A regular grid is a
tessellation
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...
of ''n''-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
by
congruent parallelotopes (e.g.
brick
A brick is a type of block used to build walls, pavements and other elements in masonry construction. Properly, the term ''brick'' denotes a block composed of dried clay, but is now also used informally to denote other chemically cured cons ...
s).
Its opposite is
irregular grid.
Grids of this type appear on
graph paper
Graph paper, coordinate paper, grid paper, or squared paper is writing paper that is printed with fine lines making up a regular grid. The lines are often used as guides for plotting graphs of functions or experimental data and drawing curves ...
and may be used in
finite element analysis
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
,
finite volume methods,
finite difference methods, and in general for discretization of parameter spaces. Since the derivatives of field variables can be conveniently expressed as finite differences,
structured grids mainly appear in finite difference methods.
Unstructured grids 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
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
s or (i, j, k) in three dimensions, and each
vertex
Vertex, vertices or vertexes may refer to:
Science and technology Mathematics and computer science
*Vertex (geometry), a point where two or more curves, lines, or edges meet
*Vertex (computer graphics), a data structure that describes the position ...
has
coordinates
in 2D or
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
In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and .
Cartesian coordinates
In a Cartesian coordin ...
s or
unit cubes, and the vertices are
points on the
integer lattice.
A rectilinear grid is a tessellation by
rectangle
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram contain ...
s or
rectangular cuboids (also known as
rectangular parallelepipeds) that are not, in general, all
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
A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a ...
graph paper
Graph paper, coordinate paper, grid paper, or squared paper is writing paper that is printed with fine lines making up a regular grid. The lines are often used as guides for plotting graphs of functions or experimental data and drawing curves ...
.
A skewed grid is a tessellation of
parallelogram
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of eq ...
s or
parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term '' rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclid ...
s. (If the unit lengths are all equal, it is a tessellation of
rhombi or
rhombohedra
In geometry, a rhombohedron (also called a rhombic hexahedron or, inaccurately, a rhomboid) is a three-dimensional figure with six faces which are rhombi. It is a special case of a parallelepiped where all edges are the same length. It can be use ...
.)
A curvilinear grid or structured grid is a grid with the same combinatorial structure as a regular grid, in which the cells are
quadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
s or
cuboid">eneralcuboids, rather than rectangles or rectangular cuboids.
See also
*
*
*
*
References
Tessellation
Lattice points
Mesh generation
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