Re: Maths Problem
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
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
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
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
Re: Maths Problem
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
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
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
* 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/
