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Ellipse and Kepler's Law in Polar Coordinates

by Septim
Tags: ellipse, kepler, polar
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Septim
#1
Apr13-12, 04:02 PM
P: 121
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/teachin...s/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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haruspex
#2
Apr14-12, 02:12 AM
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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?
Septim
#3
Apr14-12, 12:07 PM
P: 121
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
#4
Apr14-12, 06:08 PM
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Ellipse and Kepler's Law 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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