I have an ellipse centered around the origin, defined by the parametric equation: $$(x,y)=(45\cos\theta,15\sin\theta)$$
How can I modify this equation to rotate the ellipse around the top of the semi-minor ($y$) axis?
In the screenshot below, I used $30^\circ$ as an example.
But I need the rotation angle to be an input parameter - in addition to $\theta$.
If the angle should be $330^\circ$ vs. $30^\circ$, that's fine - I'm not sure which direction is which, and I'd just use CAD + trial and error to pick the correct one for my situation anyway.
$$ \pmatrix{x'\\ y'-15}= \pmatrix{\cos\alpha & \sin\alpha\\ -\sin\alpha & \cos\alpha} \pmatrix{x\\ y-15} $$