how to generate random points located on the surface of a hemisphere with its center at (2, 1, 3) and a radius of 6?

N=500;
r = 6;
center = [2,1,3];
plot3(center(1),center(2),center(3),'ro')
hold on
for i=1:N
    x1 = rand;
    x2 = rand;
    Px = sqrt(x1*(2-x1))*cos(2*pi*x2)+ center(1);
    Py = sqrt(x1*(2-x1))*sin(2*pi*x2)+ center(2);
    Pz = 1-x1 + center(3);
    plot3(Px, Py, Pz,'o'),hold on
end
 p = [Px, Py, Pz];
 dist = Ddist(center,p);

What am I missing here? I'm using the dist variable to be sure I'll get 6, but currently I get 1

Answer 1

[Jeremiah Abimbola]

N=5000;
r = 6;
center = [2,1,3];
P = zeros(N,3) ;
plot3(center(1),center(2),center(3),'ro')
hold on
for i=1:N
    x1 = rand;
    x2 = rand;
    Px = sqrt(x1*(2-x1))* cos(2*pi*x2);
    Py = sqrt(x1*(2-x1))* sin(2*pi*x2);
    Pz = 1-x1;
    Px = Px * r + center(1);
    Py = Py * r + center(2);
    Pz = Pz * r + center(3);
    P(i,:) = [Px, Py, Pz];
    plot3(Px, Py, Pz,'o'),hold on
end

Answer 2

[KSSV]

Use spherical coordinates,,,pick angles randomly and then convert into cartesian coordinates. Try this:

N=500;
r = 6;
center = [2,1,3];
plot3(center(1),center(2),center(3),'ro')
hold on

P = zeros(N,3) ;
for i=1:N
    theta = randsample(linspace(0,2*pi),1);
    phi = randsample(linspace(0,pi),1);
    Px = center(1)+r*cos(theta)*sin(phi) ;
    Py = center(2)+r*sin(theta)*sin(phi) ;
    Pz = center(3)+r*cos(phi) ;
    P(i,:) = [Px, Py, Pz];
end

hold on
plot3(P(:,1), P(:,2), P(:,3),'.r'),
% dist = Ddist(center,p);

[Jeremiah Abimbola]

Yeah, this worked, but if i wanted to go by the code I provided, how can i do that? I see what you did there, radomizing the angles, but that's not the case for me. I tried to multiply x, y, z cordinates by the radius, but I got a distance greater than 6

Answer 3

[John D'Errico]

Actually, there are easier ways than those proposed, with no loops required. The trick is easy too, basd on the symmetry of randn. I'll pick the hemisphere with x(:,1) positive.

N=500;
r = 6;
center = [2,1,3];

R = randn(N,3);
R(:,1) = abs(R(:,1));
R = r*R./sqrt(sum(R.^2,2)) + center;

The last line uses capabilities found in R2016b or later. Easrlier versions of MATLAB would use bsxfun, twice.

Plot the points to show they do indeed lie on a hemisphere.

plot3(R(:,1),R(:,2),R(:,3),'o')
axis equal
box on
grid on
xlabel x
ylabel y
zlabel z

I tried to rotate things around so the hemisphere was clear. If you do so, you will see the sphere is indeed hollow, so on the surface of a hemisphere.

Note that it took all of 3 lines to generate the list of points, far lesss than I spent in plotting them.