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Height of an ellipse above the plane in three dimensions (z parameter)

  1. Sep 26, 2011 #1
    1. The problem statement, all variables and given/known data
    What is the expression for z (the height above the xy-plane) in terms of r,f,w,i.

    r = the distance
    f = the angle between the semi-major axis and the r vector
    w = the angle that the semi-major axis makes with the y-axis
    i = the angle that the plane of the ellipse makes with the xy-plane

    This is actually an Astronomy problem, where the ellipse is the orbit of an asteroid and the xy-plane is the plane of the solar system (planetary orbital plane).


    2. Relevant equations



    3. The attempt at a solution

    I started by parametrizing the ellipse with respect to one of the foci:

    x = c+r*cos(f)
    y = r*sin(f)
    z = (c+r*cos(f))tan(i)

    This doesn't seem correct, or complete I guess. I'm not sure how w comes into play? w rotates the ellipse relative to the y-axis but I can't picture where it should go?
     
  2. jcsd
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