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.