Ellipse and Kepler's Law in Polar Coordinates

  1. Greetings everyone,

    I am having difficulties grasping the polar form of the ellipse equation, and there seems to be more than one way to express an ellipse in this form, if I am not mistaken. For example on the following webpage http://farside.ph.utexas.edu/teaching/301/lectures/node155.html the ellipse is represented in a different way than I am accustomed. How can I convert this into other forms?

    Thanks
     
  2. jcsd
  3. haruspex

    haruspex 14,322
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    One option with the equation for an ellipse is whether to set the origin at a focus or at the centre. The link you provided gives the polar equation with a focus as origin. Do you have another link for contrast?
     
  4. I do not have at the moment, I remember coming across one a year ago in a text I read. Do you have any site that I can learn conics and their equations in polar coordinates ?
     
  5. haruspex

    haruspex 14,322
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    Consider a string length 2L with endpoints fixed at (-A, 0), (+A, 0) (X-Y co-ords).
    With polar co-ordinates at the same origin, I get
    r2(L2-A2.cos2(θ)) = L2(L2-A2)
    Does that look familiar?
    Converting back to X-Y:
    (x2+y2)L2 - x2.A2 = L2(L2-A2)
    or
    x2/L2 + y2/(L2-A2) = 1
    Which does indeed appear to be an ellipse centred at the origin.
     
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