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The two infinite cones (nappes) (each 45-degree wide) have parallel axes. They are oriented in opposite directions, and the top of one is inside the other, so that the common volume V is finite. How to express V as a function of the displacement between top1 and top2?

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hardmath
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Tommaso Bolognesi
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  • You seem to be talking about truncated cones. Many Readers will think of cones that extend in both directions from their vertex, esp. when you describe them as "infinite cones". – hardmath Oct 02 '14 at 15:17
  • The two cones are infinite, but they truncate each other. The intersection is a solid with two tops, but without cylindrical symmetry (the two tops are not aligned vertically) – Tommaso Bolognesi Oct 02 '14 at 15:44
  • Here's a picture that may illustrate my confusion. I think your cones are truncated at their vertexes, and if the axes are parallel and not the same, then the picture of their intersection may be more complicated than "a solid with two tops". – hardmath Oct 02 '14 at 15:51
  • I've added a picture up in the original question. – Tommaso Bolognesi Oct 02 '14 at 16:15
  • I think that the curve defined by the intersection is an ellipsis, and therefore I can compute separately the volumes of the two finite cones with elliptical basis, and then add them up... – Tommaso Bolognesi Oct 02 '14 at 17:02
  • Note that since both generators are at the same angle to the cones' axes, there is a symmetry between these two parts. – hardmath Oct 02 '14 at 17:09

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