MaplePrimes - comments on Post, random dots on an egg

MaplePrimes - comments on Post, random dots on an egg http://www.mapleprimes.com/posts/122970-Random-Dots-On-An-Egg en-us 2026 Maplesoft, A Division of Waterloo Maple Inc. Maplesoft Document System Thu, 11 Jun 2026 11:51:44 GMT Thu, 11 Jun 2026 11:51:44 GMT The latest comments added to the Post, random dots on an egg http://www.mapleprimes.com/images/mapleprimeswhite.jpg MaplePrimes - comments on Post, random dots on an egg http://www.mapleprimes.com/posts/122970-Random-Dots-On-An-Egg Random points on a non-sphere http://www.mapleprimes.com/posts/122970-Random-Dots-On-An-Egg?ref=Feed:MaplePrimes:random dots on an egg:Comments#comment123654 Your procedure would generate uniformly-distributed random points on a sphere, but when you map this to an ellipsoid the distribution may no longer be uniform with respect to area.  Suppose you generate random points for parameters (u,v) in a rectangle (in your case [-1,1] x [0,2*Pi]) and map (u,v) to F(u,v) in 3-dimensional space.  The mapping multiplies areas by the length of the cross product of the vectors <img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=ec43b8121f017129d94613755fd9715a.gif" alt="diff(F(u,v),u)"> and <img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=b0d8bf18815b2a2579979c7933a7fab3.gif" alt="diff(F(u,v),v)">; call this C(u,v).  In order for the mapped points to be uniformly distributed with respect to area, you want the density of your points in (u,v) space to be proportional to C(u,v).  In your case <pre> F:= (u,v) -> <rX*sqrt(1-u^2)*cos(v), rY*sqrt(1-u^2)*sin(v), rZ*u>; CP:= VectorCalculus:-CrossProduct(diff~(F(u,v),u), diff~(F(u,v),v)); C:= unapply(simplify(sqrt(CP[1]^2 + CP[2]^2 + CP[3]^2)),u,v);</pre> We can obtain our (u,v) points by a rejection sampling method: start out with uniformly distributed points, and accept or reject each one, accepting with probability proportional to C(u,v).  If 0 < rX <= rY <= rZ, the maximum of C is at u=0, v=0, where C(0,0) = rY*rZ.  So we accept (u,v) with probability C/(rY*rZ).  For example: > rX, rY, rZ:= 0.5, 1, 1.5;   C0:= rY*rZ;   with(Statistics):   RawPts:= zip(`[]`,Sample(Uniform(-1,1),10000),Sample(Uniform(0,2*Pi),10000)):   AVals:= Sample(Uniform(0,1),10000):   Accepted:= select(i -> (AVals[i] < C(op(RawPts[i]))/C0), [$1..1000]):   uvPts:= RawPts[Accepted];   Pts:= map(p -> [rX*sqrt(1-p[1]^2)*cos(p[2]),rY*sqrt(1-p[1]^2)*sin(p[2]),rZ*p[1]], uvPts);   plots[pointplot3d](Pts,axes=box,scaling=constrained,colour=red); <a href="/view.aspx?sf=123654/416854/poe.jpg"><img src="/view.aspx?sf=123654/416854/poe.jpg" alt=""></a>  <a href="/view.aspx?sf=123654/416854/ptsonellipsoid.jpg"> </a> The latest comments added to the Post, random dots on an egg 123654 Fri, 08 Jul 2011 00:30:44 Z Robert Israel Robert Israel uniform http://www.mapleprimes.com/posts/122970-Random-Dots-On-An-Egg?ref=Feed:MaplePrimes:random dots on an egg:Comments#comment123802 <a href="http://www.mapleprimes.com/posts/122970-Random-Dots-On-An-Egg#comment123654">@Robert Israel</a>  Great Robert, thanks a lot (I thought there was something a little odd for a small number of points), works great, Patrick. The latest comments added to the Post, random dots on an egg 123802 Wed, 13 Jul 2011 00:20:59 Z PatrickT PatrickT How to test visually that it is not uniform with the least number of points http://www.mapleprimes.com/posts/122970-Random-Dots-On-An-Egg?ref=Feed:MaplePrimes:random dots on an egg:Comments#comment123806 To me, visually, I couldn't tell if they were uniformly distributed or not, although it did look slightly more spaced at +45 degree and -45 degree latitudes. Is there a test that would visually make you say, ... Aha!  Yes it is not a uniform distribution.  And the trick here is to do it with the least number of points.  How can we do that? **edit -add - What I mean is, is there a simple 2-d graph that would determine this?  I suppose we could plot the number of points vs. theta vs. phi?  Then see if peaks occur? The latest comments added to the Post, random dots on an egg 123806 Wed, 13 Jul 2011 04:57:29 Z Christopher2222 Christopher2222 @Christopher2222  I don't know about http://www.mapleprimes.com/posts/122970-Random-Dots-On-An-Egg?ref=Feed:MaplePrimes:random dots on an egg:Comments#comment123891 <a href="http://www.mapleprimes.com/posts/122970-Random-Dots-On-An-Egg#comment123806">@Christopher2222</a>  I don't know about graphically, I can think of two ways to test this mathematically: first, a test based on counts: for example the χ2 dispersion test; and second, a test based on distances: for example Hopkins' test.   If I have some spare time this weekend I might try to implement these for the case in hand. The latest comments added to the Post, random dots on an egg 123891 Fri, 15 Jul 2011 15:22:43 Z longrob longrob