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Are there any other ways of parametizing S?

  1. Mar 30, 2005 #1
    If S is part of the plane [tex] z = y - 2 [/tex] that lies inside the elliptic cylinder [tex]x^2 + 4y^2 = 4[/tex] and I want to parametrize S I will let
    [tex]x = 2sin(t)[/tex]
    [tex]y = cos(t)[/tex]
    [tex]z = cos(t) - 2[/tex]
    I assume this is right but let me know if not. My question is are there any other ways of parametizing S?
  2. jcsd
  3. Mar 30, 2005 #2
    I think you ahve your cosins and sins switched. Which direction does your parameter go? Im assuming counter clockwise.

    Come to think of it keeping them that way will also work, provided that you aren't going for a specific parametric starting and ending point.
  4. Mar 30, 2005 #3
    There is no specific starting point. Do you know of another way parametrizing S though?
  5. Mar 31, 2005 #4


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    Basically as he said, make every cosine a sine and the sine a cosine. I suppose you could also add an arbitrary phase shift to the trig functions as well (the sine-cosine reversal is a special case of that).
  6. Mar 31, 2005 #5
    So just switch my sin and cosin and that is another way of parametrizing S?
  7. Mar 31, 2005 #6
    I don't really understand your solution, since apparently you're parametrizing a surface (so you need 2 parameters). I suppose the solution is simply :

    [tex] x=Rcos(t)\quad y=\frac{R}{2}sin(t)\quad z=\frac{R}{2}sin(t)-2\quad R\in[0;2]\quad t\in[0;2\pi][/tex]
  8. Mar 31, 2005 #7
    Yes. It will draw the same curve but from a different starting point and direction.
  9. Mar 31, 2005 #8
    Those parameters would draw the curve of intersection. Parametrizing the entire surface, you would use klein's equation
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