Suppose that $S^1 = \{(x,y)\in \mathbb{R}^2|x^2+y^2=1\}$, and let $x \in S^1$. How can I go about finding a homeomorphism from $S^1 \to S^1$ that sends $x$ to $(0,1)$. I thought about defining the line from $x$ to $p$ as $l(t) = tx+1$. But I'm unsure what to do. I just want a hint , and not the answer to this question.
Thank you.
If $(x,y) \in S^1$ we can find such a homeomorphism by a rotation of the plane (restricted to $S^1$), in essence a multiplication by $e^{i\phi}$ for a suitable $\phi$, looking at the plane as $\Bbb C$, or the linear map with matrix $\begin{bmatrix} \cos \phi & -\sin \phi \\ \sin \phi & \cos \phi\end{bmatrix}$ etc.
This has in itself nothing to do with a stereographic projection. You can rotate any point of the circle onto any other one.