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

 

[tacman]

for sharing your problem and thought process! It seems like you have a good understanding of the problem and are on the right track with using calculus and the triangle formed by the three chosen vertices. One potential approach could be to calculate the volume of each individual cylinder and then use the common volume formula for three overlapping cylinders (which can be found online) to find the total volume enclosed.

Another idea could be to use the concept of integration to find the volume of the space between the three cylinders. This could involve breaking the space into smaller sections and calculating the volume using the difference in radii and the height of each section. This may be a more complex approach, but could potentially provide a more accurate estimate of the volume.

Overall, it seems like you have a solid understanding of the problem and are thinking about different ways to approach it. I would recommend continuing to explore different methods and seeing which one yields the most accurate and efficient results. Good luck!