I'm reading this paper and in section 4 the author proposes projective elliptic curves.
I have a doubt on why it is assumed implicitly that $c = 1$. This is done explicitely in the Mathematica computations but only implicitely in the paper when it defines:
$z_1 \oplus_1 z_2 := \tau((\tau z_1) \oplus_0 z_2) = \Big(\frac{x_1y_1-x_2y_2}{x_2y_1-x_1y_2}, \frac{x_1y_1+x_2y_2}{x_1x_2+y_1y_2}\Big)$
If you follow the definitions of $\tau$ and $\oplus_0$ in detail you would get instead:
$z_1 \oplus_1 z_2 := \tau((\tau z_1) \oplus_0 z_2) = \Big(\frac{x_1y_1-c x_2y_2}{x_2y_1-c x_1y_2}, \frac{x_1y_1+cx_2y_2}{x_1x_2+y_1y_2}\Big)$
The problem is that I'm trying to formalize this in a proof assistant and thus I aim to have as much generality as possible unless there is a good justification for setting $c = 1$.