Why does a parabola have a single point at infinity?

Question

Consider the parabola $V(zy-x^2) \subset \mathbb P^2$. This parabola has only one point at infinity which is $[0:1:0]$. On the other hand we sketch the parabola, we see that there are two asympotes, so probably it should have two points at infinity? So, why does the parabola has only one point at infinity?

Answer 1

The two “arms” of a parabola become increasingly more parallel the farther you proceed along them. In the limit, at infinity, they are parallel, and are parallel to the axis of the parabola as well. And since parallel lines pass through a common point at infinity, that's the point at infinity for both of them, and it's a point on the parabola itself as well.

Algebraically, that point is a double point: if you intersect the line at infinity with the parabola, you get two identical solutions. This means that the line at infinity is in fact tangent to the parabola. You can use this as a classification: Hyperbolas are conics which intersect the line at infinity in two distinct real points. Parabolas are conics which touch the line at infinity in a single point. And ellipses are conics which intersect the line at infinity in two complex conjugate points. For circles these are the ideal circle points $[1:\pm i:0]$.