The isomorphism between a projective line with a specified point removed and the affine line

Question

Say you're given a point with coordinates $p=(x:y) \in \mathbb{P}^1$ and you'd like to write down the isomorphism $\mathbb{P}^1-p \cong \mathbb{A}^1$. How would one define the map and it's inverse? Thanks!

Answer 1

To make @MooS answer explicit: suppose that the point $p$ has coordinates $[a: b]$ (so that I can use $x$ and $y$ as variables).

Step 1: move $[a: b]$ to the point at infinity, $[1: 0]$: $$ [x: y] \mapsto [ax+by : bx - ay] $$

Step 2: Map the punctured projective line (punctured at infinity) to the affine line: $$ [x: y] \mapsto \frac{x}{y} $$

(This is well-defined because the point at infinity, where $y = 0$, is missing).

The composition is

$$ [x: y] \mapsto \frac{ax+by}{bx-ay} $$ which does the job.