Geometric Transformations: Translation, Rotation, Reflection and Dilation

A geometric transformation is a procedure that changes a figure’s shape, size or position, producing a new figure called the image. Three of these operations belong to the same family and leave both the lengths and the angles of the starting figure unchanged. The technical name is rigid transformation, or isometry: the original figure and its image stay congruent, with the same shape and the same size, and only the position on the plane changes.

Translation

In a translation, every point of the figure moves the same distance, in the same direction. There is no center to turn around and no line to flip across.

Practical example: sliding a book across a table, without lifting it or turning it, is a translation. The book stays identical in shape and size: only its position on the table changes. Pouring a drink from one glass into another is a different operation: the liquid takes the shape of the new container, so that transformation is not rigid.

Rotation

Rotation turns the figure around a point that stays fixed, the center of rotation. Everything else about the figure moves, through an amount called the angle of rotation: the angle formed by the lines joining a point of the original figure to the center, and the center to the corresponding point of the rotated figure. Rotation, like translation, also leaves shape and size untouched: anyone reviewing the difference with an enlarged figure can compare it with the triangle similarity theorems, where shape is preserved but size is not.

Reflection

Reflection flips the figure across a line called the axis of reflection, producing the mirror image of the original. The simplest case to follow in coordinate geometry is reflection across the x-axis: the y-coordinate of every point changes sign, while the x-coordinate stays the same. Like translation and rotation, reflection leaves lengths and angles untouched, which is why the three operations share the name rigid transformation. Anyone still unsure how to read a point’s coordinates can review it on the coordinate plane, where these shifts can be followed step by step, one row at a time.

Dilation, the exception

Dilation enlarges or shrinks the figure according to a scale factor, written as k. With k between 0 and 1 the resulting figure is smaller than the original; with k greater than 1 it is larger. Unlike the rigid transformations, dilation touches the figure’s size: shape carries over, measurements don’t, so the image is similar rather than congruent to the original. This is also where the geometry of transformations meets analytic geometry, where the same operations are written as coordinates changing according to a rule. The center of dilation and what a negative k does to the image are covered in the Recap on geometric dilations.

Combining several transformations

A composition happens when several transformations are applied to the same figure, one after another. A glide reflection combines a reflection and a translation: the figure flips across an axis, then slides in a direction parallel to that axis. It belongs to the elementary isometries along with translation, rotation, reflection and the identity transformation, which leaves the figure exactly where it was.

Practical example: two reflections performed one after the other, across two lines that intersect, produce the same result as a single rotation: the center coincides with the point where the two lines meet. Two elementary operations, applied in sequence, can match a third elementary operation taken on its own.

Recognizing a transformation at a glance also helps when reading proofs in Euclidean geometry, where congruence between figures is established by comparing corresponding sides and angles — the same comparison that separates an isometry from a dilation.