Geometric Transformations — Mathematics Knowledge System

Overview

A geometric transformation is a function that moves every point of the plane (or space) to a new position according to a fixed rule, producing an image of the original figure. The four basic plane transformations are translation (sliding without rotating or resizing), reflection (flipping across a line), rotation (turning about a fixed point), and dilation (scaling by a fixed factor from a center point). Translations, reflections, and rotations are rigid motions (isometries): they preserve distances and angles, so the image is always congruent to the original. Dilation preserves angles and shape but not distance, producing a similar (not congruent) image.

Intuition

Think of each transformation as a distinct physical action on a transparent sheet with the figure drawn on it: translation slides the sheet without turning it; rotation pins one point and spins the sheet; reflection flips the sheet over like a mirror; dilation is like using a photocopier's zoom to enlarge or shrink the drawing from a fixed anchor point. The rigid motions (slide, spin, flip) never change size — only position and orientation — which is exactly why they define congruence. Dilation is the odd one out: it changes size uniformly, which is exactly why it defines similarity instead.

Interactive Graph

Formal Definition

Definition

For a point (x, y) in the coordinate plane, translation by vector (h, k), reflection across the x-axis, rotation by θ about the origin, and dilation by scale factor k about the origin are given by:

T_{(h,k)}(x,y) = (x+h,\ y+k)

Translation by (h, k)

R_{x\text{-axis}}(x,y) = (x,\ -y)

Reflection across the x-axis

\text{Rot}_{\theta}(x,y) = (x\cos\theta - y\sin\theta,\ x\sin\theta + y\cos\theta)

Rotation by angle θ about the origin (counterclockwise)

D_{k}(x,y) = (kx,\ ky)

Dilation by scale factor k, centered at the origin

Notation

Notation	Meaning
$T_{(h,k)}$	Translation transformation shifting every point by (h, k)
$R$	Reflection transformation across a specified line
$\text{Rot}_\theta$	Rotation transformation by angle θ about a specified center
$D_k$	Dilation transformation by scale factor k about a specified center
$A'$	Image of point A after a transformation (read 'A prime')

Properties

Isometry (rigid motion)

\text{Translations, reflections, and rotations preserve all distances and angles: the image is always congruent to the preimage.}

Dilation

\text{Preserves angles and shape but scales all distances by } |k|\text{: the image is similar (not congruent) to the preimage unless } k=\pm 1.

90° rotation about origin

\text{Rot}_{90^\circ}(x,y) = (-y, x)

Condition: Special case of the general rotation formula with θ = 90°.

Composition of isometries

\text{Composing any two rigid motions produces another rigid motion; a reflection composed with itself returns every point to its original position.}

Applications

Computer graphics and animation represent every on-screen movement, scaling, and rotation of objects as compositions of these transformation matrices.

Worked Examples

Add the translation vector's components to the point's coordinates.
$(2+4,\ 3+(-1)) = (6, 2)$

Answer: (6, 2)

Practice Problems

Difficulty 2/10

Reflect the point (5, -2) across the x-axis.

Difficulty 3/10

Dilate the point (-3, 8) by scale factor 3, centered at the origin.

Difficulty 5/10

A logo icon at point (6, 8) in a design app is translated by (-2, 3) and then reflected across the x-axis. Find its final coordinates.

Common Mistakes

Common Mistake

Believing dilation is a rigid motion like translation, reflection, and rotation.

Dilation changes distances (scales them by |k|) unless k = ±1, so it is NOT an isometry — it produces a similar image, not a congruent one.

Common Mistake

Assuming the order of composed transformations never matters.

Transformation composition is generally not commutative — translating then reflecting can give a different result than reflecting then translating, so order must be tracked carefully.

Quiz

Summary

The four basic plane transformations are translation, reflection, rotation, and dilation.
Translation, reflection, and rotation are rigid motions (isometries): they preserve distance and angle, so image ≅ preimage.
Dilation preserves angles and shape but scales distances by k, producing a similar (not congruent) image.
Coordinate rules: T(x,y)=(x+h,y+k); reflection across x-axis (x,y)=(x,-y); 90° rotation (x,y)=(-y,x); dilation (x,y)=(kx,ky).
Composing transformations is generally not commutative — order affects the final image.

References

WebsiteWikipedia — Transformation (function)
WebsiteWikipedia — Rigid transformation