Let $P_1 = (x_1,y_1)$ and $P_2 = (x_2,y_2)$ be points on the elliptic curve $$y^2 + a_1xy+a_3y = x^3+a_2x^2+a_4x+a_6.$$
(I'm suppressing the homogenizing variable $z$; perhaps this is a bad idea? Feel free to treat everything I say as if the above were a homogeneous polynomial in three variables.)
Suppose we're trying to find $P_1\oplus P_2$.
If I understand correctly, there's three cases:
- If $x_1 = x_2$ and $y_1+y_2+a_1x_2+a_3 = 0,$ then $P_1 \oplus P_2 = O$.
- If $x_1 = x_2$ and $y_1+y_2+a_1x_2+a_3 \neq 0,$ then there's a complicated formula.
- If $x_1 \neq x_2$, then there's a different formula.
I don't understand the need for (2). Geometrically, it seems to me that if $x_1=x_2$, then the line through $P_1$ and $P_2$ will be vertical, and hence $P_1 \oplus P_2$ will equal $O$. Am I mistaken about this?