I am going through Stillwell's the Four Pillars of Geometry and having a hard time understanding this part of the text. I know that in $\mathbb{RP^2}$, the "points" are lines through $O$ in $\mathbb{R^3}$ and it's "lines" are planes through $O$ in $\mathbb{R^3}$. How can I visualize $\mathbb{RP^1}$? Does $\mathbb{RP^1}$ consist only of a single horizontal or vertical line?
For figure 7.2 they says that 'the single line that does not meet $y=1$, namely, the x-axis, naturally gets the label $\infty$. But why doesn't the x-axis meet line y=1? My understanding so far is that lines at parallel lines meet at infinity in projective geometry. So why not here?
For figure 7.3, I have no idea what is going on here. What are they trying to show? What does 'labels' mean for these lines? What plane are these lines in? $\mathbb{R P^1}$? $\mathbb{R^2}?$
Any help would be appreciated, very confused about projective geometry in general and i'm still very much a beginner at all of this.
Best Answer
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.