Hi Peter!

Peter Santo wrote:

> I would like to render a sphere with an arbitrary number of points (or
> smaller spheres) on its surface that I want to distribute equally (so
> the distance to the neighbouring points is nearly equal for every
> point). I suppose there is no perfect solution for that problem, but I
> would like to approximate it as good as possible.

I had the same problem. After some time I realized that a factor of 5 works
very fine:
Solution 1: 7 points (1 at each pole and ring at 90degree with 5 points)
Solution 2: 22 points (1 at each pole, a ring of 5 points at 45degrees, a
ring of 10 points at 90degrees, a ring of 5 points at 135 degrees)
Solution 3: 47 points (1, 5, 10, 15, 10, 5, 1)
Solution 4: 82 points (1, 5, 10, 15, 20, 15, 10, 5, 1)
Solution 5: 127 points (1, 5, 10, 15, 20, 25, 20, 15, 10, 5, 1)
... and so on ...

You cannot choose any number of points but using a factor of 5 is *very*
simple and the output looks fine. And you can choose really high numbers of
points without any complicated calculation.
A picture using this technique can be found at
http://home.ins.de/~tobias.wiersch/htm/newyear.htm

Hope this helps ...

 ... tobias wiersch

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Distributing Points on a Sphere webpage

http://www.mhri.edu.au/~pdb/geometry/spherepoints/

steve

Peter Santo wrote:

> Hello!
>
> I would like to render a sphere with an arbitrary number of points (or
> smaller spheres) on its surface that I want to distribute equally (so
> the distance to the neighbouring points is nearly equal for every
> point). I suppose there is no perfect solution for that problem, but I
> would like to approximate it as good as possible.
>
> A funktion could look like this:
> Input: Number of points (n)
> Output: n pairs of angles (horizontal and vertical from sphere's center)
> that describe the location of the points. (alternatively <x,y,z>-coords
> on the surface)
>
> Of course, the solution is easy for n = 1(trivial), 2(line),
> 3(traingle), 4(tetrahedron), 6(octahedron), 8(cube) and some more. I
> also checked http://www.cris.com/~rjbono/html/domes.html , but geodesic
> domes always have a "magic" number of corners.
>
> Does anyone have an idea for this? Any help would be appreciated (also
> "It does not work, because..").
>
> /PETER/
>
> --
> Peter Santo (PUMP development)
> Visit: http://www.ieee.rwth-aachen.de/mp3/

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Remco de Korte wrote:
> 
> I have been wondering about a solution for this some time ago.
> I think your strategy is brilliant (in the way that it makes me say: why didn't
> _I_ thin of that? ;-)) but I think POV will parse on this forever. You'll
> probably need an external program to do the calculations.
> Anybody?

I had the idea and wrote a small C-program to do that and export
the found points to POV-speres. Bottom line: it just doesn't work,
because the spheres "fall" into some local energy minimum and
make nice patterns, but for most values for N, they are far from
being equally spaced.

Markus
-- 

 Ich nicht eine Sekunde!!!" H. Heinol in Val Thorens

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Peter Santo wrote:
> 
> Hmm, in fact, I thought about making the algorithm a little more
> efficient (adaptive step size) and implementing it as a povray macro.
> But maybe it is not a sensible thing to do, because you either render a
> single image - then it is not worth writing lots of code. Or you render
> an animation - then you do not want to spend too much time parsing.

I had my C-code to output severay POV-files so I could
render an animation and look how the sphreses spread
across the other sphere. Thus it was called sprdsphr.

If you're interested in the program, send a
mailto:mar### [at] studentuni-siegende

and I'll send you the source for it.

Markus
-- 

 Ich nicht eine Sekunde!!!" H. Heinol in Val Thorens

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Markus Becker wrote:
> 
> Remco de Korte wrote:

I've been working on something like this in a program and an include file. I
have an include that calculates points on a sphere based upon such a thing with
twelve points :), recursively, which is only for a fixed amount of points but it
takes ages to parse and I'm afraid the same will happen for the include file
that will distribute any amount of points. 
I don't really need it (anymore) but if anyone is interested I could give it a
go, but don't complain about parsing time.
(as an example: calculating some 16000 points took over a day and then I broke
it off - the same would probably take a couple of minutes in a separate program)

But, to come to my actual reply: the problem with the local minimum is something
that has been bothering me too. I think it has something to do with the points
being on a sphere. There should be workarounds, but as I said, I abandoned it
for now. I think someone will come up with a brilliant solution one of these
days.

Regards,

Remco
http://www.xs4all.nl/~remcodek/pov.html

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Remco de Korte wrote:
> 
> But, to come to my actual reply: the problem with the local minimum is something
> that has been bothering me too. I think it has something to do with the points
> being on a sphere. There should be workarounds, but as I said, I abandoned it
> for now. I think someone will come up with a brilliant solution one of these
> days.

The problem is, that there is only a solution for some speacial
values of N (ie. magic numbers...). For all other values of N
this is simply impossible. You can always minimize some energy
balance, even a global one, but don't expect to be able to
distribute the points evenly on a sphere.

Markus
-- 

 Ich nicht eine Sekunde!!!" H. Heinol in Val Thorens

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Markus Becker wrote:
> 
> Remco de Korte wrote:
> >
> > But, to come to my actual reply: the problem with the local minimum is something
> > that has been bothering me too. I think it has something to do with the points
> > being on a sphere. There should be workarounds, but as I said, I abandoned it
> > for now. I think someone will come up with a brilliant solution one of these
> > days.
> 
> The problem is, that there is only a solution for some speacial
> values of N (ie. magic numbers...). For all other values of N
> this is simply impossible. You can always minimize some energy
> balance, even a global one, but don't expect to be able to
> distribute the points evenly on a sphere.
> 
> Markus

Is that so? I'm willing to believe it, but I wonder if there's some way to prove
that or make it acceptable for a mathematician (which I'm most definitely not).

Groeten,

Remco

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Remco de Korte wrote:
> 
> Is that so? I'm willing to believe it, but I wonder if there's some way to prove
> that or make it acceptable for a mathematician (which I'm most definitely not).

Yes, this _is_ so. It is mathematically proven. A simplified (and short)
explanation is that you can only tile certain surfaces with certain
patches.
The certain surface bein the spere and the patches being triangles,
pentagons
ans so on for the sphere surface.
If you have access to Scientific American, you should be able to find
articles about it.

For some reading, look here:
http://www.rose-hulman.edu/~brought/Epubs/REU/Wabash.html

Markus
-- 

 Ich nicht eine Sekunde!!!" H. Heinol in Val Thorens

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Markus Becker wrote:
> 
> Remco de Korte wrote:
> >
> > Is that so? I'm willing to believe it, but I wonder if there's some way to prove
> > that or make it acceptable for a mathematician (which I'm most definitely not).
> 
> Yes, this _is_ so. It is mathematically proven. A simplified (and short)
> explanation is that you can only tile certain surfaces with certain
> patches.
> The certain surface bein the spere and the patches being triangles,
> pentagons
> ans so on for the sphere surface.
> If you have access to Scientific American, you should be able to find
> articles about it.
> 
> For some reading, look here:
> http://www.rose-hulman.edu/~brought/Epubs/REU/Wabash.html
> 
> Markus

Aha! Thanks! This means I won't have to feel frustrated because I can't solve
this little problem.

CU!

Remco

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I don't know why I didn't see this thread earlier... it's already one
month old...

However, I just recently wrote and published a macro for geodesic
spheres which happen to be bodies that are constructed by dividing the
surface of a sphere into triangles. The principle behind it starts from
the three basic polyhedra with triangular faces:
  tetrahedron  4 faces
  octahedron   8 faces
  icosahedron  20 faces
and divides each triangular face into smaller triangles. Though it is
NOT possible to put any number of points (or triangles/corners) onto a
sphere you can choose from quite a variety of values by just selecting
the basic polyhedron and the "frequency" of subdivision for each face.
The number of faces on the final sphere is then calculated as 

  number of faces on polyhedron * (freq^2)

so you can get the following values:

freq      tetrahedron    octahedron   icosahedron
  1             4             8            20
  2            16            32            80
  3            36            72           180
  4            64           128           320
  5           100           200           500
  6           144           288           720
  8           256           512          1280
etc.    

By defining your own "drawing" (better: object placing) routine for each
face, edge and corner, you can easily distribute whatever object you
want on the faces.

You can download the macro, documentation  and see some examples at
http://www.geocities.com/SiliconValley/Lakes/5432/povray/geodesic.html


Uwe.

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