Lines in the Real Projective Plane

Recall from The Real Projective Plane page that the set of all points $\mathbf{x} = [x_1, x_2, x_3] = [kx_1, kx_2, kx_3]$ for $k \neq 0$ and $x_1, x_2, x_3$ not all $0$ form the real projective plane denoted $\mathbb{P}^2(\mathbb{R})$. We have already described points on the real projective plane, and now we will describe lines in the projective plane.

Definition: The Line $<a_1, a_2, a_3>$ in the real projective plane $\mathbb{P}^2(\mathbb{R})$ is the set of points $\mathbf{x} = [x_1, x_2, x_3] \in \mathbb{P}^2(\mathbb{R})$ that satisfy the equation $a_1x_1 + a_2x_2 + a_3x_3 = 0$ where $a_1, a_2, a_3 \in \mathbb{R}$ and are not all zero.

Notice that we use square brackets $[]$ to denote points and angled brackets $<>$ to denote lines in the real projective plane $\mathbb{P}^2 (\mathbb{R})$.

We can alternatively write the equation of the line $<a_1, a_2, a_3>$ in terms of matrices/vectors. If $\mathbf{a} = \begin{bmatrix} a_1\\ a_2\\ a_3 \end{bmatrix}$ and $\mathbf{x} = \begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix}$ then the line $<a_1, a_2, a_3>$ is the set of solutions to $\mathbf{a}^T \mathbf{x} = 0$:

(1)

\begin{align} \quad \mathbf{a}^T \mathbf{x} = \begin{bmatrix} a_1 & a_2 & a_3 \end{bmatrix} \begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix} = a_1x_1 + a_2x_2 + a_3x_3 = 0 \end{align}

For example, the line $<1, 2, 3>$ is the set of all points $\mathbf{x} = [x_1, x_2, x_3] \in \mathbb{P}^2 (\mathbb{R})$ such that:

(2)

\begin{align} \quad x_1 + 2x_2 + 3x_3 = 0 \end{align}

The point $[3, 0, -1] = [-3, 0, 1]$ lies on the line $<1, 2, 3>$. Similarly, the point $[2, 2, -2] = [-1, -1, 1]$ lies on $<1, 2, 3>$.

With regards to homogeneous coordinates, recall that for $k \neq 0$ we say that $\mathbf{x} = [x_1, x_2, x_3] = [kx_1, kx_2, kx_3] = k\mathbf{x}$. If the point $\mathbf{x} = [x_1, x_2, x_3]$ lies on the line $<a_1, a_2, a_3>$ then $k\mathbf{x} = [kx_1, kx_2, kx_3]$, $k \neq 0$ (which we acknowledge as the same point as $\mathbf{x}$) also lies on the line $<a_1, a_2, a_3>$ since if $\mathbf{a}^T \mathbf{x} = 0$, then:

(3)

\begin{align} \quad \mathbf{a}^T (k \mathbf{x}) = k \mathbf{a}^T \mathbf{x} = k \cdot 0 = 0 \end{align}

Or equivalently, if $a_1x_1 + a_2x_2 + a_3x_3 = 0$ then:

(4)

\begin{align} \quad a_1kx_1 + a_2kx_2 + a_3kx_3 = k(a_1x_1 + a_2x_2 + a_3x_3) = k(0) = 0 \end{align}

The Line at Infinity

One important aspect of the real projective plane is the so called line at infinity. We first need to describe the points that lie on this line. Recall that all points $\mathbf{x} \in \mathbb{P}^2 (\mathbb{R})$ are of the form $\mathbf{x} = [x_1, x_2, x_3]$ where not all $x_1, x_2, x_3$ are zero. Provided that $x_3 \neq 0$ we have that:

(5)

\begin{align} \quad \left [ \frac{x_1}{x_3}, \frac{x_2}{x_3}, 1 \right ] = [x_1, x_2, x_3] \end{align}

Now suppose that $x_3 = 0$. Such points have a special name which we define below.

Definition: If $\mathbf{x} = [x_1, x_2, 0] \in \mathbb{P}^2 (\mathbb{R})$ where $x_1, x_2$ are not both zero, then the point $\mathbf{x}$ is called a Point at Infinity.

Now consider the collection of all such points $\mathbf{x} = [x_1, x_2, 0]$. Such points lie on the line $<0, 0, 1>$ since:

(6)

\begin{align} \quad 0x_1 + 0x_2 + 1(0) = 0 \end{align}

This line is given a special name which we define below.

Definition: The Line at Infinity is denoted $<0, 0, 1>$ is the collection of all points at infinity $\mathbf{x} = [x_1, x_2, 0] \in \mathbb{P}^2 (\mathbb{R})$.