# Ellipse and Kepler's Law in Polar Coordinates

1. ### Septim

125
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. ### haruspex

13,256
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?

3. ### Septim

125
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 ?

4. ### haruspex

13,256
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.