For old coordinates $(x,y)$ the new coordinates $(u,v)$ are related like this:
$x = u\cos(\theta) - v\sin(\theta)$
$y = u\sin(\theta) + v\cos(\theta)$
So would it be correct to say that to rotate the axis for a parametric equation defined by $x = f(u)$ and $y = g(u)$ I need to multiply $f(u)$ with the $x$ rotation equation and $g(u)$ with $y$ rotation equation?
I can't find any resources online for this topic.
If you have a parametric equation $$\begin{cases}x = f(u) \\ y = g (u)\end{cases}$$ and you want to rotate the image by $\theta$, you can just take $$\begin{cases}x' = x\cos \theta - y \sin \theta = (\cos\theta) f(u) - (\sin\theta)g(u) \\ y' = x \sin\theta + y\cos\theta = (\sin\theta) f(u) + (\cos\theta)g(u). \end{cases}$$