Stagnation points in polar form

Question

So I have the velocity as $V=\frac{4}{3}Kr^{\frac{1}{3}}\left[ \cos\left( \frac{4\theta}{3} \right)-i\sin\left( \frac{4\theta}{3} \right) \right]$ between $0 \leq \theta \leq \frac{3\pi}{4}$.

By setting the components of the velocity $V_1$ and $iV_2$ equal to zero I obtained the stagnation points in terms of polar coordinates $(r,\theta)$.

So, $V_1=\frac{4}{3}Kr^{\frac{1}{3}}\cos\left( \frac{4\theta}{3} \right)$,

$V_2=-\frac{4}{3}Kr^{\frac{1}{3}}\sin\left( \frac{4\theta}{3} \right)$

For $V_1=0$ the stagnation points are: $(0,\theta)$ and $(r,\frac{3\pi}{8})$.

For $V_2=0$ the stagnation points are $(0,\theta), (r,\frac{3\pi}{4})$ and $(r,0)$

How would I plot these on for example Desmos to see where these points would lie?

Answer 1

Plotting these lines in cartesian form gives you the point where they all intersect. This point is the stagnation point.