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MathGroup Archive 2000

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Random spherical troubles

  • To: mathgroup at smc.vnet.net
  • Subject: [mg25170] Random spherical troubles
  • From: Barbara DaVinci <barbara_79_f at yahoo.it>
  • Date: Tue, 12 Sep 2000 02:58:56 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

 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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