I am learning about projective geometry in my machine perception class. I am struggling to fully grasp the concept of lines at infinity, and I am hoping someone would be able to shed some light on the topic.
Here are two questions I'm struggling with (I'm including both questions because of their similarity):
When does a line have the form $(a, b, 0)$?
My intuition for this question is that if you began with a plane equation: $a(x - x0) + b(y - y0) + c(z - z0) = 0$, we know the plane must pass through the origin, so we have $ax + by + cz = 0$. At this point I am not sure where to go or if this was the correct path. For a general line in the projective plane, we could confine $z = 1$, and have $ax + by + c = 0$, but I don't think that approach applies here.
When does a line have the form $(0, 0, 1)$?
I am not sure where to begin with this.
If someone could provide both a mathematical and conceptual/intuitive explanation to these questions, that would be awesome!
There’s a model of the projective plane that might be helpful to you. Consider the plane $z=1$ in $\mathbb R^3$. A point on this plane with coordinates $(x,y,1)$ can be identified with the line through the origin that intersects the plane at this point, so that any non-zero multiple of $(x,y,1)$ represents the same point. If you then add the lines parallel to this plane, i.e., multiples of $(x,y,0)$ to the model, you get “points at infinity.”
In a similar way, identify a line in the $z=1$ plane with the plane through the origin whose intersection with $z=1$ is the line. This plane can then be identified with any normal vector to it, so just as with points, we have an equivalence class of non-zero scalar multiples of some vector in $\mathbb R^3$ that represents a line in $\mathbb P^2$. Note that this implies that, in $\mathbb P^2$, you can find the line through two points by computing their cross product. There’s a nice duality here: with this representation, the point of intersection of two lines in $\mathbb P^2$ is also given by their cross product. You can verify for yourself that parallel lines intersect at a point of the form $(x,y,0)$.
If you have the line $(a,b,0)$, then the normal to the corresponding plane in $\mathbb R^3$ lies in the $xy$-plane, and thus the plane passes through $(0,0,1)$. So, the line in $\mathbb P^2$ passes through the origin.
For the line $(0,0,1)$, its corresponding plane is normal to the $z$-axis, i.e., is the $xy$-plane. This is parallel to our $z=1$ plane, so doesn’t correspond to any line on it. We take this to be the “line at infinity” in $\mathbb P^2$. This is consistent with the interpretation of points of the form $(x,y,0)$ as points at infinity: if you take the cross product of two such points, you end up with a multiple of $(0,0,1)$.