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Confused with Space Curves

  1. Sep 27, 2016 #1
    1. The problem statement, all variables and given/known data
    Match the parametric equations with the graphs.
    In this case, I am stuck on this equation:
    x = cos t
    y = sin t
    z = 1/(1+t^2)

    2. Relevant equations


    3. The attempt at a solution
    So far I have:
    x^2 + y^2 = cos ^2 t + sin ^2 t = 1
    I know this is a circle in the xy-plane, and thus this yields a vertical, circular cylinder. I understand that there will be some form of helix going counter-clockwise around the cylinder. However, I do not understand how exactly to "graph" this using the parameter t. I know what the graph is and looks like, but I can't understand why it looks the way it does. I am having a hard time grasping this section in total.
     
  2. jcsd
  3. Sep 27, 2016 #2

    andrewkirk

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    Think of an infinitely long, infinitely thin slinky spring whose top is in the plane ##z=1##, centred on the point (0,0,1). As you follow the spring down, in the negative direction of the ##z## axis, the coils get closer and closer to one another so that it approaches but never quite touches the plane ##z=0##.
     
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