equispaced".  The most natural definition,
IMHO, would be to require that the points be the corners of a regular
polyhedron, but there are just five regular polyhedra, corresponding to
n=4, 6, 8, 12, and 20.  Of course, you can arrange the points at the
corners of a regular n-sided polygon lying on a great circle (or even a
small circle) of the sphere, which would make them "equidistant" in a
fairly natural sense, but I don't find that quite satisfying.  There are
other possibilities as well, but I suspect things may get somewhat
complicated when you explore them.
-- 
Taneli Huuskonen |   finger huuskonen@cc.helsinki.fi   | Garanteely speling
-----------------|-------------------------------------| & grammer errur free
 Did I claim | S e va'gy tala'n me'g jobban boldogi't  |---------------------
 something?  | Mint ha ott volne'k, ahol lenni va'gyok.  -Peto"fi Sa'ndor

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Article 26715 of sci.math:
Newsgroups: sci.math