I am constructing a close-packed configuration consisting of 3 identical right circular frustra or cones.

The first two are positioned in close contact and touch along the lengths of their curved surfaces, i.e. such that their vertices (or, in the case of frustra, the extrapolation to their vertices) share the same point in space. Imagining that the central axes of these two cones/frustra are in the horizontal plane, the third cone/frustrum is then positioned on top of the other two, oriented such that its curved surface touches the curved surfaces of the other two and with its vertex (or the extrapolation to its vertex) in an opposite direction to the other two.

What I am after is an analytical expression to define where the cones/frustra touch each other (and where the points of contact are close to the half-heights of the cones/frustra). What is also required to define the third cone/frustra is the straight line equation of its axis.

I'm not sure what level of mathematics is required to solve this problem, but hopefully it will not require elliptic functions. The geometry associated with this problem is very simplistic - are there any books available that bring together solutions of hundreds of such similar problems - this could be very useful to me? If I were to search for a solution - what keywords would be best to use?

Kind Regards,

tennessee