I have a question, and I hope I can word it correctly. I hope this is the right place, as I just Googled for "mathematics forum" and this was the first link on the page.
Anyway, say I have a round pipe of length 5 feet (actually, the length is irrelevant). I want to bend it at a certain radius in the horizontal direction, and also a certain radius in the vertical direction. Let's say I bend it at 20' horizontal radius, and say 30' vertical radius.
Okay, theoretically, you could just hold the pipe at one end and turn your wrist, and you've immediately altered both radii. In addition, if you lay it on the ground, it should just lay flat since it's a round pipe, so theoretically it only has one radius in a horizontal direction, right? (or a vertical direction if you hold the pipe upright).
If my conception is correct, then my question is; How do you determine the single radius based on needing both the H and V radii? I mean, it certainly has to have something to do with the angle of inclination at any given point along the pipe, right? For example, if I bend it at a 20' horizontal radius, lay it flat on the ground, and then begin to twist the pipe such that the other end rises, I've reduced the horizontal radius, and have begun to create a vertical radius. So again, if I want...say...a 20' H and 30' V radius, how do you determine the one single radius that I would have to bend the pipe at, as it lay in a fully horizontal position? I will know my exact horizontal and vertical positions at any point along the pipe, but how do you determine the single radius at which to bend the pipe in order to assure a proper perfect fit?
Am I describing it clearly? I think so, particularly with the "holding it and twisting it" analogy. Picture a short portion of a tubular roller-coaster track if that helps.
Thanks, and I'll greatly appreciate any answer. If you want to give an example, feel free to use numbers, although as I'm sure you can see, the numbers themselves aren't really important in this case.