View source for Regular grid

{{Short description|Tessellation of Euclidean space}}
{{More citations needed|date=December 2009}}
[[File:Regular grid.svg|thumb|right|Example of a regular grid]]
A '''regular grid''' is a [[tessellation]] of '''''n'''''-dimensional [[Euclidean space]] by [[Congruence_(geometry)|congruent]] [[parallelepiped#Parallelotope|parallelotope]]s (e.g. [[brick]]s).<ref>{{cite web|last=Uznanski, Dan.|title=Grid.|url=http://mathworld.wolfram.com/Grid.html|publisher=From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.|access-date=25 March 2012}}</ref> Its opposite is [[Unstructured grid|irregular grid]].

Grids of this type appear on [[graph paper]] and may be used in [[finite element analysis]], [[finite volume method]]s, [[finite difference method]]s, and in general for discretization of parameter spaces. Since the derivatives of field variables can be conveniently expressed as finite differences,<ref name="Preface, P-1.1">{{cite book|last=J.F. Thompson, B. K . Soni & N.P. Weatherill|title=Handbook of Grid Generation|year=1998|publisher=CRC-Press|isbn=978-0-8493-2687-5|url=http://www.crcnetbase.com/isbn/9781420050349}}</ref> structured grids mainly appear in finite difference methods. [[Unstructured grid]]s offer more flexibility than structured grids and hence are very useful in finite element and finite volume methods.

Each cell in the grid can be addressed by index (i, j) in two [[dimension]]s or (i, j, k) in three dimensions, and each [[vertex (geometry)|vertex]] has [[coordinate]]s <math>(i\cdot dx, j\cdot dy)</math> in 2D or <math>(i\cdot dx, j\cdot dy, k\cdot dz)</math> in 3D for some real numbers ''dx'', ''dy'', and ''dz'' representing the grid spacing.

==Related grids==
A '''Cartesian grid''' is a special case where the elements are [[unit square]]s or [[unit cube]]s, and the vertices are [[point (geometry)|point]]s on the [[integer lattice]].

{{Anchor|Rectilinear grid}}A '''rectilinear grid''' is a tessellation by [[rectangle]]s or [[rectangular cuboid]]s (also known as [[rectangular parallelepiped]]s) that are not, in general, all [[congruence (geometry)|congruent]] to each other. The cells may still be indexed by integers as above, but the mapping from indexes to vertex coordinates is less uniform than in a regular grid. An example of a rectilinear grid that is not regular appears on [[logarithmic scale]] [[graph paper]].

A '''skewed grid''' is a tessellation of [[parallelogram]]s or [[parallelepiped]]s. (If the unit lengths are all equal, it is a tessellation of [[rhombus|rhombi]] or [[rhombohedron|rhombohedra]].)

A '''curvilinear grid''' or '''structured grid''' is a grid with the same combinatorial structure as a regular grid, in which the cells are [[quadrilateral]]s or [[cuboid|[general] cuboid]]s, rather than rectangles or rectangular cuboids.

{{Gallery
|title=Examples of various grids
|width=160 | height=170 | align=center
|File:Cartesian grid.svg | 3-D Cartesian grid
|File:rectilinear grid.svg | 3-D rectilinear grid
|File:Curvilinear grid.svg | 2-D curvilinear grid
|File:Example curvilinear grid.svg | Non-curvilinear combination of different 2-D curvilinear grids
|File:Tiling_3_simple.svg | 2-D triangular grid.
}}

==See also==
*{{annotated link|Cartesian coordinate system}}
*{{annotated link|Integer lattice}}
*{{annotated link|Unstructured grid}}
*{{annotated link|Discretization}}

==References==
{{Reflist}}

{{Tessellation}}

[[Category:Tessellation]]
[[Category:Lattice points]]
[[Category:Mesh generation]]
{{Elementary-geometry-stub}}