Line at infinity definition

Question

I was wondering why z=0 defines the line at infinity in $\mathbf P_2$.

Answer 1

Because it is convenient.

Working with homogeneous coordinates $(x,y,z)$, you have two subsets: ones with $z=0$ and ones with $z\neq 0$.

Every one with $z\neq 0$ is equivalent to something of the form $(x,y,1)$. There are two degrees of freedom, and these points can be thought of as the regular Cartisian plane.

The remaining points are of the form $(x,1,0)$ and $(1,0,0)$. The former have one degree of freedom, making it a line, which together with the latter point form the complete projective line at infinity.

Everything is nice and easy to keep track of. In particular, you can talk easily about how transformations act on the line at infinity.

Now, given any projective plane, you can select any other line from it and declare that line to be "the line at infinity" instead, no matter what the $z$ coordinates of its points are. If you select that as the line at infinity, then it is not so easy to identify points that are either in the "plane" or in the "line at infinity."

Really, the coordinates you are using don't have any real significance at all for the actual projective space. They're just a way to "make the points concrete" so that you can do manipulations with them. Any other projective line would have done just as well in the role of "line at infinity."