What is the relationship between $\mathbb{R}^3$ and the projective plane $P^2$?

I'm a computer science student and I am trying to understand the relationship between $$\mathbb{R}^3$$ and the projective plane $$P^2$$. I need to learn about the projective plane as it comes up in computer graphics.

I know that $$P^2$$ can be thought of as $$\mathbb{R}^2$$ together with the line at infinity. I also know that points in $$P^2$$ look like $$[a, b, c]$$, but $$a, b, c$$ can't all be zero. What is the relationship between the point $$(a, b, c)$$ in $$\mathbb{R}^3$$ and $$[a, b, c]$$ in $$P^2$$? Also, what happens to the point $$(0, 0, 0)$$ in $$\mathbb{R}^3$$?

You can define the projective plane $$P^2$$ as the space of lines in $$\mathbb{R}^3$$ through the origin. In particular, given a line in $$\mathbb{R}^3$$ which passes through $$(0, 0, 0)$$, there is a corresponding point in the projective plane $$P^2$$.

Suppose $$\ell$$ is a line in $$\mathbb{R}^3$$ through the origin. Any line is uniquely determined by two different points on the line. We know that $$(0, 0, 0)$$ is a point on the line, so suppose $$(a, b, c) \neq (0, 0, 0)$$ is a different point on $$\ell$$. The point of $$P^2$$ corresponding to the line $$\ell$$ in $$\mathbb{R}^3$$ is denoted by $$[a, b, c]$$. Note that we could have chosen a different point on the line $$\ell$$, but every other point is of the form $$(ka, kb, kc)$$ for some non-zero real number $$k$$, so we need $$[a, b, c]$$, the point in $$P^2$$ representing $$\ell$$, to be the same as $$[ka, kb, kc]$$ because it represents the same line. That is, we require $$[a, b, c] = [ka, kb, kc]$$ for all non-zero real numbers $$k$$. This is why we use homogeneous coordinates on $$P^2$$.

With this in mind, we can now understand the relationship between points in $$\mathbb{R}^3$$, other than the origin, and points in $$P^2$$. Given any point $$(a, b, c) \neq (0, 0, 0)$$ in $$\mathbb{R}^3$$, there is a corresponding point $$[a, b, c]$$ in $$P^2$$ which represents the line in $$\mathbb{R}^3$$ which passes through $$(0, 0, 0)$$ and $$(a, b, c)$$. The reason this correspondence fails for $$(a, b, c) = (0, 0, 0)$$ is that you don't have two different points in $$\mathbb{R}^3$$, so you don't get a line (and hence, you don't get a point in $$P^2$$).

Note that a point at infinity of $$P^2$$ is a point of the form $$[a, b, 0]$$, and corresponds to a line in $$\mathbb{R}^3$$ through $$(0, 0, 0)$$ and $$(a, b, 0)$$; in particular, it lies in the $$xy$$-plane. It follows that the line at infinity of $$P^2$$ corresponds to the lines in $$\mathbb{R}^3$$ through the origin which lie in the $$xy$$-plane.

• For infinity case c=0 treated here plane or line here? Jul 31, 2021 at 17:36
• @AlokMaity: I have added an extra paragraph. I hope this addresses your question. Jul 31, 2021 at 17:38
• A line in $P^2$ corresponds to a plane in $\mathbb{R}^3$ through the origin. In particular, the line at infinity in $P^2$ corresponds to the $xy$-plane in $\mathbb{R}^3$. Jul 31, 2021 at 18:27
• A line in $P^2$ is a collection of points, and each of those points represent lines in $\mathbb{R}^3$ through the origin. If you consider all the lines together, they form a plane in $\mathbb{R}^3$ through the origin. For example, each point on the line at infinity in $P^2$ corresponds to a line through the origin in the $xy$-plane, and if you consider all of these lines together, they give you the $xy$-plane. Jul 31, 2021 at 19:24
• First of all, they are lines in different spaces. Moreover, a line in $\mathbb{R}^3$ does not give you a line in $P^2$, and a line in $P^2$ does not give you a line in $\mathbb{R}^3$. One thing you can say is that they are different topologically. A line in $\mathbb{R}^3$ is infinite in both directions, while a line in $P^2$ is a circle (the two ends of a usual line join up at a single point). Aug 1, 2021 at 0:24