The interior of the 4-sphere may equally well be any spherically-symmetric distribution. In that case, what we are doing geometrically is: 1. Normalising the vector (sending R^4 to S^3); 2. Applying the Hopf map (sending S^3 to S^2); 3. Taking one coordinate (sending S^2 to [-1, 1]). Sincerely, Adam P. Goucher
Sent: Saturday, July 05, 2014 at 7:48 PM From: "Mike Stay" <metaweta@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Gary Antonick is edging away from the following bonus puzzle
What are those points, geometrically? Somehow we're getting rid of two dimensions (from the interior of a 4-sphere to the surface of a 3-sphere).
On Sat, Jul 5, 2014 at 11:24 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
From http://mathworld.wolfram.com/SpherePointPicking.html
<< Cook (1957) extended a method of von Neumann (1951) to give a simple method of picking points uniformly distributed on the surface of a unit sphere. Pick four numbers x_0, x_1, x_2, x_3 from a uniform distribution on (-1,1) , and reject pairs with x_0^2 + x_1^2 + x_2^2 + x_3^2 >= 1. (12)
From the remaining points, the rules of quaternion transformation then imply that the points with Cartesian coordinates x = 2 (x_1x_3 + x_0x_2) / (x_0^2 + x_1^2 + x_2^2 + x_3^2) (13) y = 2 (x_2x_3 - x_0x_1) / (x_0^2 + x_1^2 + x_2^2 + x_3^2) (14) z = (x_0^2 + x_3^2 - x_1^2 - x_2^2) / (x_0^2 + x_1^2 + x_2^2 + x_3^2) (15) have the desired distribution (Cook 1957, Marsaglia 1972). >>
WFL
On 7/5/14, Eugene Salamin via math-fun <math-fun@mailman.xmission.com> wrote:
Yes, I slipped up; you do need sqrt to normalize the vector.
That z is uniform in [-1,+1] is a consequence of the solid geometry theorem that the area of a sphere between two parallel planes depends only on the separation between the planes, and is independent of what part of the sphere lies between the planes.
Also, in doing integrations over the sphere in spherical coordinates, the element of area sinθ dθ dφ under the change of coordinates μ = cosθ becomes dμ dφ.
-- Gene
________________________________ From: Cris Moore <moore@santafe.edu> To: Eugene Salamin <gene_salamin@yahoo.com>; math-fun <math-fun@mailman.xmission.com> Sent: Saturday, July 5, 2014 9:47 AM Subject: Re: [math-fun] Gary Antonick is edging away from the following bonus puzzle
Yes, of course... but the fact that z is uniform in [-1,+1] is quite surprising when you first see it.\
Cris
On Jul 5, 2014, at 9:55 AM, Eugene Salamin via math-fun <math-fun@mailman.xmission.com> wrote:
Random points on the unit n-sphere can be generated without the need for sqrt and trig. Generate n+1 independent Gaussian random numbers, and normalize the vector to unit length.
-- Gene
________________________________
From: Cris Moore <moore@santafe.edu> To: math-fun <math-fun@mailman.xmission.com> Sent: Saturday, July 5, 2014 8:38 AM Subject: Re: [math-fun] Gary Antonick is edging away from the followingbonuspuzzle
Consider also the following lovely fact: choose a point on the unit sphere uniformly at random. It's z-coordinate is uniformly distributed in the interval [-1,+1]!
To put it differently, you can generate a random point (x,y,z) like this:
choose z uniformly in [-1,+1] choose theta uniformly in [0,2pi] set x = sqrt(1-z^2) cos theta, y = sqrt(1-z^2) sin theta
This is only true for the 3-dimensional sphere, of course!
- Cris
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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