Uniform distribution of N points within a sphere

Hello, I'm trying to generate a uniform distribution of points within a spherical shell. I am able to generate a uniform distribution on the surface of a unit sphere using three gaussian random variables (normalized by sqrt(x^2+y^2+z^2), but am not sure how to convert this to an equal density distribution within the shell of some thickness, (d = r_outer - r_inner).

Answer 1

[Matt Fig]

Here is another solution which converts to Cartesian coordinates, though it does call the trig functions.

function [x,y,z] = rand_pick_sphere(n,a,b,X,Y,Z)
% Uniform points in a shell of inner radius a, outer radius b and center at
% (X,Y,Z)
% [x,y,z] = rand_pick_sphere(300,.5,.6);  % 300 points in shell between 
% r = .5 and r = .6, with center at origin.

if nargin==3
   X = 0;
   Y = 0;
   Z = 0;
end

r1 = (rand(n,1)*(b^3-a^3)+a^3).^(1/3);
phi1 = acos(-1 + 2*rand(n,1));
th1 = 2*pi*rand(n,1);
% Convert to cart.
x = r1.*sin(phi1).*sin(th1) + X;
y = r1.*sin(phi1).*cos(th1) + Y;
z = r1.*cos(phi1) + Z;

[Bruno Luong]

Few possible modifications on Matt's code:

r1 = (rand(n,1)*(b^3-a^3)+a^3).^(1/3);

cphi = -1 + 2*rand(n,1);

sphi = sqrt(1-cphi.^2);

th1 = 2*pi*rand(n,1);

z = r1.*cphi + Z;

r1 = r1.*sphi;

x = r1.*sin(th1) + X;

y = r1.*cos(th1) + Y;

[Matt Fig]

That would certainly speed things up a bit. Thanks, Bruno.

Answer 2

[Bruno Luong]

% a is inner radius
% b is outer radius
% is a number of points
a = 2;
b = 3;
n = 1000;

s = randn(3,n);
r = (rand(1,n)*(b^3-a^3)+a^3).^(1/3);
c = r./sqrt(sum(s.^2,1));
s = bsxfun(@times, s, c);

% Bruno