[betterunix2]

You are thinking in terms of affine coordinates i.e. (x,y) coordinates. The math is better explained (but not as easily to visualize) in projective coordinates. The conversions work as follows:

(x,y) --> (x : y : 1) (x : y : z) --> (x/z, y/z)

In that case, the "vertical line" will intersect the elliptic curve at (0 : 1 : 0), which is the identity element of the group.

Tangent lines are considered to intersect the curve "twice," (EDIT: "twice" at the point where the line is tangent and one more time at a third point) which is analogous to a polynomial equation having a double root. There is also a triple intersection case (EDIT: "triple" at a single point, all lines intersect the curve three times), which is possible at (0 : 1 : 0).

I'll leave it to the "reader" to work out the group law in projective coordinates based on the conversions given above. One neat trick: you can avoid inversions in the field by using projective coordinates and cleverly using the Z coordinate; this is a common optimization used in practice.