Transformation
A 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, following a rule that applies to every point of the figure at once; the four main transformations are translation (slide), reflection (flip), rotation (turn), and dilation (resize), with the original shape called the preimage and the result the image. Formally, a geometric transformation is a function $f: \mathbb{R}^2 \to \mathbb{R}^2$: isometries (translations, rotations, reflections) preserve distances and angles, dilations preserve angles but scale distances by a constant factor, and other transformations may distort shapes. In $n$ dimensions, linear transformations $T(x) = Ax$ (invertible matrix $A$) form the group $GL(n,\mathbb{R})$, affine transformations $T(x) = Ax + b$ form the affine group, and isometries form the Euclidean group $E(n)$; Klein's Erlangen Program defines geometry itself as the study of properties invariant under a given transformation group.
Example
Sliding a triangle $5$ spaces right is a translation; flipping it over a line is a reflection; spinning it $90$ degrees is a rotation; stretching it to twice its size is a dilation. The transformation $f(x, y) = (x + 3, y - 2)$ slides every point $3$ right and $2$ down, while $f(x, y) = (-x, y)$ reflects every point across the y-axis. Klein's Erlangen Program classifies geometries by their invariance groups: Euclidean geometry is invariant under $E(n)$, affine geometry under affine transformations, and projective geometry under projective transformations, each with its own invariants (distance, parallelism, cross-ratio).
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?" Composing transformations corresponds to composing functions, and the set of all rigid motions forms a mathematical group under composition. Felix Klein's 1872 Erlangen Program unified all geometries as the study of invariants under transformation groups, replacing a patchwork of separate geometric theories with a single organizing principle that now pervades modern geometry and physics.