Transformation
Geometry & MeasurementA transformation is a rule that moves, flips, turns, or resizes every point of a figure to a new position.
Definition
A transformation is a way of moving or changing a shape. The four main transformations are translation (slide), reflection (flip), rotation (turn), and dilation (resize). The original shape is called the preimage and the result is called the image.
Example
Slide a triangle $5$ spaces to the right: that is a translation. Flip it over a line: that is a reflection. Spin it $90$ degrees: that is a rotation. Stretch it to twice its size: that is a dilation.
Key Insight
Transformations are like instructions for a shape. They answer "if I apply this rule to every single point of the figure, where does each point end up?" The entire shape moves together following the same rule.
Definition
A geometric transformation is a function $f: \mathbb{R}^2 \to \mathbb{R}^2$ mapping each point of the plane to another point. Isometries (translations, rotations, reflections) preserve distances and angles; dilations preserve angles but change distances by a constant factor; other transformations may distort shapes.
Example
The transformation $f(x, y) = (x + 3, y - 2)$ slides every point $3$ right and $2$ down (a translation). The transformation $f(x, y) = (-x, y)$ reflects every point across the y-axis. Each is a rule applied to every point simultaneously.
Key Insight
Transformations are functions whose domain and range are both geometric figures. Composing transformations corresponds to composing functions, and the set of all rigid motions forms a mathematical group under composition.
Definition
A transformation of $\mathbb{R}^n$ is a bijection $T: \mathbb{R}^n \to \mathbb{R}^n$. Linear transformations $T(x) = Ax$ (where $A$ is an invertible matrix) form the group $GL(n,\mathbb{R})$. Affine transformations $T(x) = Ax + b$ form the affine group. Isometries form the Euclidean group $E(n)$. The study of geometry can be defined (Klein's Erlangen Program) as the study of properties invariant under a given transformation group.
Example
Klein's Erlangen Program classifies geometries by their invariance groups: Euclidean geometry is invariant under $E(n)$; affine geometry under affine transformations; projective geometry under projective transformations. Each geometry has its own invariants (distance, parallelism, cross-ratio).
Key Insight
Felix Klein's 1872 Erlangen Program unified all geometries as the study of invariants under transformation groups, replacing the patchwork of separate geometric theories with a single organizing principle. This perspective now pervades all of modern geometry and physics.