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

Barbara Da Vinci
barbara_79_f at yahoo.it

______________________________________________________________________
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```

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