Transformation (geometry)

A transformation is an operation that moves, resizes, or reshapes a geometric figure in space, producing a new figure from an original one. Transformations are fundamental to understanding how shapes relate to each other and form the backbone of Computer graphics, mathematical reasoning, and design.

The main types include translations (sliding without rotation or resize), rotations (spinning around a point), reflections (flipping across a mirror line), and dilations (scaling up or down). Each preserves or alters different properties: curves stay curves, but angles or distances might change depending on the transformation type.

Rigid transformations—translations, rotations, and reflections—preserve distances and angles, so the transformed figure is congruent to the original. Non-rigid transformations like dilations or shearing change the figure's size or shape while maintaining structural relationships.

Transformations reveal deep symmetries in nature and design. They're essential in Feature Extraction for recognizing patterns, in architecture for understanding spatial relationships, and in pure mathematics for proving theorems about Number Theory and abstract structures.

Related

Congruence (geometry), Similarity (geometry), Symmetry, Matrix (mathematics), Vector (mathematics), Tessellation