Compare point cloud to surrounding contour

I am currently working on a CV project and I need to determine to which extent a point cloud and a contour overlap. Please consider the image below:

The black area is a very dense cloud of points. Note that the points are not uniformly spaced. The blue line is the reference to which the point cloud has to be compared. There are four things I have to figure out:

1. The area each point takes up. Points do not have an area per se but I have to assign one.
2. The area of the contour that is filled with black.
3. The black area outside of the contour, i.e. area that should not be there.
4. The area of the contour that is filled with white, i.e. area that is missing.

Note that point cloud and contour can have arbitrary shapes. First I tried to compute the black area by Delaunay triangluating the points but it did not work because it connects the black components with each other which results in additional area. Does anybody know how to tackle this issure?

Thank you for your efforts.