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Helix - Variable Diameter with constant Pitch

  1. Sep 3, 2009 #1
    Im trying to find an equation for a helix that gets wider and thinner yet the angle of all the coils remains constant.

    Is this possible? Any ideas?

    Thank you!
    PS - I am not a math expert, but throughly enjoy the process!
    walt
     
  2. jcsd
  3. Sep 3, 2009 #2

    arildno

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    Well, you might look at parametrizations of the form:
    [tex]x=r(t)\cos\theta(t),y=r(t)\sin\theta(t),z=k\theta(t)[/tex]
    where k is a constant, and [itex]r(t),\theta(t)[/itex] are functions of t, r(t) being non-negative, and [itex]\theta(t)[/itex] a strictly increasing function.
     
  4. Sep 3, 2009 #3

    Nabeshin

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    This certainly describes a helix with constant spacing along the z-axis with variable radius, addressing the concern of a helix that gets wider, but what about the OP's question of "thinness"? I don't really know what I mean by this, perhaps he is envisioning a physical 3-dimensional coil rather than the curve you suggested.
     
  5. Sep 3, 2009 #4

    arildno

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    Well, in that case, he's after a helical surface, rather than a helix.

    He didn't ask about that.
     
  6. Sep 3, 2009 #5
    Actually, I think arildno answered the question anyway.

    Just replace [tex]r(t)=r(k \theta (t) )[/tex] where [tex] r(z) [/tex] is any positive function describing the radius of the helix (or "thinness" of it) as related to its height (z)
     
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