Isometries
An isometry is a transformation of the plane that preserves distances between every pair of points. The word comes from the Greek 'isos' (equal) and 'metron' (measure). When you apply an isometry to a geometric figure, the image you obtain is congruent to the original — same shape, same size — even if its position or orientation in the plane has changed. The three fundamental isometries of the Euclidean plane are reflections, rotations, and translations, and every isometry of the plane can be expressed as a composition of at most three reflections. Isometries are the backbone of Euclidean geometry and deductive reasoning because congruence itself is defined through them: two figures are congruent if and only if one can be mapped onto the other by an isometry. This gives a precise, algebraic meaning to the intuitive idea that two shapes 'look the same.' Isometries also form a mathematical group under composition — applying two isometries one after the other always produces another isometry, and every isometry has an inverse that 'undoes' it. Understanding isometries connects geometry to algebra through coordinate transformations, to physics through symmetry, and to art through patterns and tilings. The classification of wallpaper patterns — the 17 distinct ways to tile the plane with a repeating motif — is entirely built on the theory of isometries. In this article you will study each of the three basic isometries carefully, learn their coordinate rules, and see how they combine to produce all rigid motions of the plane.
- Translation
- Reflection
- Rotation
- Glide Reflection (Composite Isometry)
- Isometries Form a Group
Translation
A translation slides every point of the plane by the same vector. If the translation vector is (a, b), then every point P = (x, y) maps to P' = (x + a, y + b). Distances, angle measures, and orientation are all preserved. For example, translating the point (3, 1) by vector (−2, 4) gives (1, 5). A translation has no fixed points (unless the vector is the zero vector), and applying the same translation twice moves every point by twice the distance.
Reflection
A reflection flips every point across a fixed line called the line of reflection (or mirror line). The image of a point P is the point P' such that the mirror line is the perpendicular bisector of segment PP'. Reflection preserves distances and angle measures but reverses orientation — a clockwise triangle becomes counterclockwise. The coordinate rule for reflection across the x-axis is (x, y) → (x, −y); across the y-axis it is (x, y) → (−x, y); across the line y = x it is (x, y) → (y, x). Every point on the mirror line is a fixed point of the reflection.
Rotation
A rotation turns every point of the plane through an angle θ about a fixed point called the center of rotation. For a rotation of angle θ about the origin, the coordinate rule is: x' = x cos θ − y sin θ, y' = x sin θ + y cos θ. The only fixed point is the center. Rotation preserves distances, angle measures, and orientation. For example, rotating (4, 0) by 90° counterclockwise about the origin gives (0, 4), since cos 90° = 0 and sin 90° = 1. A rotation by 180° about the origin maps (x, y) to (−x, −y).
Glide Reflection (Composite Isometry)
A glide reflection is the composition of a reflection across a line followed by a translation parallel to that same line. It is the fourth type of isometry in the plane (besides the three above) and cannot be reduced to any single reflection, rotation, or translation. It reverses orientation like a reflection. Footprints walking in a straight line are a classic real-world example: each footprint is a glide reflection of the previous one.
Isometries Form a Group
The set of all isometries of the plane forms a group under composition. This means: (1) composing two isometries gives an isometry (closure); (2) composition is associative; (3) the identity transformation (leaving every point fixed) is an isometry; (4) every isometry has an inverse isometry. The subgroup of orientation-preserving isometries (translations and rotations) is called the group of direct isometries, while reflections and glide reflections are indirect (orientation-reversing) isometries.
The Key Equation
- x', y' — Image coordinates (none (length units)): The coordinates of the rotated point (image).
- x, y — Pre-image coordinates (none (length units)): The coordinates of the original point before rotation.
- θ — Angle of rotation (degrees or radians): The directed angle through which the plane is rotated; counterclockwise is positive by convention.
- \cos\theta, \sin\theta — Trigonometric components (dimensionless): The cosine and sine of the rotation angle, which scale and mix the x and y components to produce the rotation.
Worked Example
Problem. Triangle ABC has vertices A(1, 2), B(4, 2), and C(4, 5). (a) Find the image of triangle ABC after a reflection across the y-axis. (b) Then rotate that image 90° counterclockwise about the origin. State the final vertices and verify that all side lengths are preserved.
Answer. Final vertices: A''(−2, −1), B''(−2, −4), C''(−5, −4). All side lengths preserved: 3, 3, and 3√2.
Origins · 1 minFelix Klein and how this was discovered
Felix Klein revolutionized geometry with his 1872 Erlangen Program, in which he proposed that every geometry should be defined by a group of transformations and the properties that are invariant under that group. In this framework, Euclidean geometry is precisely the study of properties preserved by isometries (the group of rigid motions). This unifying vision connected geometry to group theory and clarified the relationships among Euclidean, projective, hyperbolic, and other geometries. Klein's program fundamentally changed how mathematicians think about space and symmetry. Klein is also famous for the Klein bottle — a one-sided surface with no boundary that cannot be embedded in three-dimensional space without self-intersection. He described it in 1882, and the name 'bottle' is thought to arise from a mistranslation of the German 'Fläche' (surface) as 'Flasche' (bottle).
Related Concepts
- Congruence of Triangles — Congruence is defined precisely through isometries: two triangles are congruent if and only if one can be mapped to the other by a reflection, rotation, translation, or composition of these.
- Symmetry in Geometric Figures — A figure's symmetry is described by the set of isometries that map the figure onto itself, such as lines of reflective symmetry and centers of rotational symmetry.
- Coordinate Transformations — Isometries are expressed as coordinate transformation rules, connecting geometric intuition to algebraic computation using matrices and vectors.
- Vectors and Translation — Translations are naturally described by vectors, making vector addition the algebraic counterpart of composing translations.