Random 3D points uniformly distributed on an ellipse shaped window of a sphere

Question.m

close all
clear all
clc
%%
n = 1000;
z = rand(n,1)*2-1;
t = rand(n,1)*2*pi;

x = sqrt(1. - z.*z).*cos(t);
y = sqrt(1. - z.*z).*sin(t);

figure('color','w')
scatter3(x,y,z,'ok','markerfacecolor','w')
hold on
[x,y,z] = sphere(50);
p = surf(x,y,z);
set(p,'facecolor','red','edgecolor','none','facealpha',.5);
daspect([1 1 1]);
view(3); axis tight; grid on;
camlight; lighting gouraud;
axis equal
axis off
export_fig('sphere.png')
%%
close all
clear all
clc

n = 1000;
a = .5;
cosa = cos(a);
z = rand(n,1)*(cosa - 1.) - cosa;
t = rand(n,1)*2*pi;
x = sqrt(1. - z.*z).*cos(t);
y = sqrt(1. - z.*z).*sin(t);

l = 1.2;
nn = 100;
[d,t] = meshgrid(linspace(0,l,nn),linspace(0,2*pi,nn));
r = repmat(linspace(0,tan(a)*l,nn),nn,1);

xx = r.*cos(t);
yy = r.*sin(t);
zz = -d;

figure('color','w')
p = surf(xx,yy,zz);
set(p,'facecolor','green','edgecolor','none','facealpha',.25);
hold on
plot3(xx(:,end),yy(:,end),zz(:,end),'k','linewidth',3)
scatter3(x,y,z,'ok','markerfacecolor','w')
[x,y,z] = sphere(50);
p = surf(x,y,z);
set(p,'facecolor','red','edgecolor','none','facealpha',.5);
daspect([1 1 1]);
view(3); axis tight; grid on;
view(0,15)
camlight; lighting gouraud;
axis equal
axis off
export_fig('circle1.png')
view(0,90)
export_fig('circle2.png')
%%
close all
clear all
clc

nn = 100;
r = .8;
dis = 2;
Z = -1.3;
t = linspace(0,2*pi,nn);
x = r*cos(t);
y = dis+r*sin(t);
z = Z*ones(1,nn);

figure('color','w')
plot3(x,y,z,'k','linewidth',3)
hold on
p = surf([zeros(1,nn);x],[zeros(1,nn);y],[zeros(1,nn);z]);
set(p,'facecolor','green','edgecolor','none','facealpha',.25);


inx = 0;

while inx < 200
    
    z = rand()*2-1;
    t = rand()*2*pi;
    
    xyz = [sqrt(1. - z.*z).*cos(t);sqrt(1. - z.*z).*sin(t);z];
    xyz = xyz/norm(xyz);
    d = Z/xyz(3);
    p2 = d*xyz;
    
    if inpolygon(p2(1),p2(2),x,y) && d > 0
        inx = inx+1;
%         plot3([0,p2(1)],[0,p2(2)],[0,p2(3)])
        scatter3(xyz(1),xyz(2),xyz(3),'ok','markerfacecolor','w')
    end
end