Triangulated

Triangulated is an adjective used across several disciplines to indicate that something is composed of triangles or derived from the process of triangulation. In geometric or computational contexts, a triangulated surface, polygon, or mesh is partitioned into non-overlapping triangles joined along their edges. Such meshes are common in computer graphics, finite element analysis, and geographic information systems, where triangles provide a simple, flexible primitive for representing complex shapes. Triangulation can be constrained to follow boundaries or optimized for element quality. A well-known example is the Delaunay triangulation, which tends to avoid skinny triangles.

In topology and geometry, a triangulated space or manifold is a space that admits a triangulation, i.e., it is homeomorphic to a simplicial complex. Triangulations serve as combinatorial models that enable algebraic and geometric techniques, such as calculating homology or performing piecewise-linear constructions. The existence and properties of triangulations are foundational in piecewise-linear (PL) topology; in practice, spaces are often studied via their triangulations to enable discrete methods.

In category theory, a triangulated category is an additive category equipped with an autoequivalence (often called the shift functor) and a class of distinguished triangles satisfying specific axioms. This abstraction underpins derived categories and stable homotopy theory, linking homological algebra to geometry and representation theory.

The term derives from Latin triangulus, meaning a three-cornered figure; its use spans surveying, mathematics, and computer science.