Vierecksnetze

Vierecksnetze, or quadrilateral nets, describe tilings or meshes of a plane or surface whose faces are quadrilaterals. They serve as a discrete representation of a grid and are used in mathematics, computer graphics, and geographic information systems. A Vierecksnetz consists of vertices connected by edges to form four-edged faces; in a regular rectangular grid each interior vertex has four incident edges and each cell is a rectangle; in more general grids the cells are arbitrary quadrilaterals.

On a two-dimensional manifold, a Vierecksnetz can be seen as the image of a lattice Z^2 under a map to the surface, where the two coordinate directions correspond to two families of grid lines. In Euclidean plane grids these directions are orthogonal; on curved surfaces the grid may be deformed while preserving the quadrilateral topology. A common mathematical variant is a Q-net, in discrete differential geometry, where each elementary quadrilateral is planar. Such nets can be used to approximate conjugate or isothermal coordinate nets on surfaces.

Types of Vierecksnetze include regular square grids, rectangular grids with non-equal side lengths, and more general quadrilateral meshes with varying edge lengths and angles. Quality criteria in applications often target aspect ratio of cells, smoothness of the mesh, and alignment with features of the underlying domain.

Applications and use: computational meshes for finite element methods, texture mapping and remeshing in computer graphics, planar or surface parameterizations in GIS and cartography, architectural design, and the study of tilings in mathematics.

See also: square lattice, quadrilateral mesh, tiling, discrete differential geometry.