title: Calculate the area of points
body: <p>I was not able to find good answers in Google, or perhaps I am just missing the correct key words. Any help or suggetions are welcome!</p>
  
  <p>My problem is the following:
  I want to calculate the area a certain point cloud covers (in 2D). I know that mathematically speaking the area is 0, but I am only able to take sample points out of the correct distribution. Additionally I do not have any information about the boundary of the point cloud, every shape is possible, including holes etc. So algorithms using the boundary of a manifold will not work?!. </p>
  
  <p>Since the functions I am working with are smooth I can assume that the space in between points also belongs to the area I want to calculate. </p>
  
  <p>At the moment I divide the space into a lot of small boxes and count how many boxes are populated with one or more points. The count multiplied with the box size gives me an area. </p>
  
  <p>Is there a more elegant solution to this? Any ideas?</p>
  
  <p>Thanks 
  Thomas</p>
  
  <hr>
  
  <p>EDIT:</p>
  
  <p>What I do is projecting high dimensional points to a low dimensional embedding. I can determine the number of points in the high dimensional space and therefore also the number of points in the low dimensional space which form the area I want to calculate. If I increase the number of points it turns out that they are positionned between the "old" points, that is what I mean by smooth. Given a certain point I can assume that in some proximity around that point I will be able to find new points belonging to the area if I sample more dense. </p>
  
  <p>Additionally I have a threshold value at which I can consider two points to be "equal", or in other words I know which resulution I want to achieve. </p>
  
  <hr>
  
  <p>EDIT 2:</p>
  
  <p>I use GPLVM's to do the mapping from high dimensional space to low dimensional space. So I think analysing that directly is two difficult/not possible. They are not very intuitive and I think in that case it is easier to work directly with the two dimensional points...</p>