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Radius in two dimensions

  1. Jul 10, 2011 #1
    I have a question, and I hope I can word it correctly.

    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 now have a pipe that has both a horizontal and 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 know my exact horizontal and vertical positions (coordinates) 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.
     
  2. jcsd
  3. Jul 11, 2011 #2

    Mentallic

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    I find this question interesting. Sadly I can't help out, because I'm struggling to picture what bending a tube in the horizontal then vertical direction would be like. If we take a long straight pipe and bend it with some radius in one direction, and the bend makes the furthest end of the pipe move a displacement of x units to the horizontal, when we next make the vertical bend, will the pipe still be x units across in the horizontal if we keep it in the same position it was initially and don't lay it down?

    I too hope I'm making sense :yuck:
     
  4. Jul 11, 2011 #3
    I want to make sure I understand your post correctly, so allow me to contruct a (what I think is) similar situation.
    Take a tube with square cross section. Our aim is to bend the tube so that it has some radius along one set of parallel edges and some other radius along the other set of parallel edges.
    Am I anywhere close?
     
  5. Jul 11, 2011 #4
    Looking at it from the top; yes. However, after bending it vertically as you describe (and it's already bent horizontally), you can still just lay the pipe down flat on the ground, at which point looking at it from the top, technically there's only one radius - a horizontal radius.

    As you turn the curved pipe upward from a flat horizontal position, the horizontal X position continues to change. In fact (and if you can picture it), if you were to turn the pipe from a flat horizontal position 90 degrees upward, then looking at it from the top, there would be zero horizontal bend, and the pipe would be curving upward towards you.

    Hmm, I'll have to try and get a little piece of pipe or stiff tube or something and make a small video and convert it to an animated GIF. I'll have to confirm that indeed the pipe will behave this way. It's certainly an interesting problem, huh? lol
     
  6. Jul 11, 2011 #5
    Exactly. Except we're dealing with a round pipe, so there really are no "sides" to it, lol.
     
  7. Jul 11, 2011 #6
    P.S. - My name is supposed to be 'FrigginEngineer'. Maybe the site has a limit on the number of characters your name can be, and they just truncated mine, lol.
     
  8. Jul 11, 2011 #7

    chiro

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    Hello FrigginEngine and welcome to the forums.

    Unfortunately I can not visualize what you are trying to do, but it sounds like you might be able to form the question in terms of a calculus problem, and use constraints to help you solve for your unknown parameter.

    If you could post a picture, hopefully myself and other posters will get a better idea of your problem.
     
  9. Jul 11, 2011 #8
    Alright. Assuming it's a smooth pipe, we want to bend it such a way that projection of pipe in one direction is a circular arc of some radius and projection in a perpendicular direction is a circular arc of some other radius.
    Is that right?
     
  10. Jul 11, 2011 #9
    Yeah, I'm gonna make some type of visual. It's really rather simple, though apparently I'm not explaining it clearly. I mean, I'm sure we've all seen instances of what I'm talking about.

    Picture a roller coaster on one of the tubular tracks that they have, going around a horizontal curve, and the curve is also going upward/uphill. Well, picture the track itself - just one of the round tubular pipes that make up the track - not the coaster itself. Also, we're only going to bend the pipe at about a 45^ deflection angle or so, not an entire 360^, or 180^, or even a 90^ angle, only about 45^ or so.

    lol I hope I didn't make it worse by saying that. I'll definitely make some type of visual, but I'm sure you know what I'm talking about once you can visualize it.
     
  11. Jul 11, 2011 #10
    Yes. We're definitely in 3 dimensions here too.

    If the pipe behaves the way I'm envisioning, it's definitely an interesting mathematics/geometry problem.
     
  12. Jul 11, 2011 #11
    That was the way I understood it, too. But I think this is a more complicated question than the OP realizes. The projection of a circle onto a plane is not a circle (unless the circle is parallel to the plane), but an ellipse. So if you have a pipe bent in the way described, it will not actually be a circular arc, but some more complicated curve. And that means it's unlikely to have a single radius of curvature, and may not even lie in a plane.
     
  13. Jul 11, 2011 #12
    Exactly! But think in the opposite direction - projection of a (suitable) ellipse on a (suitable) plane would be a circle. My intuition says, imagine an ellipse whose projection on some plane (call it horizontal) is a circle and on a perpendicular plane (call it vertical) is also a circle.
    I'm still working on the mathematical backing for that last sentence.
     
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