Geometry

Geometric transformations

Maps of the plane such as translations, rotations, reflections and dilations that preserve or scale shape.

IntuitionSliding, Spinning and Scaling Shapes

Every day you see shapes move without changing size (a book slides across a table, a wheel spins on its axle) and shapes that grow or shrink while keeping the same outline (a photograph enlarged for printing, a shadow that lengthens at sunset). A geometric transformation is a rule that turns each point $M$ of the plane into a new point $M'$ . The four basic transformations you meet in school, translation, rotation, reflection and homothety (dilation), are the building blocks for describing symmetry, congruence and similarity precisely.

A plane transformation matrix: when

\det A = 1

with orthogonal columns (such as

(0.8, 0.6)

and

(-0.6, 0.8)

) it is a pure rotation preserving lengths and areas; scaling the columns produces a dilation.

SchoolThe Four Basic Transformations

Definition: Translation

Given a fixed vector $\vec{v} = (a, b)$ , the translation $T_{\vec{v}}$ sends each point $M(x, y)$ to the point $M'(x', y')$ satisfying $\vec{MM'} = \vec{v}$ . In coordinates this is simply adding the vector's components to every point.

x' = x + a, \qquad y' = y + b

Here $(x, y)$ are the coordinates of $M$ and $(x', y')$ the coordinates of its image $M'$ ; the pair $(a, b)$ are the horizontal and vertical components of the translation vector $\vec{v}$ . A translation preserves distances, angles and orientation: it is an isometry that never rotates or flips the plane.

Definition: Rotation

Fix a center $I(a, b)$ and an angle $\alpha$ . The rotation $Q_{(I, \alpha)}$ sends $M$ to the point $M'$ such that $IM' = IM$ and the angle from $\overrightarrow{IM}$ to $\overrightarrow{IM'}$ equals $\alpha$ (measured counter-clockwise).

\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{pmatrix}\begin{pmatrix} x - a \\ y - b \end{pmatrix} + \begin{pmatrix} a \\ b \end{pmatrix}

The matrix rotates the vector from the center $I(a, b)$ to $M(x, y)$ by angle $\alpha$ , and the result is shifted back so the rotation truly fixes $I$ . When $\alpha$ is $180°$ , the rotation coincides with the point reflection through $I$ .

Definition: Reflection (axial symmetry)

Given a line $d$ , the reflection $R_d$ sends each point $M$ to the point $M'$ such that $d$ is the perpendicular bisector of segment $MM'$ (points on $d$ are fixed). Across the horizontal axis this is simply $R_{Ox}(x, y) = (x, -y)$ .

R_{Ox}(x, y) = (x, -y)

Definition: Homothety (dilation)

Given a center $I(a, b)$ and a ratio $k \neq 0$ , the homothety $V_{(I, k)}$ sends $M$ to the point $M'$ with $\overrightarrow{IM'} = k\,\overrightarrow{IM}$ . Unlike the previous three maps, a homothety with $|k| \neq 1$ changes lengths, so it is not an isometry.

x' = a + k(x - a), \qquad y' = b + k(y - b)

A map obtained by composing an isometry (translation, rotation or reflection) with a homothety is called a similarity transformation: it preserves angles and multiplies every length by the same ratio $|k|$ , so it sends any figure to a similar one.

Comparing the four basic plane transformations
Transformation	Formula	Scales distances by	Scales areas by
Translation	$x' = x + a, \qquad y' = y + b$	1 (preserved)	1 (preserved)
Rotation	center $I$ , angle $\alpha$	1 (preserved)	1 (preserved)
Reflection	$R_{Ox}(x, y) = (x, -y)$	1 (preserved)	1 (preserved)
Homothety $V_{(I, k)}$	$x' = a + k(x - a), \qquad y' = b + k(y - b)$	$\|k\|$	$k^2$

UndergraduateComposition of Transformations and the Isometry Group

Theorem: Composition of Two Reflections Across Intersecting Lines

Let $d_1$ and $d_2$ be two lines intersecting at a point $I$ such that the directed angle from $d_1$ to $d_2$ is $\alpha$ . Then the composition of the reflection across $d_1$ followed by the reflection across $d_2$ is the rotation about $I$ by angle $2\alpha$ : $R_{d_2} \circ R_{d_1} = Q_{(I, 2\alpha)}$ .

Why is it true?

A single reflection flips orientation (left hand becomes right hand), so doing two reflections in a row restores the original orientation and must be a rigid motion without flipping. Since the intersection point $I$ lies on both mirrors, neither reflection moves $I$ , so the combined motion must be a rotation around $I$ , and each mirror contributes twice the angle between the point and that mirror.

Proof

First, since $I$ lies on both $d_1$ and $d_2$ , both reflections fix $I$ . Take any point $M$ distinct from $I$ , and let $M_1 = R_{d_1}(M)$ and $M' = R_{d_2}(M_1)$ .

Because $d_1$ is the perpendicular bisector of $MM_1$ and $d_2$ is the perpendicular bisector of $M_1M'$ , we have $IM = IM_1 = IM'$ , so $M$ and $M' = R_{d_2}(M_1)$ lie on the same circle centered at $I$ .

Let $\varphi_1$ be the directed angle from ray $\overrightarrow{IM}$ to $\overrightarrow{Id_1}$ , and $\varphi_2$ be the directed angle from $\overrightarrow{Id_1}$ to $\overrightarrow{Id_2}$ . Reflection across $d_1$ sends $\overrightarrow{IM}$ to $\overrightarrow{IM_1}$ by rotating through $2\varphi_1$ , and reflection across $d_2$ sends $\overrightarrow{IM_1}$ to $\overrightarrow{IM'}$ by rotating through $2\varphi_2$ . Adding the directed angles gives $(\overrightarrow{IM}, \overrightarrow{IM'}) = 2\varphi_1 + 2\varphi_2 = 2(\varphi_1 + \varphi_2) = 2\alpha$ , which is independent of $M$ . Hence every point is rotated around $I$ by $2\alpha$ , proving the composition equals $Q_{(I, 2\alpha)}$ .

Theorem: Scaling of Distances and Areas Under Homothety

Under a homothety $V_{(I, k)}$ with center $I$ and ratio $k \neq 0$ , for any two points $A, B$ with images $A' = V_{(I,k)}(A)$ and $B' = V_{(I,k)}(B)$ , we have $\overrightarrow{B'A'} = k\,\overrightarrow{BA}$ and hence $A'B' = |k|\cdot AB$ . Consequently, the area of any triangle $ABC$ scales by the square of the ratio: $[A'B'C'] = k^2 \cdot [ABC]$ .

Why is it true?

Because every point is pushed away from (or pulled toward) the center $I$ by the same factor $k$ , the triangle formed by $I$ and any two points $A, B$ is scaled uniformly in both radial sides, so by Thales' theorem the third side $A'B'$ stays parallel to $AB$ and scales by $|k|$ . Area is two-dimensional (base times height), and since both base and height are multiplied by $|k|$ , their product is multiplied by $|k| \cdot |k| = k^2$ .

Proof

By the definition of homothety, $\overrightarrow{IA'} = k\,\overrightarrow{IA}$ and $\overrightarrow{IB'} = k\,\overrightarrow{IB}$ . Subtracting the second vector equation from the first yields $\overrightarrow{IA'} - \overrightarrow{IB'} = k(\overrightarrow{IA} - \overrightarrow{IB})$ .

Using the head-to-tail rule $\overrightarrow{IA} - \overrightarrow{IB} = \overrightarrow{BA}$ on both sides, we obtain $\overrightarrow{B'A'} = k\,\overrightarrow{BA}$ . Taking lengths of both vectors immediately gives $A'B' = |\overrightarrow{B'A'}| = |k| \cdot |\overrightarrow{BA}| = |k| \cdot AB$ .

Now consider any triangle $ABC$ . Its altitude $h$ from $C$ to line $AB$ is the distance between $C$ and its orthogonal projection $H$ on $AB$ . Since homothety preserves parallelism and angles, it preserves perpendicularity, so $H' = V_{(I,k)}(H)$ is the foot of the altitude of $A'B'C'$ . Thus both the base $A'B' = |k|\cdot AB$ and the altitude $C'H' = |k|\cdot CH$ scale by $|k|$ , giving $[A'B'C'] = \tfrac{1}{2} A'B' \cdot C'H' = |k|^2 \cdot \tfrac{1}{2} AB \cdot CH = k^2 \cdot [ABC]$ .

UndergraduatePractical Applications and Worked Examples

Geometric transformations are the everyday language of computer graphics (every frame of a 2D or 3D game applies translation, rotation and scaling matrices to thousands of vertices on the GPU), robotics (a robot arm's hand position is the composition of rotations at each joint), crystallography (atoms in a crystal lattice repeat under a discrete group of translations, rotations and reflections) and cartography (map projections and pantographs use homothety to scale terrain accurately).

Example: Rotating a Robot Sensor by 90 Degrees

A planar robot wrist pivots at the origin $O(0, 0)$ . A laser sensor mounted on the hand is currently at $P(3, 4)$ . Find the new coordinates $P'(x', y')$ of the sensor after the wrist rotates counter-clockwise by $\alpha = 90°$ .

Solution

Use the rotation formula around the origin $(a, b) = (0, 0)$ with $\alpha = 90°$ : $x' = x\cos 90° - y\sin 90°$ and $y' = x\sin 90° + y\cos 90°$ .

Since $\cos 90° = 0$ and $\sin 90° = 1$ , the formula simplifies to the standard quarter-turn rule $(x', y') = (-y, x)$ .

Substituting $(x, y) = (3, 4)$ gives $P' = (-4, 3)$ . Notice that the distance to the pivot is preserved: $OP' = \sqrt{(-4)^2 + 3^2} = 5 = OP$ .

Example: Scaling an Architectural Blueprint by Homothety

On a coordinate grid with origin at the corner $I(1, 2)$ of a courtyard, a pillar is located at $A(4, 6)$ and a triangular flowerbed has area $S = 5$ . The architect enlarges the drawing by the homothety $V_{(I, 3)}$ centered at $I(1, 2)$ with ratio $k = 3$ . Find the new coordinates $A'$ of the pillar and the new area $S'$ of the flowerbed.

Solution

Apply the homothety coordinate formula $x' = a + k(x - a)$ , $y' = b + k(y - b)$ with center $(a, b) = (1, 2)$ , point $(x, y) = (4, 6)$ and ratio $k = 3$ .

We compute $x' = 1 + 3(4 - 1) = 10$ and $y' = 2 + 3(6 - 2) = 14$ , so the pillar moves to $A'(10, 14)$ .

By the area scaling theorem, areas multiply by $k^2 = 3^2 = 9$ , so the enlarged flowerbed has area $S' = 9 \cdot 5 = 45$ .

What is the image of the point $M(3, -2)$ under the translation $T_{\vec{v}}$ by vector $\vec{v} = (-1, 5)$ ?

What is the image of $M(0, 2)$ under the counter-clockwise rotation $Q_{(O, 90°)}$ about the origin?

Two mirror lines $d_1$ and $d_2$ intersect at $I$ with a directed angle of $35°$ from $d_1$ to $d_2$ . Reflecting across $d_1$ and then across $d_2$ is equivalent to which single transformation?

A park has area $6$ on a city map. If the map is scaled by a homothety $V_{(O, -4)}$ with ratio $k = -4$ , what is the area of the park's image?

References

H. S. M. Coxeter (1969). Introduction to Geometry
Wikipedia contributors (2026). Geometric transformation — Wikipedia