A spherical set of n spheres of equal radius r casts a shadow/silhouette on a plane. If they are far enough apart the area of the silhouette is equal to the sum of the silhouettes of the individual spheres. When they are so compressed (dense) they all occupy the space of a single sphere the area of the silhouette is of course equal to that of a single sphere. I'm looking for everywhere in between. I'm not a mathematician and I don't know the right terms to look up on here but I couldn't find anything on point using terms I know. A formula would be great, but even a reference to the term for this kind of problem or a link to a discussion of it somewhere would be great. Thanks.
Edit: Below is a link to an image which hopefully displays full size when you click it. The formula for the shadow of a sphere is the same for the area of circle of the same radius: pi times r squared (3.14 for r=1). On the left is a dispersed cloud of 100 spheres (all r=1) where the total shadow area is roughly the sum of the individual shadows since little overlap is happening. On the right is a cloud of radius 0 (a point) so all the spheres occupy the same space, and the total shadow is 3.14. The middle one is much harder. I'm looking for a general formula for these problems. 1
