Trying to understand real projective space ${\mathbb {R}}P^{3}$.
Quoting this article:
The line at infinity can be visualized as a circle which surrounds the affine plane. However, diametrically opposite points of the circle are equivalent—they are the same point. The combination of the affine plane and the line at infinity makes the real projective plane $\mathbb{R}P^2$.
This makes me wonder if the following is true:
In $\mathbb{R}P^3$, the plane at infinity is a real projective plane $\mathbb{R}P^2$ that can be visualized as a 2-sphere which surrounds the affine 3D space. However, diametrically opposite points of the 2-sphere are equivalent—they are the same point. The combination of the affine 3D space and the projective plane at infinity (which is $\mathbb{R}P^2$) makes the real projective plane ${\mathbb {R}}P^{3}$.
Some other questions:
- Is $\mathbb{R}P^2$ "made of" projective lines $\mathbb{R}P^1$ or of "regular" ("Euclidean"?) lines?
- Is the plane at infinity $\mathbb{R}P^2$ unique (in some sense) in $\mathbb{R}P^3$, or is $\mathbb{R}P^3$ made of uncountably many stacked copies of $\mathbb{R}P^2$, just like regular 3D Euclidean space is made of uncountably made stacked copies of Euclidean 2D space?
- Is the plane at infinity $\mathbb{R}P^2$ unique (in some sense) in $\mathbb{R}P^3$, or, if you apply some kind of "rotation", then some other plane would become the new plane at infinity, and then the old plane at infinity would become a regular plane not-at-infinity?
Not sure what "a 2-sphere which surrounds the affine plane" is, but yes: $\mathbb{R}P^3$ can be seen as a 3-disk whose points of the boundary $S^3$ are glued via the antipodal involution. So, $\mathbb{R}P^3$ is ''a disk with $\mathbb{R}P^2$ on the boundary''. Analytically, this can be seen as points of the form $[a:b:c:0]$ sitting inside of the set of points of the form $[a:b:c:d]. $
As our undergraduate algebra professor loved to repeat, in $\mathbb{R}P^2$ all lines "have the same rights". That is, there are no special cases like parallel lines, all lines intersect at 2 points, etc. So, the projective plane can be seen as a "symmetric" version of the Euclidean plane. In particular, there is nothing special or unique about the line at infinity. Analytically, you can see that there are points of the form $[a:b:0]$ in $\mathbb{R}P^2,$ but there are also, for example, points of the form $[a:0:b]$, which also form a projective line. All lines can be transformed into one another. Same goes for copies of $\mathbb{R}P^2$ in $\mathbb{R}P^3$ and so on. If you go back to the model of $\mathbb{R}P^3$ as a quotient of $S^3$ under the antipodal involution, the symmetry should be clear. There are different copies of big $S^2$'s inside of $S^3,$ each giving a plane $\mathbb{R}P^2 \subset \mathbb{R}P^3.$ This also shows how $\mathbb{R}P^3$ can be sliced into copies of $\mathbb{R}P^2$: you encode big circles $S^2$ in $S^3$ by $\mathbb{R}P^1$ as they rotate covering the whole $S^3.$ As an exercise I suggest checking this from the homogeneous coordinates perspective.