1. What a transformation is
A function from the plane to itself: a rule that takes each point $(x, y)$ as input and returns another point $(x', y')$ as output. Apply it to every point of a figure and you get the figure's image.
That's the whole frame. Geometry used to be a collection of static shapes; once you have transformations, it becomes a study of mappings. A square doesn't just sit there — it's something you can slide, spin, flip, or stretch, and the interesting questions are about which of those moves leave the shape recognizably the same.
The four moves below cover most of what you'll encounter. The first three are rigid: they move a figure without distorting it. The fourth — dilation — changes size on purpose.
2. The four core moves
Each of these acts on a small triangle below. The pale outline is the original; the solid orange is the image. Watch what changes and what doesn't.
Translation, rotation, and reflection all preserve the triangle's edge lengths and interior angles. Dilation preserves the angles but multiplies every distance by $2$ — same shape, bigger.
3. Coordinate rules at a glance
Once you fix a coordinate system, every transformation becomes a tiny algebraic rule. Memorize these — they're the working alphabet for everything below.
| Transformation | Coordinate rule | What it does |
|---|---|---|
| Translation by $(a, b)$ | $(x, y) \to (x + a,\; y + b)$ | Slide by the vector $(a, b)$. |
| Rotation $90°$ CCW about origin | $(x, y) \to (-y,\; x)$ | Quarter-turn left. |
| Rotation $180°$ about origin | $(x, y) \to (-x,\; -y)$ | Half-turn (also called point reflection). |
| Rotation $270°$ CCW about origin | $(x, y) \to (y,\; -x)$ | Three-quarter turn left (or quarter-turn right). |
| Rotation by $\theta$ about origin | $(x, y) \to (x\cos\theta - y\sin\theta,\; x\sin\theta + y\cos\theta)$ | The general formula. The three rules above are the $\theta = 90°,\,180°,\,270°$ specializations. |
| Reflection across $x$-axis | $(x, y) \to (x,\; -y)$ | Flip $y$. |
| Reflection across $y$-axis | $(x, y) \to (-x,\; y)$ | Flip $x$. |
| Reflection across $y = x$ | $(x, y) \to (y,\; x)$ | Swap coordinates. |
| Dilation by $k$ about origin | $(x, y) \to (k x,\; k y)$ | Scale all distances from the origin by $|k|$. |
Plug $\theta = 90°$ into the general rotation formula: $\cos 90° = 0$, $\sin 90° = 1$, so $(x, y) \to (x \cdot 0 - y \cdot 1,\; x \cdot 1 + y \cdot 0) = (-y, x)$. The "special cases" are just the general formula in disguise — once you trust it, the others fall out.
Rotations about a centre that isn't the origin take one extra step: translate so the centre sits at the origin, apply the rotation, then translate back. Same trick works for any "do this thing, but somewhere else."
4. Isometry vs similarity
The four moves split into two camps based on what they preserve.
A transformation that preserves all distances. Equivalently, it preserves all lengths, all angles, and all areas. Translations, rotations, and reflections are the three isometries you've met; there's a fourth, the glide reflection, mentioned below.
A transformation that preserves shape but may scale size: an isometry followed by a dilation. It preserves angles and ratios of distances, but multiplies every length by a fixed factor $|k|$.
This vocabulary cleans up two ideas you've already met in earlier topics:
- Two figures are congruent iff one can be mapped onto the other by an isometry. SSS, SAS, ASA — those familiar congruence criteria are just convenient ways of guaranteeing an isometry exists.
- Two figures are similar iff one can be mapped onto the other by a similarity transformation. AA similarity, SAS similarity, and the rest are similarly disguised.
A remarkable classification theorem says that every isometry of the plane is one of four types:
- Translation — slide.
- Rotation — turn about a fixed centre.
- Reflection — flip across a fixed line.
- Glide reflection — reflect across a line, then translate along that same line.
There are no others. Whatever rigid motion you can dream up is one of these four in disguise. Translations and rotations preserve orientation (a clockwise loop stays clockwise); reflections and glide reflections reverse it.
It's tempting to think reflection + any translation is just "still a reflection" — but a reflection has a fixed mirror line, and if you slide afterward, no point stays put. Glide reflections are a genuinely distinct fourth species. Footprints in sand exhibit glide-reflection symmetry: left foot, right foot, left foot is reflect-then-step-forward, not just step.
5. Composition — order matters
Apply one transformation, then another. The result is itself a transformation, called the composition. Composition is associative — grouping three transformations into "first two, then the third" vs "first, then the last two" gives the same answer — but it is not commutative. The order you do them in changes where you end up.
Take a single concrete example. Start at $(0, 0)$. Define
- $T$: translation by $(1, 0)$, so $(x, y) \to (x + 1, y)$.
- $R$: rotation by $90°$ CCW about the origin, so $(x, y) \to (-y, x)$.
Apply $T$ first, then $R$:
$$ (0, 0) \xrightarrow{\;T\;} (1, 0) \xrightarrow{\;R\;} (0, 1) $$Now reverse the order — $R$ first, then $T$:
$$ (0, 0) \xrightarrow{\;R\;} (0, 0) \xrightarrow{\;T\;} (1, 0) $$Same two transformations, same starting point, different endpoint. The intuition: $R$ rotates around the origin, but after $T$ the point has moved, so $R$ now rotates it around a different "feels-like" centre.
Two patterns of composition come up so often they're worth memorizing:
- Two reflections across parallel lines compose to a translation — perpendicular to the lines, with length twice the distance between them.
- Two reflections across intersecting lines compose to a rotation about the intersection point, by twice the angle between the lines.
Every direct isometry (translation or rotation) is therefore secretly two reflections. This is why classifying the four isometries turns out to be tractable: reflections are the atoms, and everything else is built from pairs of them.
6. Symmetry as invariance
Here's the payoff of the transformation viewpoint. A figure has a symmetry if some non-identity transformation maps it to itself — same set of points, possibly with the labels shuffled.
A transformation $T$ such that $T(F) = F$ as a set. The figure looks identical before and after.
The familiar flavors of symmetry are all just naming the type of $T$:
- Line (reflective) symmetry — the figure is invariant under some reflection. A line of symmetry is the mirror.
- Rotational symmetry — invariant under a rotation by some angle less than $360°$. The order is how many distinct rotations (including the identity) leave the figure unchanged.
- Point symmetry — the special case of rotational symmetry with a $180°$ rotation.
- Translational symmetry — only possible for infinite patterns like a tiling or a frieze.
A regular hexagon has six lines of symmetry — three through opposite vertices, three through opposite edge midpoints — and rotational symmetry of order $6$, since rotations by $60°, 120°, 180°, 240°, 300°$, and $360°$ all map it to itself. A regular pentagon: five lines, order $5$. A non-square parallelogram: zero lines, order $2$ (the $180°$ rotation works).
The set of all symmetries of a figure, under composition, forms an algebraic structure called a group. The symmetries of a regular $n$-gon form the dihedral group $D_n$ with $2n$ elements ($n$ rotations and $n$ reflections). You'll meet groups properly when this thread reaches abstract algebra; the seed of the idea is planted here.
7. Common pitfalls
When people write $T \circ R$, the convention is "apply $R$ first, then $T$" — the function on the right hits the point first. It reads in the opposite order of how it sounds in English. If a problem says "rotate, then translate," the composition is $T \circ R$, not $R \circ T$.
"Rotate by $90°$" is ambiguous without a centre. Rotating by $90°$ about the origin sends $(1, 0)$ to $(0, 1)$; rotating by $90°$ about $(1, 0)$ leaves $(1, 0)$ fixed. Two rotations about different centres usually compose to a third rotation about a new centre — not the average of the two, and not even on the line between them.
Dilation by factor $k$ multiplies every distance by $|k|$, so it preserves shape but not size. Only $k = \pm 1$ is rigid — and $k = -1$ is the same as a $180°$ rotation, not a "negative scaling" in any geometric sense.
It is tempting to confuse "flip across the origin" (a $180°$ rotation, sends $(x, y) \to (-x, -y)$) with "reflect across the $x$-axis" (sends $(x, y) \to (x, -y)$). They behave alike on points of the $y$-axis but disagree everywhere else. The rotation preserves orientation; the reflection reverses it.
If a problem asks "list the plane isometries," there are four kinds, not three. Glide reflection is the easy one to drop — it doesn't appear in everyday talk — but a reflection composed with a translation along the mirror is genuinely its own thing, not just "still a reflection."
8. Worked examples
Compute each one yourself first. The point isn't the answer — it's whether your steps match the coordinate rule.
Example 1 · Translate $(3, 4)$ by $(2, -1)$
Apply the rule $(x, y) \to (x + a, y + b)$ with $(a, b) = (2, -1)$:
$$ (3, 4) \to (3 + 2,\; 4 - 1) = (5, 3) $$Example 2 · Rotate $(3, 4)$ by $90°$ CCW about the origin
The $90°$ CCW rule is $(x, y) \to (-y, x)$. Plug in:
$$ (3, 4) \to (-4,\; 3) $$Sanity check. Distance from origin: $\sqrt{3^2 + 4^2} = 5$ before, and $\sqrt{(-4)^2 + 3^2} = 5$ after. Rotations preserve distance to the centre, so the magnitudes match.
Example 3 · Reflect the triangle with vertices $(1, 1), (4, 1), (1, 5)$ across the $x$-axis
Apply $(x, y) \to (x, -y)$ to each vertex:
$$ (1, 1) \to (1, -1), \quad (4, 1) \to (4, -1), \quad (1, 5) \to (1, -5) $$The image is the triangle $(1, -1), (4, -1), (1, -5)$ — same shape, mirrored below the $x$-axis.
Example 4 · Dilate $(4, 6)$ by factor $\tfrac{1}{2}$ about the origin
Apply $(x, y) \to (k x, k y)$ with $k = \tfrac{1}{2}$:
$$ (4, 6) \to \left(\tfrac{1}{2} \cdot 4,\; \tfrac{1}{2} \cdot 6\right) = (2, 3) $$Distance from origin halves from $\sqrt{52}$ to $\sqrt{13}$, as expected for a dilation by $\tfrac{1}{2}$.
Example 5 · Composition order — translate then rotate vs rotate then translate
Let $T$ be translation by $(1, 0)$ and $R$ be rotation $90°$ CCW about the origin. Start at $(2, 0)$.
Translate first, then rotate:
$$ (2, 0) \xrightarrow{T} (3, 0) \xrightarrow{R} (-0, 3) = (0, 3) $$Rotate first, then translate:
$$ (2, 0) \xrightarrow{R} (0, 2) \xrightarrow{T} (1, 2) $$Same input, same two transformations, two different outputs. This is composition non-commutativity in action.
Example 6 · Two reflections compose to a rotation
Reflect $(3, 0)$ across the $x$-axis, then across the $y$-axis.
Step 1. Across the $x$-axis: $(x, y) \to (x, -y)$, so $(3, 0) \to (3, 0)$ (point is on the axis).
Step 2. Across the $y$-axis: $(x, y) \to (-x, y)$, so $(3, 0) \to (-3, 0)$.
Net effect: $(3, 0) \to (-3, 0)$, which is what a $180°$ rotation about the origin would do. This matches the general rule: two reflections across intersecting lines compose to a rotation about the intersection by twice the angle between them — and the $x$-axis and $y$-axis meet at $90°$, so the composition is a $180°$ rotation.