Reflection, Translation and Rotation

Reflection, translation and rotation are the three basic rigid transformations: they move a figure without changing its shape or size. In brief: a reflection flips a figure across a line, a translation slides it by a vector and a rotation turns it about a point by an angle. For example, reflecting the point (3, 2) in the y-axis gives (−3, 2), translating it by (4, −1) gives (7, 1) and rotating it 90° counterclockwise about the origin gives (−2, 3).

Transformations preserve lengths and angles, so the image is congruent to the original. They are central to the idea of congruence and to computer graphics. This guide gives the rules for the coordinate plane, worked examples for each transformation and a table of the standard cases.

What is a translation?

A translation moves every point of a figure the same distance in the same direction. It is described by a vector (a, b): add a to each x-coordinate and b to each y-coordinate, so (x, y) → (x + a, y + b). The figure is not turned or flipped, and each side is parallel to its image.

Example 1

Triangle with vertices A(1, 1), B(4, 1) and C(2, 3) is translated by (−3, 2). The images are A′(−2, 3), B′(1, 3) and C′(−1, 5). The side lengths are unchanged: AB = A′B′ = 3.

What is a reflection?

A reflection in a line m maps each point P to P′ so that m is the perpendicular bisector of PP′. Points on the line stay fixed. A reflection reverses the orientation: a clockwise order of vertices becomes counterclockwise.

Rules in the coordinate plane: reflection in the x-axis is (x, y) → (x, −y), reflection in the y-axis is (x, y) → (−x, y), reflection in the line y = x is (x, y) → (y, x), and reflection in y = −x is (x, y) → (−y, −x). For a vertical line x = k, the rule is (x, y) → (2k − x, y).

Example 2

Reflect P(5, −2) in the line x = 1. Then P′ = (2 · 1 − 5, −2) = (−3, −2). Check: the midpoint of PP′ is (1, −2), which lies on the line ✓.

What is a rotation?

A rotation turns every point about a fixed center through a given angle. By convention, positive angles are counterclockwise. For rotations about the origin, the rules are: 90° counterclockwise (x, y) → (−y, x), 180° (x, y) → (−x, −y) and 270° counterclockwise (or 90° clockwise) (x, y) → (y, −x).

Example 3

Rotate A(4, 1) by 90° counterclockwise about the origin: A′ = (−1, 4). Rotating by 180° gives (−4, −1). The distance from the origin is the same in all cases: √17.

Example 4: rotation about another center

Rotate P(5, 3) by 90° counterclockwise about C(2, 1). Subtract the center: (3, 2). Rotate: (−2, 3). Add the center back: (0, 4). So P′ = (0, 4). Check the distance from C: before √(9 + 4) = √13, after √(4 + 9) = √13 ✓.

Coordinate rules for common transformations

What happens when you combine transformations?

Doing one transformation after another gives a composition. Two reflections in parallel lines produce a translation of twice the distance between the lines. Two reflections in intersecting lines produce a rotation about their intersection by twice the angle between them. A reflection followed by a translation along the axis is a glide reflection, as seen in footprints. The order usually matters: a rotation followed by a translation generally gives a different result from the same translation followed by the rotation.

Example 5

Reflect (3, 2) in the line x = 1 to get (−1, 2), then in the line x = 4 to get (2 · 4 − (−1), 2) = (9, 2). The combined effect is a translation by (6, 0), which is twice the distance 3 between the lines ✓.

Where do these ideas come from?

Euclid used the idea of moving a figure onto another to prove congruence, although he did so with some reluctance. The systematic treatment of transformations as functions of the plane developed in the nineteenth century. Felix Klein’s Erlangen program of 1872 proposed to define geometries by the transformations that leave their properties unchanged, which made transformations a central tool.

Where are they used today?

Computer graphics and video games move objects on the screen by translations and rotations. Robotics describes the position of an arm by a sequence of rotations. Tessellations and ornaments are created by reflecting and translating a basic pattern, and the idea of congruence is based on rigid motions.

Which mistakes are common?

Mixing up the direction of rotation

Counterclockwise is positive. A 90° clockwise rotation is the same as 270° counterclockwise.

Using the origin as the center by default

If the center is another point, translate the center to the origin first, rotate and translate back.

Reflecting only one vertex

Transform every vertex and then connect the images in the same order.

Confusing reflection in y = x with reflection in the y-axis

(x, y) → (y, x) swaps the coordinates, while (x, y) → (−x, y) changes only the sign of x.