Point Cloud Generation

I have a 3-D geometrical shape which I have to convert into a point cloud.
The resultant point cloud can be considered equivalent to a point cloud output from a Laser Scan of the object.
No mesh generation is neeeded
The points generated may be evenly spaced, or maybe just randomly spaced - doesn't matter
The 3-D shape can be provided in the form of a 3-D mathematical formula
This has to be done using MATLAB

Solution

It's difficult to answer without an example but it sounds like you just want to perform a montecarlo simulation?

Lets say your shape is defined by the function f and that you have X, Y limits stored in two element vector e.g. xlim = [-10 10] i.e. all possible x values of this shape lie between x = -10 and x = 10 then I would suggest that you make f return some sort of error code if there is no value for a specific x-y pair. I'm going to assume that will be NaN. So f(x,y) is a function you are writing that either returns a z if it can or NaN if it can't

n= 10000;
counter  = 1;
shape = nan(n, 3)
while counter < n
   x = rand*diff(xlim) + mean(xlmin);
   y = rand*diff(ylim) + mean(ylim);
   z = f(x,y)
   if ~isnan(z)
      shape(counter, :) = [x, y, z];
      counter = counter + 1    
   end
end

So the above code will generate 10000 (non unique, but that's easily adapted for) points randomly sample across your shape.

Now after typing this I realise that perhaps your shape is actually not all that big and maybe you can uniformly sample it rather than randomly:

for x = xlim(1):xstep:xlim(2)
   for y = ylim(1):ystep:ylim(2)
       shape(counter, :) = [x, y, f(x,y)];
   end
end

or if you write f to be vectorized (preferable)

shape = [(xlim(1):xstep:xlim(2))', (ylim(1):ystep:ylim(2))', f(xlim(1):xstep:xlim(2), ylim(1):ystep:ylim(2));

and then either way

shape(isnan(shape(:, 3), :) = []; %remove the points that fell outside the shape