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I think Wikipedia's polar coordinate elliptical equation isn't correct. Here is my explanation: Imagine constants $a$ and $b$ in this format - image

Where $2a$ is the total height of the ellipse and $2b$ being the total width. You can then find the radial length, $r$, at any degree, $\theta$, as...

$$r(\theta) = \sqrt{(b \cos(\theta))^2 + (a \sin(\theta))^2}$$

...by just following the Pythagorean theorem. Yet Wikipedia's equation for the polar coordinate ellipse is as follows:

$$r(\theta) = \frac{ab}{\sqrt{(b \cos(\theta))^2 + (a \sin(\theta))^2}}$$

Here is the link to the Wikipedia page: http://en.wikipedia.org/wiki/Ellipse#Polar_form_relative_to_center Can someone explain this, please? Why divide by the hypotenuse? Why the $ab$? Thank you!

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The point $(b\cos\theta,a\sin\theta)$ is not at angle $\theta$. –  Rahul Feb 27 '13 at 0:18
    
How is it not?? I'm pretty sure it's at angle $\theta$ moving ccw from $y = 0$, $x = b$ EDIT: Oh shit you're right!!! $\theta$ changes at the constant rate of a circle, not at the rate of an ellipse! Thank you!!!! –  Athan Clark Feb 27 '13 at 0:42
    
You may also want to look at my answer to math.stackexchange.com/questions/493104/… –  barrycarter Feb 23 at 18:04

2 Answers 2

It's easiest to start with the equation for the ellipse in rectangular coordinates:

$$(x/a)^2 + (y/b)^2 = 1$$

Then substitute $x = r(\theta)\cos\theta$ and $y = r(\theta)\sin\theta$ and solve for $r(\theta)$.

That will give you the equation you found on Wikipedia.

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According to Wolfram Alpha, your version - because of the difference in theta as described, gives an interesting shape that looks like a slightly distorted ellipse:

http://www.wolframalpha.com/input/?i=r%3Dsqrt%28%282cos%28theta%29%29%5E2%2B%283sin%28theta%29%29%5E2%29

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While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference. Link-only answers can become invalid if the linked page changes. –  naslundx May 8 at 12:56

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