Re: Random spherical troubles

*To*: mathgroup at smc.vnet.net*Subject*: [mg25184] Re: Random spherical troubles*From*: "Allan Hayes" <hay at haystack.demon.co.uk>*Date*: Tue, 12 Sep 2000 21:24:27 -0400 (EDT)*References*: <8pkl8u$m80@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Barbara, Noting that cylindrical projection preserves area we get, for a unit sphere, randomOnCylinder[n_] := Table[{Random[Real, {0, 2N[Pi]}], Random[Real, {-1, 1}]}, {n}] toSphere[{a_, z_}] := {#Cos[a], # Sin[a], z} &[Sqrt[1 - z^2]] randomOnSphere[n_] := toSphere /@ randomOnCylinder[n] Show[Graphics3D[Point /@ randomOnSphere[500]]] -- Allan --------------------- Allan Hayes Mathematica Training and Consulting Leicester UK www.haystack.demon.co.uk hay at haystack.demon.co.uk Voice: +44 (0)116 271 4198 Fax: +44 (0)870 164 0565 "Barbara DaVinci" <barbara_79_f at yahoo.it> wrote in message news:8pkl8u$m80 at smc.vnet.net... > 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 > > > > ______________________________________________________________________ > Do You Yahoo!? > Il tuo indirizzo gratis e per sempre @yahoo.it su http://mail.yahoo.it >