Is it possible to determine if three known polar coordinates $r(0)$, $r(\frac{\pi}{4})$ and $r(\frac{\pi}{2})$ define an ellipse centred at the origin?
Correct me if I'm not wrong, but this translates to finding $a$, $b$ and $\theta_{0}$ so that the following is true for $r(0)$, $r(\frac{\pi}{4})$ and $r(\frac{\pi}{2})$:
$r\left( \theta \right)=\frac{ab}{\sqrt{a^{2}\sin ^{2}\left( \theta -\theta _{0} \right)+b^{2}\cos ^{2}\left( \theta -\theta _{0} \right)}}$
Something like this:
The equation of an ellipse centered at the origin can be written as: $$ Ax^2+2Cxy+By^2=1, $$ where numbers $A$, $B$ and $C$ must satisfy: $AB-C^2>0$. Insert here the coordinates of the three given points $P_1=(r(0),0)$, $P_2={r(\pi/4)\over\sqrt2}(1,1)$, $P_3=(0, r(\pi/2))$ and solve for $A$, $B$ and $C$. If $AB-C^2>0$ then you have an ellipse.