I am reading the text Multiple-View Geometry in Computer Vision where the author is introducing the projective plane $\mathbb{P}^2$ as the augmentation of the homogeneous co-ordinates of $\mathbb{R}^2$ of the form $(x_1, x_2, x_3)$ with $x_3=0$ for points at infinity, and the line at infinity defined as $(1,0,0$), and regular euclidean points in $\mathbb{R}^2$ as $(x_1/x_3, x_2/x_3)$ for $x_3 \neq 0$.
The author writes on page $29$:
A fruitful way of thinking of $\mathbb{P}^2$ is as a set of rays in $\mathbb{R}^3$. The set of all vectors $k(x_1, x_2, x_3)^T$ as $k$ varies forms a ray through the origin. Such a ray may be thought of as representing a single point in $\mathbb{P}^2$. In this model the lines in $\mathbb{P}^2$ are planes passing through the origin.
This last sentence is giving me difficulty. Take two rays that are not identical say $k(x_1, x_2, x_3)$ and $t(x'_1, x'_2, x'_3)$, then since they both contain the origin, they span a plane. I'm not sure what the corresponding "line" is in $\mathbb{P}^2$, I'm having a bit of difficulty understanding the terminology/concepts. Any insights appreciated.
Best Answer
First, "ray through the origin" is (imo) a bad term to use. Indeed a point in $\mathbb{P}^2$ corresponds to a set of the form $\{k (x_1, x_2, x_3) : k \in \mathbb{R}\}$ for some nonzero vector $(x_1, x_2, x_3)$, but this is a line, not a ray! The distinction is very important, since $(x_1, x_2, x_3)$ and $(-x_1, -x_2, -x_3)$ trace out the same line (and thus correspond to the same point in $\mathbb{P}^2$).
Anyway, this is really a way to define "line in $\mathbb{P}^2$". A line in $\mathbb{P}^2$, by this definition, is the set of points in $\mathbb{P}^2$ corresponding to lines through the origin in $\mathbb{R}^3$ which are contained in a given plane through the origin.