Projective transformation at infinity

Question

Suppose I have the line $2x+3y+w=0$ which contains the point $[1, -1, 1]$. After projective transformation given by multiplying by a matrix, it goes to infinite point $[1, -1, 0]$ which lies on the line $x+y+w =0$ at infinity. But the line at infinity is the line $w=0$ which contains the infinite point $[1, -1, 0]$, and we know that the ideal line (the line at infinity) is just the collection of ideal points (points at infinity).

My first question is that is it possible that $x+y+w =0$ is also treated as the ideal line since it contains the ideal point $[1, -1, 0]$?

My second question is that some websites say that in the projective plane, $w=0$ is the equation of a plane and some say that it is the equation of a line. Which one is right?

My final question is that on Wikipedia, it says that affine space is the complement of the hyperplane at infinity in the projective plane. But my understanding is that they should write the complement of the line at infinity (or $w=0$). Why did they write the hyperplane at infinity instead of the line at infinity?

Answer 1

The line $x + y + w = 0$ is not the line at infinity; the line at infinity is $w = 0$. You seem to think that because the point $[1, -1, 0]$ is on the line at infinity and the line $x + y + w = 0$, then the two lines must be the same; that is not the case. The line $x + y + w = 0$ contains a point at infinity, but it does not contain all of them (for example, $[1, 0, 0]$), whereas the line at infinity does. In fact, every line $ax + by + cz = 0$ (other than the line at infinity), intersects the line at infinity in exactly one point, namely $[b, -a, 0]$.

In the projective plane, the equation $w = 0$ is the line at infinity. However, one can also form $n$-dimensional projective space, in which case the equation $w = 0$ defines the hyperplane at infinity. There is no contradiction here; when $n = 2$, the two notions coincide, i.e. $2$-dimensional projective space is the projective plane, and a hyperplane in the projective plane is a line.

Similarly, the complement of the line at infinity of the projective plane is the affine plane. On the other hand, the complement of the hyperplane at infinity of $n$-dimensional projective space is the $n$-dimensional affine space. Again, when $n = 2$, the two notions coincide, i.e. the $2$-dimensional affine space is the affine plane.

Answer 2

For the first question, if you have a projective plane and have identified the ideal points in an appropriate way, every line contains at least one ideal point. The difference between the ideal line and every other line is that every other line contains exactly one ideal point (and no others), whereas every point on the ideal line is an ideal point.

A projective plane has only one ideal line, so no, you cannot have an ideal line defined by $w = 0$ and also have an ideal line defined by $x + y + w = 0.$

Regarding whether $w = 0$ is the equation of a line, a plane, or something else, it is highly dependent on context. Without going to the specific place on the specific page of the specific website where a statement was made that $w = 0$ is the equation of something, it is impossible to say whether the statement was correct. It is as if a friend of yours told you he read on a website somewhere that the author's car was blue, and he asks you if it was true that the car is blue. How would you know? You can't know unless your friend tells you where to find the statement on the website. Otherwise it could be any car, of practically any color.

When discussing geometry and projective geometry, even the meanings of apparently simple words are highly context-dependent. We might be discussing four-dimensional Euclidean space one day, within which we identify various subsets of points that form one-dimensional lines, two-dimensional planes, and three-dimensional hyperplanes. But in a different conversation the word "hyperplane" might mean any of those things--it could have three dimensions, or two, or one.

The meaning of "hyperplane" on the linked Wikipedia page is one of those any-number-of-dimensions hyperplanes, not a necessarily-three-or-more-dimensional kind of hyperplane. It could be a line, and if the projective space under discussion is a projective plane, the hyperplane at infinity is a line.

It is also possible that someone explaining projective geometry might use three-dimensional Euclidean space with Cartesian coordinates $x,y,w$ in order to model a two-dimensional projective plane. In one such model, each line through the origin of the three-dimensional Euclidean space is declared to be a point of the projective plane. The equation $w = 0$ describes a plane in the Euclidean space which also contains all the ideal points of the projective plane and therefore represents the ideal line of the projective plane. Therefore when we speak of the object defined by $w = 0$ in this particular model of a projective space, we might be referring to the line (of the projective plane) defined by this equation or the plane (of the Euclidean space) defined by this equation. And in some contexts we might mean both things at the same time, since that plane is a model of that line.

All of this may be confusing if you are just getting introduced to projective geometry. It's fine not to consider projective spaces other than projective planes or to identify points in a projective plane with lines in a Euclidean space. But if you don't want to consider such things, you had better ignore websites that do consider such things, because the only way to make sense of what they are saying is to consider the possible dimensions and models that they are considering.