I am trying to figure out how to take a parameterized equation of a circle with radius 1 centered at $(0,1)$ and rotate it 90 degrees clockwise.
I have found the parameterized equation to be $$(x,y) \in \mathbb{R}^2 = (cos(2\pi{t}),1+sin(2\pi{t}))$$ for $t\in (0,1]$.
However, I want $t=1$ to map to $(0,0)$, $t=1/4$ to map to $(1,1)$, and so on.
How could I achieve this?
Apply a rotation by $-\dfrac{\pi}{2}$ to the circle about its center $(0,1)$, then the image is
$\begin{equation} \begin{split} (x', y') &= (0, 1) + R(-\dfrac{\pi}{2}) ( \cos (2 \pi t ), \sin (2 \pi t ) ) \\ &= (0, 1) + (\cos (2 \pi t - \dfrac{\pi}{2} ), \sin (2 \pi t - \dfrac{\pi}{2} ) \\ &= (0, 1) + (\cos (2 \pi (t - \frac{1}{4} ) ), \sin ( 2 \pi (t - \frac{1}{4} )) ) \\ &= (\cos (2 \pi (t - \frac{1}{4})), 1 + \sin (2 \pi (t - \frac{1}{4})) ) \end{split} \end{equation}$