# Ellipse in polar coordinates

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

• 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 '14 at 18:04
• 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/… – user148686 May 8 '14 at 12:29

## 3 Answers

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.

You're making the common mistake of using the polar coordinate instead of the eccentric anomaly which is the parameter in the ellipse coordinates.

• 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. – Athan Clark Dec 28 '16 at 1:06

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.