Random points on the Earth

The problem with puzzles is that you keep it in your head for days, until you find an answer. Or at least some ideas about a possible answer. This is what happened to me a few weeks ago, when a colleague of mine asked me the following question : Consider points uniformly distributed on a sphere. What is the probability that the points lie on a same hemisphere, for some hemisphere (there is no south or north here) ?

Analogously, what is the probability to see the points on the Earth, at the same time, from somewhere in the galaxy ? (even extremely far away, so we can see a complete hemisphere) I wanted to use Monte Carlo simulations to estimate that probability, for some . But it was difficult. I mean, given points on the sphere, in  can you easily determine if they lie on a common hemisphere, or not ? I did try with distance, or angle, but I could not find a simple answer. So I tried a technique I did learn a few years ago : if you cannot do something in dimension 3, try first in dimension 2.

Again, I could not find a simple answer. But there is a simple technique.

1. Draw points on the unit sphere, which simply means generate random variables 
2. Try to find such that, after a rotation (with angle ) all the points lie in the upper part (the North hemisphere, for instance)

So the question here is simply : is

equal to ? Of course, from a computational point of view, it is slightly more complex, since this function is not differentiable.

n=5
Theta=runif(n)*2*pi
top=Vectorize(function(theta) sum(sin(Theta+theta)>=0))

So a simple strategy can be to compute those values on a finite grid, and to check if, for some , all the points lie in the upper part

max(top(seq(0,2*pi,length=6001)))==n

Hence, with the following sample, all the points cannot be on a common hemisphere

set.seed(2)
Theta=runif(5)*2*pi

while, for another sample of points, it was possible

set.seed(7)
Theta=runif(5)*2*pi

With this simple code, we get get the probability, but only in dimension 2 (so far)

SIM=Vectorize(function(n) simul(n,1000))
plot(3:10,SIM(3:10))

In dimension 3, it is still possible to use also a polar representation. Things are easier to generate, but also it is simple to consider rotations. And again, a simple algorithm can be derived,

But there is a simple technique.

1. Draw points on the unit sphere, which simply means generate  random variables  and
2. Try to find  and such that, after a rotation (with angles and ) all the points lie in some given part (say  )

As mentioned by Dominque, using this technique, points are not uniformly distributed on the sphere. Instead we can use

1. Draw  points on points from a trivaraite Gaussian distribution, and normalize it
2. Get the polar coordinates, and try to find  and  such that, after a “rotation” (with angles  and ) all the points lie in some given part (say  )

So the question here is simply : is

(I am not sure about the set of angles for the rotations, so I tried a larger one, just in case). Again, it would be complex, or more complex than before, because we need here a joint grid. For instance

MZ=matrix(rnorm(n*3),n,3)
d=apply(MZ,1,function(z) sqrt(sum(z^2)))
X=MZ[,1]/d; Y=MZ[,2]/d; Z=MZ[,3]/d; 
Theta=acos(Z)
Phi=acos(X/sqrt(X^2+Y^2))*(Y>=0)+(2*pi-acos(X/sqrt(X^2+Y^2)))*(Y<0)
top=function(theta,phi) sum(sin(Theta+theta)*cos(Phi+phi)>=0)
TOP=mapply(top,rep(seq(0,2*pi,length=1001),1001),rep(seq(-pi,pi,length=1001),each=1001))
max(TOP)==n

As we can see with the red curve, there might be some problems here. Because, (as mention in kmath327), this problem was solved in any dimension in Wendel (1962) with the following simple expression : in dimension , the probability that the points (uniformly distributed on a sphere) lie on a same hemisphere is exactly

p=function(d,n) .5^(n-1) * sum(choose(n-1,0:(d-1)))

Note that Leonard Savage proved (a few years before) that

(which can be obtained easily actually). I do not see what’s wrong with my Monte Carlo simulations… and if anyone has a nice Monte Carlo strategy to get that probability, I’d be glad to hear it !