Invariance in Transformations

What can change, and what stays the same?

A transformation maps each point of a shape to a new position. With a combination of transformations, you carry out one movement, then transform its image again. The final result may look like a single translation, reflection or rotation, even though it took two steps to produce it. In this Higher-tier topic, the task is to describe both the changes and the invariance: what has stayed unchanged.

Translations slide a shape, rotations turn it about a centre, and reflections flip it across a mirror line. None of these three changes its size or shape. Corresponding side lengths and angles remain equal, so the final image is congruent to the original; its perimeter and area remain equal too. Its position can change, and a rotation or reflection can change the direction in which it appears to face. A reflection also reverses the shape's orientation in the precise sense of its clockwise or anticlockwise order of vertices. Translations and rotations preserve that order. After two reflections, it is preserved again.

Following a sequence

Start with the original shape, perform the first transformation and mark an intermediate image. Apply the second transformation to that image—not to the original. Then compare the original directly with the final image to describe their overall relationship. The order can matter: reflecting in one line and then another may produce a different result if the order is reversed.

For example, reflect a point $(2,1)$ in the $x$-axis to get $(2,-1)$, then reflect it in the $y$-axis to get $(-2,-1)$. The same sequence takes every point $(x,y)$ to $(-x,-y)$. Comparing start and finish shows that the two reflections together are equivalent to a $180^\circ$ rotation about the origin. The intermediate point helps you carry out the steps; it is the original and final positions that reveal the overall transformation.

Describing translations with column vectors

A translation by $\begin{pmatrix}a\\b\end{pmatrix}$ moves every point $a$ units horizontally and $b$ units vertically: $(x,y)\mapsto(x+a,y+b)$. A positive top entry means right and a negative one means left; a positive bottom entry means up and a negative one means down. For instance, $\begin{pmatrix}3\\-2\end{pmatrix}$ means three units right and two down.

If two translations follow one another, add their horizontal movements and their vertical movements separately. A translation by $\begin{pmatrix}4\\-3\end{pmatrix}$ followed by $\begin{pmatrix}-1\\7\end{pmatrix}$ gives $\begin{pmatrix}3\\4\end{pmatrix}$ overall. Every point finishes three units right and four units up from where it started. Unlike many other combinations, swapping the order of these two translations gives the same result.

Two reflections can also combine into a translation. Reflecting first in $x=1$ and then in $x=5$ moves every point eight units to the right, so the overall translation is $\begin{pmatrix}8\\0\end{pmatrix}$. More generally, reflections in parallel lines a distance $d$ apart produce a translation of $2d$ perpendicular to the lines, from the first mirror towards the second. Reflections in intersecting lines produce a rotation about their intersection; its angle is twice the angle through which the first mirror line turns to the second. Thus the mirror lines and their order affect the final result.

Invariant points and invariant shapes

An invariant point ends exactly where it started. Under a single reflection, every point on the mirror line is invariant. Under a rotation through an angle other than a full turn, the centre is invariant. A translation by a non-zero vector has no invariant points. For a combination, judge invariance from the whole sequence: a point might move in the first step but return to its starting place after the second. Reflecting twice in the same line, for example, returns every point.

An invariant shape is a different idea: its final outline occupies the same region as its original outline. A square rotated $90^\circ$ about its centre lands on itself, but its corners move to different corners. The shape is invariant as a whole, while only its centre is an invariant point. This distinction lets you describe accurately both what the combined transformation changes and what it leaves unchanged.

Combined transformations

Carry out each step in order, then compare the original with the final image. Translations, reflections and rotations preserve lengths, angles, perimeter and area. A reflection reverses clockwise/anticlockwise vertex order; a translation or rotation preserves it.

Useful results

Translation $\begin{pmatrix}a\\b\end{pmatrix}$ means $(x,y)\mapsto(x+a,y+b)$. Add column vectors component by component for successive translations. Two reflections in parallel lines give a translation perpendicular to the lines, twice their separation; two reflections in intersecting lines give a rotation about their intersection, through twice the angle between the lines. Order can change the result.

Invariance

An invariant point finishes where it started: points on a reflection line, the centre of a non-trivial rotation, or none under a non-zero translation. For a sequence, check the final position. A shape can map onto itself even when its individual points move.