geometryprojective-geometry
I am learning about projective planes. A projective plane is a affine plane with an added point at infinity for each set of parallel lines such that each set of parallel lines intersect at a unique point at infinity.
Is there another point at negative infinity? or does the plane start from some set of axis? or are the parallel lines parallel at negative infinity even though they meet at positive infinity?
Suppose that your eye is at the origin, and the "canvas" on which you draw is in front of your eye, and is the $z = 1$ plane. Then the point $(2, 3, 5)$ in space will project, in your "drawing" or "seeing" of the world, to the point $(2/5, 3/5, 1)$ in the $z = 1$ plane. In general, any point $(x, y, z)$ will project to $(x/z, y/z, 1)$.
Now consider two parallel lines; the first one consisting of all points of the form $(1, t, 4)$ and the second consisting of points $(3, t, 2)$. The projections of these to the drawing plane will consist of points of the form $(1/4, t/4, 1)$ and $(3/2, t/2, 1)$, respectively, i.e., they'll still be parallel.
Now look at the lines $(-1, 0, t)$ and $(1, 0, t)$. These project to $(-1/t, 0, 1)$ and $(1/t, 0, 1)$, which "meet" at the point $(0, 0, 1)$ when $t$ goes to $\infty$.
Does that help at all?
The purpose of figures 7.2 and 7.3 is to establish the connection $1D \leftrightarrow 2D$ by placing a "screen" at $y=1$.
What do I mean by "screen" ? It will be better understood on the equivalent correspondance with one more dimension (connection $2D \leftrightarrow 3D$) as shown on the right figure below (an excerpt of my lectures) where the eye is at the origin and "sees" the cube (or any other object as "projectively projected" on the screen positionned at height $z=1$. Please note that a "projective point" is completely identified with the corresponding (red) line issued from the origin.
The left hand side figure accounts for the connection between this screen plane, which is the plane of (projective) points, and the parallel plane at $z=0$ which is the plane of vectors. Moreover, it justifies barycentric expressions like $C=\tfrac23 A+\tfrac23 B$ by a 3D interpretation.
Best Answer
It is easier to start with the definition of a real projective line, which is just a normal line with one point added, that we call infinity. You can think of this as a circle: either way you go off in the line to infinity, you hit the same point, so there is no difference between positive and negative infinity.
Now we go up a dimension, and start with a normal plane. As you say, we want to add one point at infinity for each set of parallel lines in the plane. You should think of each of these points at infinity as corresponding to slopes in the obvious way: all parallel lines with slope, say, $1/2$, intersect at a point at infinity that we might call "$1/2$". And just like in the line case, we think of this point at infinity as connected to the lines at both endpoints -- so if you go off to infinity either direction on a line with slope $1/2$, you hit the point "$1/2$." In this sense, there is no difference between "positive and negative."
The way I like to think of this is that we can go off to infinity in many directions, corresponding to our slope, and each lands at a different point at infinity, with the exception that going off in two exactly opposite directions lands at the same infinite point.
So how many points at infinity are there? Well, there should be one for each possible slope, so we should think of there as being $\mathbb{R}$'s-worth of points at infinity. But we also have vertical lines, with slope infinity! So we also need a point at infinity that we just call "infinity." If you look back at the first paragraph, this is exactly the definition of a projective line. So the points at infinity together define one projective line, which we can think of as all the ways we can go off to infinity.