Polygonnetzes

Polygonnetzes is a term used in geometry to describe planar networks formed by polygons that share edges in order to partition a region into polygonal faces. In practice, polygonnetzes are commonly known as polygonal meshes or tessellations, and they are central to fields such as computational geometry, computer graphics, and finite element analysis. A polygonnetz is defined by its vertices, edges, and faces. Each edge connects two vertices and is incident to one or two faces; each face is a polygon, and the union of all faces constitutes the region.

The dual graph of a polygonnetz—constructed by placing a vertex in each face and joining vertices for adjacent faces—offers a useful perspective on the structure of the network. Polygonnetzes can be regular, semi-regular, or irregular. Regular tilings use congruent polygons in uniform patterns (such as triangles, squares, and hexagons). Semi-regular tilings employ more than one polygon type arranged in uniform patterns. Irregular or adaptive meshes arise from subdivision or refinement processes that respond to boundaries, features, or solution requirements.

Construction methods include tessellation of a region by a prescribed polygon type, triangulation and quadrangulation via Delaunay or constrained algorithms, and adaptive refinement that preserves mesh quality while conforming to geometry or physics. Quality metrics such as minimum angle, aspect ratio, and size gradation guide the generation and refinement of polygonnetzes to avoid poorly shaped elements.

Applications of polygonnetzes span finite element analysis, computer graphics, geographic information systems, and architectural design. Related concepts include polygonal mesh, tiling, planar graphs, and Voronoi diagrams.