A. Old pieces

A..1 Digression (was in introduction)

In this context, carefully stated trans-typing are certainly useful, but not all trans-typing in all circumstances. For example, wording $f\left(z\right)=a\, z+b$ as "integral linear" because transformations $h\left(z\right)=\left(a\, z+b\right)/\left(c\, z+d\right)$ have been worded as "linear" seems misleading. Neither $h\left(z_{1}+z_{2}\right)=h\left(z_{1}\right)+h\left(z_{2}\right)$ nor $f\left(z_{1}+z_{2}\right)=f\left(z_{1}\right)+f\left(z_{2}\right)$ are identities. If you want to speak at elementary level, $f$ is a similitude and $h$ an homography. If you want to speak at an upper level, $h$ belongs to $\mathbb{P_{\mathrm{\mathbb{C}}}GL\left(C^{\mathrm{2}}\right)}$ acting over $\mathbb{P}_{\mathbb{C}}\mathbb{C}^{2}$ and $f$ belongs to the subgroup fixing $\infty\in\mathbb{P}_{\mathbb{C}}\mathbb{C}^{2}$.

Doing otherwise will lead to define $3\, A_{1}$ in $\mathbb{P}_{\mathbb{R}}\mathbb{R}^{4}$ by $\left(3a_{1},3b_{1},3c_{1},d_{1}\right)$ and pursue by defining $3A_{1}+2A_{2}$ by $\left(3a_{1}+2a_{2},\,3b_{1}+2b_{2},\,3c_{1}+2c_{2},\, d_{1}+d_{2}\right)$. But the later is an unspecified point of the projective line through $A_{1}$and $A_{2}$ that depends on the representations in $\mathbb{R}^{4}$ chosen for the points. In other words, it is better to use a language that awakens the good reflexes in everyone's minds.

To fulfill these requirements, we will use the following notations..