Re: Maths Problem

2001-06-18 Thread Philip Newton

Chris Benson wrote:
 Mmmm, so if there are 3 water lilies with circular leaves, what
  is the largest they can grow on the surface of a sphere without
 overlap? On a circle it's easy to see it's just less than the
 radius of the circle.  Not so easy with a sphere.

Well, first off, the circles won't be circles as we know them since
they're not 2D circles but have a 3D component (or they wouldn't be on the
surface of the sphere but rather cutting a slice through it).

However, I'd imagine that with three such bulgy circles, the best you can do
is space them equally around the equator.

Cheers,
Philip
-- 
Philip Newton [EMAIL PROTECTED]
All opinions are my own, not my employer's.
If you're not part of the solution, you're part of the precipitate.



Re: Maths Problem

2001-06-18 Thread Roger Burton West

On Mon, Jun 18, 2001 at 12:01:05AM +0100, Chris Benson wrote:
Mmmm, so if there are 3 water lilies with circular leaves, what is the
largest they can grow on the surface of a sphere without overlap?  
On a circle it's easy to see it's just less than the radius of the 
circle.  Not so easy with a sphere.

Looks like evenly-spaced around the equator. With only three points,
they'll _have_ to be coplanar by definition. And, of course, a belt
of n points around the equator is even spacing, but doesn't look good...

Roger



Re: Maths Problem

2001-06-18 Thread Chris Benson

On Mon, Jun 18, 2001 at 07:29:28AM +0100, Roger Burton West wrote:
 On Mon, Jun 18, 2001 at 12:01:05AM +0100, Chris Benson wrote:
 Mmmm, so if there are 3 water lilies with circular leaves, what is the
 largest they can grow on the surface of a sphere without overlap?  
 
 Looks like evenly-spaced around the equator. With only three points,
 they'll _have_ to be coplanar by definition. And, of course, a belt
 of n points around the equator is even spacing, but doesn't look good...

But that limits the diameter of each to 1/3 the circumference ...
I was sort of thinking that you'd get a larger area by offsetting
them 
O O
 O
, that doesn't really cut it does it?   

I'm going off to cut out some circles and look for a ball ...
-- 
Chris Benson



Re: Maths Problem

2001-06-18 Thread Chris Benson

On Mon, Jun 18, 2001 at 11:56:59AM +0100, David Cantrell wrote:
 On Mon, Jun 18, 2001 at 08:29:18AM +0200, Philip Newton wrote:
  Chris Benson wrote:
   Mmmm, so if there are 3 water lilies with circular leaves, what
is the largest they can grow on the surface of a sphere without
  
  Well, first off, the circles won't be circles as we know them since
  they're not 2D circles but have a 3D component (or they wouldn't be on the
  surface of the sphere but rather cutting a slice through it).

Leaves aren't that strong -- they'd flop into curve to fit the sphere :-)

  However, I'd imagine that with three such bulgy circles, the best you can do
  is space them equally around the equator.
 
 Yes.  However you arrange them they're going to be on a plane, and so to
 have them the maximum distance apart you make sure the plane also contains
 the centre of the sphere.  It gets interesting for N3

I thought N=4 was the easy one: points of a tetrahedron!
-- 
Chris Benson



Maths Problem

2001-06-17 Thread Greg McCarroll


I was working on my talk for YAPC::Europe and I got a little distracted,
with the following problem and I also thought some of you might like to 
think about it.

First of all, consider the problem of distributing N points around the
origin evenly in 2D, so they are all the same distance from the origin.

Now this is quite easy, you can simply imagine a circle and the points
placed around the circle, each 360/N degrees apart in terms of projections
from the origin.

Simple huh?

Ok, now how can you distribute N points around the origin in _3_ dimensions,
again all of them at the same distance from the origin? Obviously
there will be an imaginary sphere again, but where do you put the points.

Thoughts are welcome, i'm currently trying to solve it and having
lots of gotchas. However if you have a complete solution please put
in some *spoiler* space.  

Greg

-- 
Greg McCarrollhttp://217.34.97.146/~gem/



Re: Maths Problem

2001-06-17 Thread Roger Burton West

On Sun, Jun 17, 2001 at 06:52:04PM +0100, Greg McCarroll wrote:

Ok, now how can you distribute N points around the origin in _3_ dimensions,
again all of them at the same distance from the origin? Obviously
there will be an imaginary sphere again, but where do you put the points.

Best general treatment of this I've seen is at
http://www.math.niu.edu/~rusin/known-math/index/spheres.html

which has the summary:

* uniformly distributed has more than one meaning;
* for most n there is no answer which is particularly elegant;
* quick-and-dirty approximations are easy.

R



Re: Maths Problem

2001-06-17 Thread Mindaugas Genutis

How about drawing a 3D shape (depending upon the value of N) with equal
distances between neighbour nodes and equal angles between the edges? All the
nodes lie on the imaginary sphere and the distance to the center is the same.
Thus you get one and only one shape for each value of N. You can rotate it
inside the sphere.

How about putting them randomly on that sphere? Or use one of the well-known
distributions (Poisson distribution for example)? (use 3 coordinate versions of
these distributions)

What about using a random/stochastic process (Markov, for instance). (use 3
coordinate version of these processes)

Greg McCarroll wrote:

 I was working on my talk for YAPC::Europe and I got a little distracted,
 with the following problem and I also thought some of you might like to
 think about it.

 First of all, consider the problem of distributing N points around the
 origin evenly in 2D, so they are all the same distance from the origin.

 Now this is quite easy, you can simply imagine a circle and the points
 placed around the circle, each 360/N degrees apart in terms of projections
 from the origin.

 Simple huh?

 Ok, now how can you distribute N points around the origin in _3_ dimensions,
 again all of them at the same distance from the origin? Obviously
 there will be an imaginary sphere again, but where do you put the points.

 Thoughts are welcome, i'm currently trying to solve it and having
 lots of gotchas. However if you have a complete solution please put
 in some *spoiler* space.

 Greg

 --
 Greg McCarrollhttp://217.34.97.146/~gem/

--
Mindaugas Genutis
Department of Communication Technology
Aalborg University of Technology





Re: Maths Problem

2001-06-17 Thread Chris Benson

On Sun, Jun 17, 2001 at 06:58:03PM +0100, Roger Burton West wrote:
 On Sun, Jun 17, 2001 at 06:52:04PM +0100, Greg McCarroll wrote:
 
 Ok, now how can you distribute N points around the origin in _3_ dimensions,
 again all of them at the same distance from the origin? Obviously
 there will be an imaginary sphere again, but where do you put the points.

Neat question for a Sunday evening: I've been wondering about that for a 
while.
 
 Best general treatment of this I've seen is at
 http://www.math.niu.edu/~rusin/known-math/index/spheres.html
 

and that page also has a link to Easy method for a fairly good point
distribution  at http://www.math.niu.edu/~rusin/known-math/97/spherefaq

An excellent site.
-- 
Chris Benson



Re: Maths Problem

2001-06-17 Thread Greg McCarroll

* Chris Benson ([EMAIL PROTECTED]) wrote:
 On Sun, Jun 17, 2001 at 06:58:03PM +0100, Roger Burton West wrote:
  On Sun, Jun 17, 2001 at 06:52:04PM +0100, Greg McCarroll wrote:
  
  Ok, now how can you distribute N points around the origin in _3_ dimensions,
  again all of them at the same distance from the origin? Obviously
  there will be an imaginary sphere again, but where do you put the points.
 
 Neat question for a Sunday evening: I've been wondering about that for a 
 while.

the main problem is for low values of N, i.e. the ones you can imaginine
in your head, you can figure out regular convex polyhedra whose points 
lie of the sphere and whose sides are all the same shape (i.e. a triangular
pyramid, a cube or diamond, etc. however i'm not convinced you can 
construct such shapes for all values of N

  Best general treatment of this I've seen is at
  http://www.math.niu.edu/~rusin/known-math/index/spheres.html
  
 
 and that page also has a link to Easy method for a fairly good point
 distribution  at http://www.math.niu.edu/~rusin/known-math/97/spherefaq

yes, but it leaves an unpleasant taste in your mouth afterwards,
or is that just me?

-- 
Greg McCarrollhttp://217.34.97.146/~gem/