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Can you give me some specifics on how you would design the lathe you are describing? If there is no shaft down the center, how are the two ends of the part anchored to the lathe, and how can you be sure different sections of the rocket are perfectly collimated with each other, and the faces of each section are perfectly square with each other?
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Regretfully, my pen turning lathe is mounted on a flip-table to save space and it isn't up and theres a bunch of stuff on it, so I can't easily flip it.
I went to youtube and found some pics and did screenshots that hopefully explain the concept.
In tbc0.png, you see the blank getting ready to be turned. It is a square tube with a hole in it. Normally, there is a brass tube glue'd in the hole, but not in this pic. Either way, assume your pink foam is the square blank with the hole in it and a cardboard tube (38mm, as an example) is glued in, like the speedmotion example.
tbc1.png shows him turning the pen. The left is just along for the ride, while the right is where the power comes in. For your case, you would have already used the foam cutter with the template (just like speedmotion) and would be spinning it to sand smooth, not using lathe tools.
tbc2.png shows what the centers look like. In your case, the one on the left just spins freely - it doesn't need to be a fancy bearing etc, just a good smooth spinning rod. It could be a wooden rod with a point cut on it and it could run between a couple of 2x4's with tight holes instead of bearings. Lube it with wax, since you're going to be spinning fairly slowly.
The right side is the drive side. Same general requirements, a cone spinning with simple support. It can be spun by hand, with a drill or with a hand crank etc.
Things need to be pretty concentric, so to make the points, you would assemble everything and then spin them, using a sander or similar to form the point. Then the point is pretty concentric.
The rest of the assembly is similar to what speedmotion showed as far as the frame goes etc., but it is made with mostly random wood junk instead of shafts, bearings etc. It is pretty 'Fred Flintsone', but would trade some cost for materials to do things 'right' and get decent results with more labor.
Without question, I think using the hot wire to get the cone nearly the correct shape is the number one priority. Foam dust sux. . .
I hope that's clear. If not, I can try to explain more, but it would take 2 hours of cleaning to get to the lathe to actually show that I'm saying. . .
Sandy.