# Projectivities $\pi:\mathbb KP^1\rightarrow\mathbb KP^1$

I am a little bit confused conerning the following example of projectivities $\pi:\mathbb KP^1\rightarrow\mathbb KP^1$.

On the affine part $\mathbb K\subseteq \mathbb KP^1$ they are exactly the rational maps of the form $x\rightarrow \frac{ax+b}{cx+d}, x\in\mathbb K$

My question is: How can I wirte down explicitly the six projectivities of the form above that permutes the points $0,1,\infty$ ?

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I dislike the formulation as a fraction, as it doesn't handle $\infty$ too well. I prefer homogenous coordinates and a matrix notation. Doesn't change the core of the issue, though.