MATLAB: How to Generate 100 Random points inside the figure

100 random points

[x,y,z]=sphere;
mesh(x,y,z+9);
hold on;
% cylinder
r = 1.0;
h = 9.0;
m = h/r;
[R,A] = meshgrid(linspace(0,r,11),linspace(0,2*pi,41));
X = R .* cos(A);
Y = R .* sin(A);
Z = m*R;
% Cone around the z-axis, point at the origin
mesh(X,Y,Z)
hold on
axis equal
phi = -pi/3;
X1 = X*cos(phi) - Z*sin(phi);
Y1 = Y;
Z1 = X*sin(phi) + Z*cos(phi);
[x,y,z]=sphere;
mesh(x*0.6,y*0.6,z*0.6);

Best Answer

Create the 3 shapes at first:
warning('off', 'MATLAB:alphaShape:DupPointsBasicWarnId');
Shape = cell(1, 3);
% Top sphere:
[x, y, z] = sphere;
Alpha     = 2;
Shape{1}  = alphaShape(x(:), y(:), z(:) + 9, Alpha);
% Cone:
r = 1.0;
h = 9.0;
m = h/r;
[R,A] = meshgrid(linspace(0,r,11), linspace(0,2*pi,41));
X2 = R .* cos(A);
Y2 = R .* sin(A);
Z2 = m * R;
Shape{2} = alphaShape(X2(:), Y2(:), Z2(:), Alpha);
% Bottom sphere:
Shape{3} = alphaShape(x(:) * 0.6, y(:) * 0.6, z(:) * 0.6, Alpha);
warning('on', 'MATLAB:alphaShape:DupPointsBasicWarnId');
% Surrounding cuboid:
coorMin   = [-1, -1, -0.6];  % Minimal x,y,z value
coorRange = [2, 2, 10.4];    % Max - Min of x,y,z value
Now choose random points inside the surrounding cuboid and collect the points, which are in any of the shapes:
% Get points inside the shapes:
maxIter = 1e6;  % Security limit
iter    = 0;
n       = 100;  % Number of output points
P       = zeros(n, 3);
iP      = 0;
while iP < n && iter <= maxIter
   Q = rand(1, 3) .* coorRange + coorMin;
   if inShape(Shape{1}, Q) || inShape(Shape{2}, Q) || inShape(Shape{3}, Q)
      iP       = iP + 1;
      P(iP, :) = Q;
   end
   iter = iter + 1;
end
if iter == maxIter  % Increase maxIter on demand:
   error('Cannot find enough points.');
end
This approach is simpler than creating one shape only, because you do not have to consider the overlap or know anything about the geometry or relative position of the objects. Even the sharp kink between the lower sphere and the cone is not problem here.
The smaller the volume of the shape containing the accepted points compared to the surrounding cuboid, the higher is the number of rejected points. Set maxIter accordingly.
Display the shape and the points:
% Show the shape and the points:
figure;
subplot(1,2,1, 'NextPlot', 'add');
for k = 1:3
   plot(Shape{k});
end
axis([-1,1, -1,1, -1,10], 'equal')
view(3)
subplot(1,2,2, 'NextPlot', 'add', 'XLim', [-1,1], 'YLim', [-1,1]);
plot3(P(:,1), P(:,2), P(:,3), '.');
axis([-1,1, -1,1, -1,10], 'equal')
view(3)
Here with n=1000 points:
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