Understanding why the lines in $\mathbb{P}^2$ can be seen as planes passing through the origin of $\mathbb{R}^3$

Question

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.

Answer 1

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.

Answer 2

Hartley & Zisserman define $\mathbb{P}^2$ as the quotient space $(\mathbb{R}^3 \setminus \{0\}) / \sim$, where two points are equivalent iff the one dimensional subspace spanned by the points is the same. Let $[x]$ denote the equivalence classes.

So, not only is it fruitful to think of points as 'rays' (they seem to mean $k \neq 0$ as opposed to the more usual meaning of ray, $k\ge 0$) but that is essentially how they define them. (This interpretation is consistent with other texts.)

As far as I can determine, the authors use the sentence "In this model..." to define what they mean by a line in $\mathbb{P}^2$. That is, a line in $\mathbb{P}^2$ corresponds to a plane (passing through the origin) in $\mathbb{R}^3$.

If $[a],[b]$ are two distinct elements of $\mathbb{P}^2$, then $a,b$ are linearly independent elements of $\mathbb{R}^3$ and define a unique plane $\{ta+sb| s,t \in \mathbb{R} \} \subset \mathbb{R}^3$ which is, by definition, a line in $\mathbb{P}^2$.

Similarly, if you are given two distinct lines in $\mathbb{P}^2$, these correspond to two distinct planes passing through the origin. Their intersection defines a line (in $\mathbb{R}^3$) passing through the origin which is a point in $\mathbb{P}^2$.

(Frankly I find that particular section of the book to be a notational morass.)