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MathGroup Archive 2000

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RE: Random spherical troubles

  • To: mathgroup at smc.vnet.net
  • Subject: [mg25198] RE: [mg25170] Random spherical troubles
  • From: PNichols at cornell-iowa.edu
  • Date: Tue, 12 Sep 2000 21:24:43 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Barbara--

I think this will do what you need; some explanation follows later.

============================================

  phi := Random[Real, {0, 2 Pi}; (* uniform distribution, for longitude *)
  theta := ArcCos[1 - 2 Random[]]; (* non-uniform distribution, for latitude
*)

Note the use of :=, not =, for then we can make a list of random points
thus:

  sphericalData = Table[{phi, theta}, {5000}];

  ListPlot[sphericalData]

The ListPlot will clearly show the sparseness of the points near theta=0 or
theta=Pi.

To map the points into the sphere, I'll use

  sphereMap[{phi_, theta_}] := {Sin[theta] Cos[phi], Sin[theta] Sin[phi],
Cos[theta]}

  mappedSphericalData=sphereMap /@ sphericalData;

To display this as a "cloud" of 3D points, use

  Show[Graphics3D[Point /@ mappedSphericalData]];

Here's the contrasting "wrong" case -- uniform distribution on the rectangle
-- in similar style:

  rectangularData = Table[{Random[Real, {0, Pi}], Random[Real, {0, Pi}]},
{5000}];

  ListPlot[rectangularData];

(Note uniform distribution on the rectangle.)

  mappedRectangularData = sphereMap /@ rectangularData;

  Show[Graphics3D[Point /@ mappedRectangularData]];

If you have Mathematica version 4, evaluate

  <<RealTime3D` 

before the Show statements above, and then you can move the pictures around
with your mouse.  That makes the 3D pictures easier to understand.  (Use
<<Default3D` to revert to ordinary 3D display.)

============================================
DISCUSSION (more mathematics than Mathematica)

The formula

  theta := ArcCos[1 - 2 Random[]];

needs explanation.  I'm not an expert in probability theory, but here goes.

Because the Jacobian of the sphere map measures the distortion of the area,
we need random values of theta with a probability density
f[theta_]:=Sin[theta]/2 on [0, Pi].  The Jacobian is independent of phi,
reflecting the fact that the "pull-back" area-measure from the sphere to the
rectangle is uniform along the phi-direction.  (There is also a factor of
rho^2, where rho is the radius of the sphere; we use rho=1.  The division by
2 makes the total probability 1.)

There is a theorem (not terribly difficult to prove) which states that if
you have a random variable X uniformly distributed on [0,1]  (like
Random[]), and you want a random variable Y with density f, then you can do
this (warning:  pseudo-Mathematica pseudo-code follows!):

(1)  Find the "cumulative distribution function",
        cdf[y_]=Integrate[f[t],{t,0,y}];
(2)  solve x==cdf[y] for y (i.e. find the inverse function of the cdf);
(3)  Substitute the random variable X for the dummy variable x in the
inverse function just found.

(All this works well if f is non-negative and zero only at isolated points,
so that the cdf is continuous and strictly increasing.  More general cases
require a generalized type of "inverse function", but that's not relevant
for your problem.)

Since the functions in this problem are so nice, Mathematica can do nearly
everything for us:

  Solve[x == Integrate[Sin[y]/2, {y, 0, y0}], y0] 

  {{y0 -> -ArcCos[1 - 2*x]}, {y0 -> ArcCos[1 - 2*x]}}

There, in the second solution, is the function your problem needs.

I have made a notebook containing more detailed notes on this problem; let
me know if you want a copy.

Best regards,

Preston Nichols
Department of Mathematics
Cornell College


============================================

-----Original Message-----
From: Barbara DaVinci [mailto:barbara_79_f at yahoo.it]
To: mathgroup at smc.vnet.net
Subject: [mg25198] [mg25170] Random spherical troubles


 Hi MathGrouppisti

 This time, my problem is to generate a set of
directions randomly
 distributed over the whole solid angle. 

 This simple approach is incorrect (spherical
coordinates are assumed) :

 Table[{Pi Random[], 2 Pi Random[]} , {100}]

 because this way we obtain a set of point uniformly
distributed
 over the [0 Pi] x [0 2Pi] rectangle NOT over a
spherical surface :-(

 If you try doing so and plot the points {1,
random_theta , random_phi}
 you will see them gathering around the poles because
that simple
 transformation from rectangle to sphere isn't
"area-preserving" . 

 Such a set is involved in a simulation in statistical
 mechanics ... 
 and I can't get out this trouble.

 May be mapping [0 Pi] x [0 2Pi] in itself , using an
suitable 
 "non-identity" transformation, can spread points in a
way balancing
 the poles clustering effect.
 

====================================================================

 While I was brooding over that, an intuition flashed
trought my mind :
 since spherical to cartesian transformation is
  
  x = rho Sin[ theta ] Cos[ phi ]
  y = rho Sin[ theta ] Sin[ phi ]
  z = rho Cos[ theta ]
 
 perhaps the right quantities to randomly spread
around are Cos[ theta ] and
 Cos[ phi ] rather than theta and phi for itself. Give
a glance at this : 

 Table[{
 ArcCos[ Random[] ], 
 ArcCos[ Random[] Sign[ 0.5 - Random[] ]
 } , {100}] 
 
 Do you think it is close to the right ? Do you see a
better way ?
 Have you just done the job in the past ? Should I
reinvent the wheel ?


====================================================================


 I thanks you all for prior replies and in advance
this time.

 Distinti Saluti
 (read : "Faithfully yours")

 Barbara Da Vinci
 barbara_79_f at yahoo.it
 


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