I want to transform cartesian space to polar space to draw an ellipse.
For circle the polar space is $(r,\theta)$ and $(x,y)$ being cartesian space. How do we represent polar space for an ellipse? $$x=i \cos (t) \cos (\theta)-j \sin(t) \sin (\theta)$$
$$y=i \sin (t) \cos (\theta)+j \cos (t) \sin (\theta) $$
in the formula above is ellipse parametrized by $(i,j,t,\theta)$ as this doesnt look like polar space?
Note that $\theta$ is not the polar angle $\phi$.
Since $i,j$ can be regarded as unit vectors, quaternion and so on, use $a,b$ instead.
As $t$ varies,
$$x^2+y^2=a^2\cos^2 \theta+b^2\sin^2 \theta \tag{1a}$$
which is a circle of with radius $$r(\phi)=\sqrt{a^2\cos^2 \theta+b^2\sin^2 \theta} \tag{1b}$$
As $\theta$ varies,
$$\frac{(x\cos t+y\sin t)^2}{a^2}+\frac{(x\sin t-y\cos t)^2}{b^2}=1 \tag{2a}$$
which is a rotated ellipse with radius $$r(\phi)=\frac{ab}{\sqrt{a^2\sin^2 (\phi-t)+b^2\cos^2(\phi-t)}} \tag{2b}$$
For both $t$ and $\theta$ vary, we get annular region $b \le r \le a$.