What does the Inclusion-Exclusion formula applied to balls look like?

If is a set of balls and | . | denotes the volume of an object, then this is the formula.


You just add and subtract more and more intersections till you run out. The same applies for surface area of spheres. If you're interested, here is a way too detailed proof.

This method is simple to understand. The problem is that if you have more than 4 balls you run into trouble. You are stuck with figuring out if all the intersections exists and what they are.

Conclusion: This method is great if you have a small number of balls or if you know you don't have any intersections of 4 or more balls.
If you do, then you're going to have to think a little harder.

If you're new, check out part 1. First, means union and means intersections. Just so you know.

So how do you start computing the intersection of a union of balls? Lets look at some simple examples.

If you have two balls, you can start with the volume of 2 balls. But then you need to subtract their intersection because that part was added in twice.
The resulting formula looks like this


If you have three balls, you subtracted too much so you need to add in the 3-sphere intersection.


You may see a pattern here, and there is. The resulting formula is the Inclusion-Exclusion Formula. More on that next time.

Again, mathematical balls.

Made you look, but I mean balls in the mathematical sense. Sorry.

My wife always complains that I do too much cycling and not enough research. I don't see the problem, but I'll humor her.

This is the first in a multi-part series on measuring the surface and volume of a union of spheres in 3-dimensions, like the object on the right.

The work is motivated by applications in Computational Biology. It has been shown that the surface area of a molecule is proportional to its interaction with the surrounding water.

Specifically, we are interested in the Accessible Surface Area (ASA). We are looking for how much area is available to a probe representing the water. The probe is modeled by a sphere of radius 1.4 angstrom, the approximate radius of a water molecule.

Bellow is an example is 2-dimensions. The atoms are red, the probe is green and the ASA is blue.
As you can see, the ASA can then be computed as the surface area of the spheres if each radius is increased by the probe radius.

Usually we are looking for the accessible surface of each atom individually. We then weight the values based on the atom type. Doing this provides more meaningful results than just using the total area.

I passed my qualifying exam today!

Now back to riding my bike.