Elliptic curve representation
When $c=1$ and $d = r^2$, an Edwards curve $x^2+y^2=c^2(1+dx^2y^2)$ becomes $$x^2+y^2=1+r^2x^2y^2 \iff x^2(1-r^2 y^2) = 1 - y^2$$
With the $(Y,Z)$ notation, a point $P = (x,y)$ is represented as a pair $(Y:Z)$ satisfying  $ry = Y/Z$.  Note that this representation does not allow to distinguish $P = (x,y)$ from $-P = (-x,y)$.  

Given a pair $(Y:Z)$, one can recover $\pm P = (\pm x, y)$ where $y = Y/(rZ)$ and $\pm x$ is the square root of $(1-y^2)/(1-r^2y^2)$.