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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: Can someone explain this, please? Why divide by the hypotenuse? Why the $ab$? Thank you!

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    $\begingroup$ The point $(b\cos\theta,a\sin\theta)$ is not at angle $\theta$. $\endgroup$ – Rahul Feb 27 '13 at 0:18
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    $\begingroup$ 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!!!! $\endgroup$ – Athan Clark Feb 27 '13 at 0:42
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    $\begingroup$ You may also want to look at my answer to math.stackexchange.com/questions/493104/… $\endgroup$ – barrycarter Feb 23 '14 at 18:04
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    $\begingroup$ 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: wolframalpha.com/input/… $\endgroup$ – user148686 May 8 '14 at 12:29
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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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You're making the common mistake of using the polar coordinate instead of the eccentric anomaly which is the parameter in the ellipse coordinates.

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    $\begingroup$ Would you be willing to flesh out more detail, supplying definitions for your vocabulary and clear implication of your reasoning? Just so this answer is self-contained. $\endgroup$ – Athan Clark Dec 28 '16 at 1:06
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There are two $\theta$ s . One is used in polar coordinates, it is starting at the unsymmetrical focal point on major axis.It is is Newton ellipse used for planet motion.

The symmetric or center origin $\theta $ is not the same angle, as also mentioned by Rick.Here you convert cartesian to polar.

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