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Math Help - Parametric equation for a helix winding around an arbitrary line

  1. #1
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    Parametric equation for a helix winding around an arbitrary line

    Let l(t)=p+vt where l,p,v are vectors. Say p=r0=<1,1,1> and v=<1,2,3> and the helix has radius 2.
    So I have l(t)=<1+t,1+2t,1+3t> and that the equation of a helix would be <2cost,2sint,t>
    How do I find the x(t),y(t), and z(t)?
    I also found that sin(theta)=2/r(t), but I know that theta will get smaller as t goes on.

    I'm studying for my multivariable calc final and this one is just stumping me! It feels like I should know the answer to this.
    Thank you!
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  2. #2
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    Re: Parametric equation for a helix winding around an arbitrary line

    I think I got it

    Would you simply add the parametric equation of the helix to the parametric equation of the line?

    l(t)=<1+t,1+2t,1+3t> and h(t)=<cost,sint,t>

    so x(t)=1+t+cost, y(t)=1+2t+sint, z(t)=1+3t+t=1+4t

    I believe this works. Someone correct me if I'm wrong.
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  3. #3
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    Re: Parametric equation for a helix winding around an arbitrary line

    Hi,
    No, what you are suggesting is to translate the given curve, but with a varying t in the translation. You don't get a helix. Here's a drawing of what you get:

    Parametric equation for a helix winding around an arbitrary line-mhfnothelix.png

    See the following post for a correct solution.
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  4. #4
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    Re: Parametric equation for a helix winding around an arbitrary line

    Hi,
    First, I've attached a discussion about how to change coordinates to produce the desired helix. The second attachment shows the "standard" helix as a dotted curve, the line (1,1,1) + t(1,2,3) and the helix which wraps around this line:

    Parametric equation for a helix winding around an arbitrary line-mhfhelix.png

    Parametric equation for a helix winding around an arbitrary line-mhfhelixa.png
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