Ellipse and Kepler's Law in Polar Coordinates

[Septim]

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

 

[haruspex]

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?

 

[Septim]

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 ?

 

[haruspex]

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.