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.(adsbygoogle = window.adsbygoogle || []).push({});

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

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# Estimating a difficult volume

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