How to Calculate the Maximum Volume of Overlapping Cylinders on a 3D Grid?

[dzza]

Imagine that you are free to choose any 3 vertices from a 4 x 4 x 4 grid that has 16 equally distributed vertices, as if made up of 16 1x1x1 cubes. Now, having chosen your 3 vertices, these 3 vertices each serve as the "middle" of a circular cylinder. By middle I mean it is located at half the height of the cylinder, at the origin of the circle. Each of these cylinders' radii depend on the z coordinate of the vertice you picked. Given this info, I'd like to compute the maximum volume enclosed by any given configuration of 3 vertices. This is easy if they do not overlap, but trickier if two or even all three overlap. I'm sure I can use calculus to find the volume common to all 3 and subtract that from the sum of the three volumes. That is, if I wasn't trying to write this in c++, of which I just barely know enough to get by.

I wanted to get some comments about how I might go about designing this problem. My current thought was that each any three points will form a triangle, and I could somehow use the area of this triangle along with some sort of correction factor that accounted for the differences in radii that come from the different z coordinates to make a good estimate of the actual volume. If this idea were to lead anywhere, I'd use the area of the triangle connecting the points and the correction factor to form a cylinder whose 'middle' is the centroid of the triangle. This is an overly simplified version of the problem that I am trying to work with, but any thoughts will be greatly appreciated.
Thanks

 

[benorin]

Some comments: I assume that you meant to truncate each of the three cylinders at the edges of the cube (or else the volume is clearly infinite!) Also, the cylinders lack orientation: e.g. to describe a right circular cylinder you need to know i) the radius, and ii) the axis of the cylinder; you have specified only that the radius depends on the z-coordinate of it's point, and that the axis of the cylinder passes through this point (whereas it takes two points to determine a line, namely the axis)--I mean to ask is the cylinder vertical, horizontal, or other?. Also there are 4x4x4=64 verticies. Post a little more to clarify