Math Core

Real motions are rarely a single slide or a single turn. A car on a curving road, a tile pattern, or a figure in a proof may need several transformations in a row. Performing one transformation after another is called a composition, and a few surprising shortcuts let you replace a whole chain with a single, simpler move.

Doing one transformation after another

Definition

Composition of transformations

A composition applies one transformation and then applies a second transformation to the image. The final figure is written with double primes: $A \rightarrow A' \rightarrow A''$ .

Compositions are written like composed functions. The notation $(r_{x\text{-axis}} \circ T_{\langle 2, 3 \rangle})(P)$ means "translate $P$ by $\langle 2, 3 \rangle$ , then reflect over the x-axis." You read it from right to left, because the transformation closest to $P$ happens first.

Worked example: A translation followed by a reflection

Triangle $ABC$ has vertices $A(-5, 1)$ , $B(-2, 1)$ and $C(-4, 3)$ . Translate it by $\langle 6, 1 \rangle$ , then reflect the image over the x-axis.

Step 1: translate. $(x, y) \rightarrow (x + 6,\ y + 1)$ gives $A'(1, 2)$ , $B'(4, 2)$ and $C'(2, 4)$ .

Step 2: reflect. $(x, y) \rightarrow (x, -y)$ gives $A''(1, -2)$ , $B''(4, -2)$ and $C''(2, -4)$ .

Triangle ABC, its translation A′B′C′, and the final image A″B″C″ after the reflection.Open in grapher →

Order matters

Changing the order of two transformations can change the result.

Worked example: Swapping the order

Start with $P(2, 1)$ . Compare reflecting over the y-axis then translating by $\langle 4, 0 \rangle$ , with translating first and reflecting second.

Reflect, then translate: $(2, 1) \rightarrow (-2, 1) \rightarrow (2, 1)$ .
Translate, then reflect: $(2, 1) \rightarrow (6, 1) \rightarrow (-6, 1)$ .

Different answers, so these two compositions are different transformations.

Common mistake

In notation like $(R \circ T)(P)$ , do $T$ first. It's tempting to work left to right like reading a sentence, but compositions work inside out, just like $f(g(x))$ evaluates $g$ first.

Compositions of rigid motions

Each rigid motion preserves distance and angle measure, so a chain of them does too.

Rigid motions compose to rigid motions

A composition of translations, reflections and rotations is again a rigid motion. The final image is congruent to the pre-image. Each reflection in the chain reverses orientation, so an even number of reflections keeps orientation and an odd number reverses it.

This is the modern definition of congruence you'll use in the next unit: two figures are congruent when some sequence of rigid motions maps one onto the other.

Glide reflections

A translation followed by a reflection over a line parallel to the translation vector is called a glide reflection. Footprints in sand are the classic picture: each print is the previous one slid forward and flipped. The first example above is not a glide reflection, because the vector $\langle 6, 1 \rangle$ isn't parallel to the x-axis. Translating by $\langle 6, 0 \rangle$ and then reflecting over the x-axis would be.

Two reflections make a simpler motion

Reflecting twice is always the same as a single translation or a single rotation.

Reflecting in two lines

Parallel lines. Reflecting over line $\ell$ and then over a parallel line $m$ is a translation. It moves points perpendicular to the lines, in the direction from $\ell$ toward $m$ , by twice the distance between the lines.
Intersecting lines. Reflecting over $\ell$ and then over a line $m$ that crosses $\ell$ at point $P$ is a rotation about $P$ by twice the angle from $\ell$ to $m$ .

Worked example: Two parallel mirrors

Reflect $Q(-1, 3)$ over the line $x = 1$ , and then over the line $x = 4$ .

Using $(x, y) \rightarrow (2h - x,\ y)$ twice:

(-1, 3) \rightarrow (2 - (-1),\ 3) = (3, 3) \rightarrow (8 - 3,\ 3) = (5, 3)

The point moved $6$ units right. The lines are $3$ units apart, and $2 \cdot 3 = 6$ : the composition is the translation $\langle 6, 0 \rangle$ . Any other point moves the same way; try $(0, 0) \rightarrow (2, 0) \rightarrow (6, 0)$ .

Worked example: Two crossing mirrors

Show that reflecting over $y = x$ and then over the x-axis is a rotation.

Follow a general point: $(x, y) \rightarrow (y, x) \rightarrow (y, -x)$ .

That's the rule for a $90^\circ$ clockwise rotation about the origin. It matches the theorem: the lines meet at the origin, the angle from $y = x$ down to the x-axis is $45^\circ$ clockwise, and twice that is $90^\circ$ clockwise.

Tip

To identify a composition, follow a general point $(x, y)$ through each rule and simplify. If the final rule matches a known rule, you've found the single transformation.

Practice

Practice 1

Translate $(3, -2)$ by $\langle -5, 4 \rangle$ , and then reflect the image over the y-axis. What is the final image?

Enter a point like (2, -3)

Practice 2

Rotate $(4, 1)$ $90^\circ$ counterclockwise about the origin, and then reflect the image over the x-axis. What is the final image?

Enter a point like (2, -3)

Practice 3

Now do it in the other order: reflect $(4, 1)$ over the x-axis, and then rotate the image $90^\circ$ counterclockwise about the origin. What is the final image?

Enter a point like (2, -3)

Practice 4

Find $(r_{y\text{-axis}} \circ T_{\langle 2, 3 \rangle})(1, -4)$ , where $r_{y\text{-axis}}$ is the reflection over the y-axis and $T_{\langle 2, 3 \rangle}$ is the translation by $\langle 2, 3 \rangle$ .

Enter a point like (2, -3)

Practice 5

Reflecting a figure over the y-axis and then over the x-axis is the same as which single transformation?

Practice 6

A figure is reflected over the line $x = -2$ and then over the line $x = 3$ . The result is a translation. How many units, and in which direction, does each point move? Enter the number of units.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

Two lines intersect at point $P$ , forming a $35^\circ$ angle. A figure is reflected over the first line and then over the second. The result is a rotation about $P$ . What is the angle of rotation, in degrees?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

A glide reflection translates by $\langle 5, 0 \rangle$ and then reflects over the x-axis. Apply this glide reflection to $(-1, 2)$ , and then apply it again to the result. Where does the point end up?

Enter a point like (2, -3)