meshtopology

Meshtopology is the study of the connectivity structure of a mesh—the way its elements are linked to one another—independent of the geometric positions of its vertices. It describes how vertices, edges, faces, and cells form a discrete representation of a domain, and encapsulates properties such as boundary, interior, and genus. A valid mesh topology supports consistent data association for simulations and rendering and can influence solver behavior and numerical stability.

In two dimensions, a mesh consists of vertices connected by edges to form polygonal faces, most commonly triangles or quadrilaterals. In three dimensions, cells (such as tetrahedra or hexahedra) are bounded by faces, which meet at edges and vertices. A mesh can be manifold, meaning certain incidences are limited in a well-defined way, or non-manifold, with more complex incidences. Topology also includes the specification of boundary components and the overall genus.

Data structures for representing meshtopology separate geometric embedding from connectivity. Common representations include the half-edge (or edge-graph) structure, the winged-edge structure, and the DCEL. These support rapid queries about adjacency, incidence, and boundary, and they enable topology-aware editing operations.

Operations of interest include refinement and coarsening, vertex and edge splits or merges, and hole creation or filling. Topology-preserving edits modify geometry without changing connectivity; topology-changing edits alter the mesh’s combinatorial structure. Applications span finite element analysis, computational fluid dynamics, computer graphics, and computer-aided design, where the embodied topology determines feasible discretizations and affects solver performance and visual correctness.