A geodesic grid is a technique used to model the surface of a sphere (such as the Earth) with a subdivided polyhedron

Polyhedron

In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

, usually an icosahedron

Icosahedron

In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....

.

Introduction

A geodesic grid is a global Earth reference that uses cells or tiles to statistically represent data encoded to the area covered by the cell location. The focus of the discrete cells in a geodesic grid reference is different from that of a conventional lattice-based Earth reference where the focus is on a continuity of points used for addressing location and navigation.

In biodiversity science, geodesic grids are a global extension of local discrete grids that are staked out in field studies to ensure appropriate statistical sampling and larger multi-use grids deployed at regional and national levels to develop an aggregated understanding of biodiversity. These grids translate environmental and ecological monitoring data from multiple spatial and temporal scales into assessments of current ecological condition and forecasts of risks to our natural resources. A geodesic grid allows local to global assimilation of ecologically significant information at its own level of granularity.

When modeling the weather, ocean circulation, or the climate

Climate prediction

Climate prediction is a subset of numerical weather prediction dealing with generalized forecasts beyond the usual short-range and medium-range forecast periods...

, partial differential equation

Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s are used to describe the evolution of these systems over time. Because computer programs are used to build and work with these complex models, approximations need to be formulated into easily computable forms. Some of these numerical analysis

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....

 techniques (such as finite differences) require the area of interest to be subdivided into a grid — in this case, over the shape of the Earth

Geodesy

Geodesy , also named geodetics, a branch of earth sciences, is the scientific discipline that deals with the measurement and representation of the Earth, including its gravitational field, in a three-dimensional time-varying space. Geodesists also study geodynamical phenomena such as crustal...

.

Geodesic grids have been developed by subdividing a sphere to developing a global tiling (tessellation) based on a geographic coordinates (longitude/latitude) where a rectilinear cell is defined as the intersection of a longitude and latitude line. This approach is easily understood in terms of accepted Earth reference, accessible using the longitude and latitude as an ordered pair, and implemented in a computer coding as a rectangular grid. However, such a pattern does not conform to many of the main criteria for a statistically valid discrete global grid, primarily that the cells' area and shape are not generally similar; this is especially noticeable as the cells are developed towards the poles.

Another approach gaining favour uses geodesic sphere grids generated by the subdivision of a platonic solid

Platonic solid

In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...

 into cells or by iteratively bisecting

Bisection

In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a bisector. The most often considered types of bisectors are the segment bisector and the angle bisector In geometry, bisection is the division of something into two equal...

 the edges of the polyhedron

Polyhedron

In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

 and projecting the new cells onto a sphere

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

. In this geodesic grid, each of the vertices in the resulting geodesic sphere corresponds to a cell. One implementation uses an icosahedron

Icosahedron

In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....

 as the base polyhedron, hexagonal cells, and the Snyder equal area projection

Map projection

A map projection is any method of representing the surface of a sphere or other three-dimensional body on a plane. Map projections are necessary for creating maps. All map projections distort the surface in some fashion...

 is known as the Icosahedron Snyder Equal Area (ISEA) grid. Another method, using the intersection of a tetrahedron

Tetrahedron

In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

 into triangular quadtrees, is known as the Quaternary Triangular Mesh (QTM). A triangular mesh conforms well to representation in a graphics pipeline, and its dual cells are hexagons, convenient for encoding data. The hexagonal geodesic grid inherits many of the virtues of 2D hexagonal grids, and specifically avoids problems with singularities and oversampling near the poles. Along the same line, different Platonic solid

Platonic solid

In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...

s could also be used as a starting point instead of an icosahedron or tetrahedron — e.g. cubes are common in video games.

The quadrilateralized spherical cube

Quadrilateralized spherical cube

In mapmaking, a quadrilateralized spherical cube, or quad sphere for short, is an equal-area mapping and binning scheme for data collected on a spherical surface...

 is a kind of geodesic grid based on subdividing a cube into equal-area cells that are approximately square.

Positive traits

• Largely isotropic.
• Resolution can be easily increased by binary division.
• Does not suffer from over sampling near the poles like more traditional rectangular longitude/latitude square grids.
• Does not result in dense linear systems like spectral method
  Spectral method
  Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain Dynamical Systems, often involving the use of the Fast Fourier Transform. Where applicable, spectral methods have excellent error properties, with the so called "exponential...
  
  s do (see also Gaussian grid
  Gaussian grid
  A Gaussian grid is used in the earth sciences as a gridded horizontal coordinate system for scientific modeling on a sphere...
  
  ).
• No single points of contact between neighboring grid cells. Square grids and isometric grids suffer from the ambiguous problem of how to handle neighbors that only touch at a single point.
• Cells can be both minimally distorted and near-equal-area. In contrast, square grids are not equal area, while equal-area rectangular grids vary in shape from equator to poles.

Negative traits

• More complicated to implement than rectangular longitude/latitude grids in computers

History

The earliest use of the (icosahedral) geodesic grid in geophysical modeling dates back to 1968 and the work by
Sadourny, Arakawa, and Mintz
and Williamson.

Later work expanded on this base.

External links

• BUGS climate model page on geodesic grids
• Discrete Global Grids page at the Computer Science department at Southern Oregon University
• http://www.pyxisinnovation.com/pyxwiki/index.php?title=How_PYXIS_Worksthe PYXIS innovation Digital Earth Reference Model
  Digital Earth Reference Model
  The term Digital Earth Reference Model was coined by Tim Foresman in context with a vision for an all encompassing geospatial platform as an abstract for information flow in support of Al Gore’s vision for a Digital Earth...
  
  ].