Composite translations and reflections

25 min

Two mirrors facing each other

In a hair salon with mirrors on two opposite walls, you see a long row of copies of yourself. The copy seen "through" both mirrors, a reflection of a reflection, faces the same way as you and is simply further along the row: two reflections have made a slide. In this lesson you will see why, and learn which single transformation replaces two translations or two reflections.

Definition

The composite $g \circ f$ of two transformations means: apply $f$ , then apply $g$ to the result. If a single transformation $h$ has the same effect on every point, we write $g \circ f = h$ .

Example: translating by $\begin{pmatrix} 2 \\ 0 \end{pmatrix}$ and then by $\begin{pmatrix} 0 \\ 3 \end{pmatrix}$ has the same effect as translating by $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ .

Important

1. Two translations: $T_{\vec{w}} \circ T_{\vec{v}} = T_{\vec{v} + \vec{w}}$ . The order does not matter, because $\vec{v} + \vec{w} = \vec{w} + \vec{v}$ .

2. Two reflections in parallel mirrors $m_{1}$ then $m_{2}$ , a distance $d$ apart: a translation through $2 d$ , at right angles to the mirrors, in the direction from $m_{1}$ to $m_{2}$ .

3. Two reflections in mirrors meeting at $X$ at an angle $\alpha$ : a rotation about $X$ through $2\alpha$ , turning from $m_{1}$ towards $m_{2}$ . If the mirrors are perpendicular, it is the half-turn $S_{X}$ .

Why a reflection twice gives no reflection: each reflection reverses orientation, so two of them restore it. The result is a direct isometry: a translation or a rotation.

Worked example 1: two translations

Triangle $A (1, 1)$ , $B (3, 1)$ , $C (1, 4)$ is translated by $\vec{v} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}$ and then by $\vec{w} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}$ . Find the final image and the single transformation.

Solution The single transformation is $T_{\vec{v} + \vec{w}}$ with $\vec{v} + \vec{w} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}$ . Final image: $A^{''} (3, 6)$ , $B^{''} (5, 6)$ , $C^{''} (3, 9)$ . Step by step gives the same: $A(1, 1) \to (4, 2) \to (3, 6)$ .

Worked example 2: two parallel mirrors

A flag is reflected in $m_{1} : x = 1$ and then in $m_{2} : x = 4$ . Find the single transformation.

Solution $M_{m_1} : (x, y) \mapsto (2 - x,\; y)$ and $M_{m_2} : (x, y) \mapsto (8 - x,\; y)$ . So

(x, y) \;\longmapsto\; (2 - x,\; y) \;\longmapsto\; \big(8 - (2 - x),\; y\big) = (x + 6,\; y) .

This is the translation by $\begin{pmatrix} 6 \\ 0 \end{pmatrix}$ : twice the distance 3 between the mirrors, from $m_{1}$ towards $m_{2}$ . The pole at $x = - 1$ goes to $x = 3$ and then to $x = 5$ .

Worked example 3: mirrors that cross

Triangle $A (3, 1)$ , $B (5, 1)$ , $C (5, 2)$ is reflected in the $x$ -axis and then in the line $y = x$ . Find the images and the single transformation.

Solution In the $x$ -axis: $A^{'} (3, - 1)$ , $B^{'} (5, - 1)$ , $C^{'} (5, - 2)$ . Then in $y = x$ (swap the coordinates): $A^{''} (- 1, 3)$ , $B^{''} (- 1, 5)$ , $C^{''} (- 2, 5)$ .

In general: $(x, y) \mapsto (x, -y) \mapsto (-y, x)$ , the rule for $R_{O, +90^\circ}$ . The mirrors meet at $O$ at $45^\circ$ , and $2 \times 45^\circ = 90^\circ$ , turning from the $x$ -axis towards $y = x$ : anticlockwise.

Worked example 4: the order matters

Reflect first in $y = x$ and then in the $x$ -axis. What single transformation results?

Solution $(x, y) \mapsto (y, x) \mapsto (y, -x)$ : the rotation $R_{O, -90^\circ}$ , a quarter-turn clockwise. Swapping the order of the mirrors reverses the direction of the rotation.

Worked example 5: two horizontal mirrors

Find the single transformation equal to the reflection in $y = 2$ followed by the reflection in $y = - 1$ .

Solution $(x, y) \mapsto (x,\; 4 - y) \mapsto \big(x,\; -2 - (4 - y)\big) = (x,\; y - 6)$ : the translation by $\begin{pmatrix} 0 \\ -6 \end{pmatrix}$ . Check: the mirrors are 3 apart, and the second is below the first, so the slide is 6 units down.

Warning

Twice the gap, twice the angle. Two parallel mirrors 3 units apart give a translation of 6 units, not 3; mirrors at $45^\circ$ give a rotation of $90^\circ$ , not $45^\circ$ .

For reflections, the order matters: $M_{m_2} \circ M_{m_1}$ and $M_{m_1} \circ M_{m_2}$ are translations (or rotations) in opposite directions.

Tip

To name a composite, push a general point $(x, y)$ through both rules, one after the other, and then read the final rule against the table of isometries from the last lesson.

Mario's Math Tutoring: Composition of reflections over 2 parallel lines. Watch the distance between the mirrors being doubled to give the length of the translation, as in Worked example 2.

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Practice

The last two are harder.

Find the single transformation equal to $T_{\vec{v}}$ followed by $T_{\vec{w}}$ , where $\vec{v} = \begin{pmatrix} 4 \\ -2 \end{pmatrix}$ and $\vec{w} = \begin{pmatrix} -7 \\ 5 \end{pmatrix}$ , and the image of $(0, 0)$ .
Find the single transformation equal to the reflection in $x = 2$ followed by the reflection in $x = 5$ .
Find the single transformation equal to the reflection in the $x$ -axis followed by the reflection in the $y$ -axis.
Reflect $P (4, 1)$ in the $y$ -axis and then in $y = x$ . Name the single transformation and check it on $P$ .
Find the single transformation equal to the reflection in $y = 3$ followed by the reflection in the $x$ -axis.
A translation by $\begin{pmatrix} 8 \\ 0 \end{pmatrix}$ is to be made from two reflections in vertical mirrors. The first mirror is $x = 1$ . Where must the second be?
Show that reflecting in $y = - x$ and then in the $y$ -axis is a rotation about $O$ , and find its angle.

Show answersHide answers

$T_{\vec{v} + \vec{w}}$ with $\vec{v} + \vec{w} = \begin{pmatrix} -3 \\ 3 \end{pmatrix}$ ; $(0, 0) \mapsto (-3, 3)$ .
$(x, y) \mapsto (4 - x,\; y) \mapsto (10 - (4 - x),\; y) = (x + 6,\; y)$ : translation by $\begin{pmatrix} 6 \\ 0 \end{pmatrix}$ .
$(x, y) \mapsto (x, -y) \mapsto (-x, -y)$ : the half-turn $S_{O}$ (perpendicular mirrors).
$(4, 1) \mapsto (-4, 1) \mapsto (1, -4)$ . In general $(x, y) \mapsto (-x, y) \mapsto (y, -x)$ : $R_{O, -90^\circ}$ , and indeed $(4, 1) \mapsto (1, -4)$ .
$(x, y) \mapsto (x,\; 6 - y) \mapsto (x,\; y - 6)$ : translation by $\begin{pmatrix} 0 \\ -6 \end{pmatrix}$ .
The mirrors must be $\frac{8}{2} = 4$ apart, with the second to the right: $x = 5$ .
$(x, y) \mapsto (-y, -x) \mapsto (y, -x)$ : $R_{O, -90^\circ}$ . The mirrors meet at $45^\circ$ , and turning from $y = - x$ to the $y$ -axis is clockwise, so the angle is $-90^\circ$ .

Sources and credits

Written by Ganzaa.
Video: Composition of Reflections over 2 Parallel Lines, Mario's Math Tutoring, embedded from YouTube.

The diagrams, worked examples and practice items were written by Ganzaa.

1Two mirrors facing each other

Two mirrors facing each other