Preservation of Collinear Points and Concurrent Lines with Collineations

# The Preservation of Collinear Points and Concurrent Lines with Collineations

Recall from the Collineations of Projective Planes on Points page that if $F$ is any field, $\mathbb{P}^2(F)$ be the projective plane over $F$, and $M$ be any $3 \times 3$ invertible matrix whose entries are from $F$ then a collineation of $\mathbb{P}^2(F)$ is a bijective function $\phi_M : \mathbb{P}^2(F) \to \mathbb{P}^2(F)$ defined for each point $\mathbf{x} = [x_1, x_2, x_3] \in \mathbb{P}^2(F)$ by $\phi_M(\mathbf{x}) = \mathbf{x}M$.

One important aspect about collineations of the projection plane $\mathbb{P}^2(F)$ is that the bijective functions $\phi_M$ that define the collineation preserve sets of collinear points and sets of concurrent lines. In other words, if $\mathbf{p}, \mathbf{q}, \mathbf{r}$ are collinear points, then $\phi_M(\mathbf{p}), \phi_M(\mathbf{q}), \phi_M(\mathbf{r})$ are collinear points, and furthermore, if $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are concurrent lines, then $\phi_M(\mathbf{a}), \phi_M(\mathbf{b}), \phi_M(\mathbf{c})$ are concurrent lines.

We prove these statements in the following theorems.

 Theorem 1 (The Preservation of Collinear Points with Collineations): Let $F$ be any field, $\mathbb{P}^2(F)$ be the projective plane over $F$, and $M$ be any $3 \times 3$ invertible matrix whose entries are from $F$. Then $\mathbf{p}, \mathbf{q}, \mathbf{r}$ are collinear points if and only if $\phi_M(\mathbf{p}), \phi_M(\mathbf{q}), \phi_M(\mathbf{r})$ are collinear points.
• Proof: Let $\mathbf{p} = [p_1, p_2, p_3], \mathbf{q} = [q_1, q_2, q_3], \mathbf{r} = [r_1, r_2, r_3] \in \mathbb{P}^2(F)$ and consider the augmented $3 \times 3$ matrix:
(1)
\begin{align} \quad \begin{bmatrix} \mathbf{p}\\ \mathbf{q}\\ \mathbf{r} \end{bmatrix} = \begin{bmatrix} p_1 & p_2 & p_3\\ q_1 & q_2 & q_3\\ r_1 & r_2 & r_3 \end{bmatrix} \end{align}
• From the collineation $\phi_M : \mathbb{P}^2(F) \to \mathbb{P}^2(F)$ we get that $\phi_M(\mathbf{p}) = \mathbf{p} M$, $\phi_M(\mathbf{q}) = \mathbf{q}M$ and $\phi_M(\mathbf{r}) = \mathbf{r}M$. This system is representable by the matrix system:
(2)
\begin{align} \quad \begin{bmatrix} \mathbf{p}\\ \mathbf{q}\\ \mathbf{r} \end{bmatrix} M = \begin{bmatrix} \phi_M(\mathbf{p})\\ \phi_M(\mathbf{q})\\ \phi_M(\mathbf{r}) \end{bmatrix} \end{align}
• Let $A = \begin{bmatrix} \mathbf{p}\\ \mathbf{q}\\ \mathbf{r} \end{bmatrix}$ and $B = \begin{bmatrix} \phi_M(\mathbf{p})\\ \phi_M(\mathbf{q})\\ \phi_M(\mathbf{r}) \end{bmatrix}$. Then $AM = B$ and:
(3)
\begin{align} \quad \det (AM) = \det (B) \\ \quad \det (A) \cdot \det (M) = \det (B) \end{align}
• Since $M$ is an invertible matrix we have that $\det (M) \neq 0$, so $\det (A) = 0$ if and only if $\det (B) = 0$, i.e., the points $\mathbf{p}, \mathbf{q}, \mathbf{r}$ are collinear if and only if $\phi_M(\mathbf{p}), \phi_M(\mathbf{q}), \phi_M(\mathbf{r})$ are collinear. $\blacksquare$

Similarly, the analogous result holds for concurrent lines.

 Theorem 2 (The Preservation of Concurrent Lines with Collineations): Let $F$ be any field, $\mathbb{P}^2(F)$ be the projective plane over $F$, and $M$ be any $3 \times 3$ invertible matrix whose entries are from $F$. Then $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are concurrent lines if and only if $\phi_M(\mathbf{a}), \phi_M(\mathbf{b}), \phi_M(\mathbf{c})$ are collinear lines.