I've been navigating through many of StackExchange's history questions about point at infinity, and there's something that still doesn't "click" for me.
Let me share my understanding (and its logic) and maybe become apparent where I fail.
Let's say I have an elliptic curve equation, particularly a Twisted Edwards Curve equation. The first thing to note, contrary to 90% of texts on the internet, the identity point isn't the point at infinity. (I think it's true in other kind of curves).
OK, so if the points at infinity aren't relevant to define the identity element, the question I'm asking myself is: why are they relevant at all?
I think in this case (Twisted Edwards), points at infinity are solutions in the equation in projective form with $Z=0$. (Is this correct?).
I'd guess we call those solutions "points at infinity" because we can't project them to $Z=1$ (i.e: affine coordinates). (Right?).
If I set $Z=0$ in the projective form of Twisted Edwards $(x^2+y^2).z^2=z^4+d.x^2.y^2$, what I get is $0=d.x^2.y^2$, thus any point $(x, 0, 0)$ or $(0, y, 0)$ is a point at infinity. (Right?)
OK, so my understanding of "why is this relevant?", is that we want to be sure that when we apply the group law in projective form, we would like to never get one of these points as they don't map to affine coordinates. (Right?).
Is that the reason why Twisted Edwards curve might want to find a way to avoid points at infinity?
