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Spiral parametrization in 3D

  1. Oct 24, 2014 #1
    Hello!
    There is a parametric way of defining a spiral curve:
    z = a*t;
    x = r1*cos(w*t)
    y = r2*sin(w*t).
    Is there a way to define the thickness of spiral?
     
  2. jcsd
  3. Oct 24, 2014 #2
    Wow, that's an interesting idea. If there is, the thickness is an inequality. I don't know about any 3d parametric graph generators that handle inequality though, so it'd be hard to play with.

    Think about a 2d parametric system, defining a circle. Is there a way to give the circle 'width', such that it resembles a flat torus? Ah, in this case what you need is two parametric curves, and then define another system that lies between the earlier curves (Although, I don't think this is what you wanted).

    width.png
    I defined this system implicitly, not parametrically, but only due to how my graphing software works (Winplot)
    $$
    x^2 + y^2 \leq (1-A/2)^2
    $$
    $$
    x^2 + y^2 \geq (1+A/2)^2
    $$

    Unfortunately, I'm not that experienced with 3d parametric equations. I know, though, that a spiral curve with thickness... i.e. a spiral 'cord' is possible to be represented parametrically if the torus is possible. But, unfortunately, I've been having trouble getting the torus to work on Winplot.
     
    Last edited: Oct 24, 2014
  4. Oct 29, 2014 #3
    This seems like a reasonable attempt:

    z=a*t + thickness*cos(u)
    x=(r+thickness*cos(u))*cos(w*t)
    y=(r+thickness*cos(u))*sin(w*t)

    For a surface you'll need two parameters u and t of course.
     
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