I looked at some posts on this website and on Wikipedia for a derivation on the general form of a 2D rotated ellipse, but I've only come across an explanation for the parametric form. Could someone please walk me through the solution to the general form as shown by ankidat dennis (2nd equation shown) here:
What is the general equation of the ellipse that is not in the origin and rotated by an angle?
Thanks,
reacher33
If you want to rotate first (by $\theta$ CCW), then translate (by $(h,k)$), you get \begin{align*} (x,y)&\xrightarrow{\text{rotate}}(x\cos\theta+y\sin\theta, -x\sin\theta+y\cos\theta)\\&\xrightarrow{\text{translate}}(x\cos\theta+y\sin\theta-h, -x\sin\theta+y\cos\theta-k). \end{align*} If you want to translate first, then rotate, you get \begin{align*} (x,y)&\xrightarrow{\text{translate}}(x-h,y-k)\\&\xrightarrow{\text{rotate}}((x-h)\cos\theta+(y-k)\sin\theta, -(x-h)\sin\theta+(y-k)\cos\theta) \end{align*}