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.

 

[sardar]

for your question. I understand that the polar form of the ellipse equation can be confusing, as there are multiple ways to express it. However, all of these forms are essentially different representations of the same underlying concept.

To convert between different forms, you can use Kepler's laws, which describe the motion of planets in an elliptical orbit around the sun. These laws were derived by Johannes Kepler in the early 17th century and are based on observations made by Tycho Brahe. Kepler's first law states that the orbit of a planet is an ellipse, with the sun at one of the two foci. This means that the polar coordinates of any point on the ellipse can be expressed in terms of the distance from the focus (r) and the angle from the major axis (θ).

Using this information, you can derive the polar form of the ellipse equation, which is r = a(1-ε²)/(1+εcos(θ)), where a is the semi-major axis and ε is the eccentricity of the ellipse. This form is different from the traditional Cartesian form (x²/a² + y²/b² = 1), but they are equivalent representations of the same ellipse.

If you are having trouble understanding the polar form of the ellipse equation, I suggest practicing with different values of a and ε to see how they affect the shape of the ellipse. Additionally, familiarizing yourself with the concept of eccentricity and how it relates to the shape of an ellipse can also help with understanding the polar form.

I hope this helps clarify the polar form of the ellipse equation and how it relates to Kepler's laws. If you have any further questions, please do not hesitate to ask.