Ellipse and Kepler's Law in Polar Coordinates

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Discussion Overview

The discussion revolves around the polar form of the ellipse equation, exploring different representations and conversions between forms. Participants are examining the implications of setting the origin at different points, specifically a focus versus the center of the ellipse, and seeking resources for further understanding of conics in polar coordinates.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant expresses difficulty in understanding the polar form of the ellipse equation and notes that there seem to be multiple representations.
  • Another participant points out that the choice of origin (focus versus center) affects the polar equation of the ellipse.
  • A request is made for additional resources on conics and their equations in polar coordinates.
  • A participant presents a mathematical derivation involving a string length and polar coordinates, suggesting it leads to an equation that resembles an ellipse centered at the origin.

Areas of Agreement / Disagreement

Participants appear to agree on the existence of multiple forms of the ellipse in polar coordinates and the significance of the origin's placement, but no consensus is reached on specific representations or resources.

Contextual Notes

There are limitations regarding the availability of contrasting resources for the polar equation of the ellipse, and the discussion does not resolve the conversion methods or the implications of different forms.

Who May Find This Useful

Individuals interested in conic sections, polar coordinates, and mathematical representations of ellipses may find this discussion relevant.

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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