Wikipedia gives the following equation for the conic sections in the polar coordinate system:
$r = \frac{l}{1+e\cos\varphi}$.
According to the article on conic sections, in case of an ellipse $e = \sqrt{1-\frac{b^2}{a^2}}$ and $l = \frac{b^2}{a}$, where $a$ is the semi-major axis and $b$ is the semi-minor axis.
However, when I try to substitute concrete values into the equation, I don't get the results I expect. For example, let $a = 2$ and $b = 1$. It seems clear to me that for $\varphi = 0$ I should get the rightmost point of the ellipse, that is $r = a = 2$, correspoding to $(a, 0) = (2, 0)$ in the Cartesian coordinates. However:
$l = \frac{b^2}{a} = \frac{1}{2}$
$e = \sqrt{1-\frac{b^2}{a^2}} = \sqrt{1 – \frac{1^2}{2^2}} = \frac{\sqrt{3}}{2}$
$r = \frac{\frac{1}{2}}{1 + \frac{\sqrt{3}}{2}} = \frac{1}{2 + \sqrt{3}}$.
It's hardly two. I am sorry if I am making a fool of myself, but where do I make a mistake?
Best Answer
The polar center for your equation is a focus of the ellipse, not its center.
Polar form relative to the center of the ellipse is $$r(\phi)=\frac{ab}{\sqrt{(a\sin\phi)^2+(b\cos\phi)^2}}$$