Ellipse and Kepler's Law in Polar Coordinates

AI Thread Summary
The discussion focuses on understanding the polar form of the ellipse equation, highlighting that there are multiple representations. One key point is the difference in setting the origin at a focus versus the center of the ellipse. A user seeks additional resources for learning about conics and their equations in polar coordinates. They provide a derived equation in polar coordinates related to a string length and convert it back to Cartesian coordinates, confirming it represents an ellipse centered at the origin. The conversation emphasizes the importance of understanding different forms and origins in ellipse equations.
Septim
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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
 
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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?
 
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 ?
 
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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