I am trying to make planet orbit in a rotated ellipse fashion. Inside this desmos graph is the rotated ellipse. How do I have a point orbit along that path?
I have used $(r \sin ( \frac{1}{p} 2 \pi t)+h,r \cos ( \frac{1}{p} 2 \pi t)+k)$ to make ordinary planets orbit in a circle, but can't figure out how how to apply this to a rotated ellipse. Any help is appreciated.
Let $a$ and $b$ be the major and minor axes of the ellipse, respectively, and $\alpha$ be the "declination" (rotation angle) of the ellipse.
The parametric equation of the ellipse centered at the origin of coordinates with the main axes directed along the coordinate axes reads: $$ \begin{pmatrix}x'\\y'\end{pmatrix}= \begin{pmatrix}a\cos\omega t\\ b\sin\omega t\end{pmatrix}. $$
In the coordinate system rotated at angle $\alpha$ about the origin it will read: $$ \begin{pmatrix}x\\y\end{pmatrix}= \begin{pmatrix}\cos\alpha&-\sin\alpha\\ \sin\alpha&\cos\alpha\end{pmatrix} \begin{pmatrix}x'\\y'\end{pmatrix} =\begin{pmatrix}a\cos\alpha\cos\omega t-b\sin\alpha\sin\omega t\\ a\sin\alpha\cos\omega t+b\cos\alpha\sin\omega t\end{pmatrix}. $$
Further modifications such as shift of the coordinate origin or time are obvious and are left to you.