The Real Projective Plane

Definition: Let $\sim \: \subseteq \mathbb{R}^3 \times \mathbb{R}^3$ be the equivalence relation such that for $\mathbf{x} = (x_1, x_2, x_3), \mathbf{y} = (y_1, y_2, y_3) \in \mathbb{R}^3$ we have that $\mathbf{x} \sim \mathbf{y}$ if $(x_1, x_2, x_3) = (ky_1, ky_2, ky_3)$ for some $k \in \mathbb{R}$, $k \neq 0$. Let $[\mathbf{x}] = \{ \mathbf{a} \in \mathbb{R}^3 : \mathbf{x} \sim \mathbf{a} \}$ be the equivalence class of $\mathbf{x} \in \mathbb{R}^2$ and let $\bigcup_{\mathbf{x} \in \mathbb{R}^3} [\mathbf{x}]$ be the union of these equivalence classes. The Real Projective Plane denoted $\mathbb{P}^2 (\mathbb{R})$ is defined to be the collection of these equivalence classes. The collection of elements in $\mathbb{P}^2 (\mathbb{R})$ are called Points on the real projective plane.

The definition above may be a little cloudy, so let's try to gain some intuition on the matter.

As we note above, a general point in $x \in \mathbb{P}^2(\mathbb{R})$ is denoted in the following form:

(1)

\begin{align} \quad \mathbf{x} = [x_1, x_2, x_3] \end{align}

The equivalence class of $\mathbf{x}$ with respect to the equivalence relation $\sim$ is the collection of scalar multiples of $\mathbf{x}$ for some $k \in \mathbb{R}$ and $k \neq 0$:

(2)

\begin{align} \quad [\mathbf{x}] = \{ [kx_1, kx_2, kx_3], k \in \mathbb{R}, k \neq 0 \} \end{align}

Therefore we say that the points $\mathbf{x} = [x_1, x_2, x_3]$ and $k\mathbf{x} = [kx_1, kx_2, kx_3]$, $k \neq 0$ are equivalent or the same since:

(3)

\begin{align} \quad [x_1, x_2, x_3] \sim [kx_1, kx_2, kx_3] \end{align}

Now consider any point $(x_1, x_2) \in \mathbb{R}^2$. We can acknowledge this point on the real projective plane by adjoining a third coordinate. If we choose the third coordinate $x_3 = 1$ then the point $(x_1, x_2) \in \mathbb{R}^2$ is represented by $\mathbf{x} = [x_1, x_2, 1] \in \mathbb{P}^2 (\mathbb{R})$. In such case, we see that if $k \neq 0$ then a line $L$ in $\mathbb{R}^2$ is:

(4)

\begin{align} \quad L = \{ (kx_1, kx_2) : k \in \mathbb{R} \} \subset \mathbb{R}^2 \end{align}

The line above can be given by the parametric equations $x = kx_1$ and $y = kx_2$ and provided that $x_1 \neq 0$ (i.e., the line is not horizontal in $\mathbb{R}^2$) the equation of the line with these parametric equations is $y = \frac{x_2}{x_1}x$ having slope $m = \frac{x_2}{x_1}$.

Correspondingly, the points on the line $L$ in the real projective plane $\mathbb{P}^2 (\mathbb{R})$ are of the form $[kx_1, kx_2, k]$ for $k \neq 0$ for $\mathbf{x} \sim [x_1, x_2, 1]$ that the equivalence class of $\mathbf{x}$ with respect to the equivalence relation $\sim$ is $[ \mathbf{x} ] = [kx_1, kx_2, k]$ i.e., the collection of points $(kx_1, kx_2) \in L \subset \mathbb{R}^2$ on the line converted to coordinates $[kx_1, kx_2, k]$, $k \neq 0$ in the projective plane represent a single point in the real projective plane $\mathbb{P}^2 (\mathbb{R})$.

As we have seen, any point $\mathbf{x} = [x_1, x_2, x_3]$ where $x_1 \neq 0$ has an analogue in $\mathbb{R}^2$ since $[x_1, x_2, x_3] \sim \left [\frac{x_1}{x_3}, \frac{x_2}{x_3}, 1 \right ]$ corresponds to the point $\left ( \frac{x_1}{x_3}, \frac{x_2}{x_3} \right ) \in \mathbb{R}^2$.

So what exactly happens when $x_3 = 0$? In such cases, the point $\mathbf{x} = [x_1, x_2, x_3] = [x_1, x_2, 0]$ cannot be written in the form above. We will look at these points and the collection of these points on the Lines in the Real Projective Plane page.