Quadrilateralized spherical cube
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mapmaking Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an im ...
, a quadrilateralized spherical cube, or quad sphere for short, is an equal-area
polyhedral map projection A polyhedral map projection is a map projection based on a spherical polyhedron. Typically, the polyhedron is overlaid on the globe, and each face of the polyhedron is transformed to a polygon or other shape in the plane. The best-known polyhedral ...
and
discrete global grid A discrete global grid (DGG) is a mosaic that covers the entire Earth's surface. Mathematically it is a space partitioning: it consists of a set of non-empty regions that form a partition of the Earth's surface. In a usual grid-modeling strate ...
scheme for data collected on a
spherical A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
surface (either that of the Earth or the
celestial sphere In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth. All objects in the sky can be conceived as being projected upon the inner surface of the celestial sphere, ...
). It was first proposed in 1975 by Chan and O'Neill for the Naval Environmental Prediction Research Facility. This scheme is also often called the COBE sky cube, because it was designed to hold data from the
Cosmic Background Explorer The Cosmic Background Explorer (COBE ), also referred to as Explorer 66, was a NASA satellite dedicated to cosmology, which operated from 1989 to 1993. Its goals were to investigate the cosmic microwave background radiation (CMB or CMBR) of th ...
(COBE) project. A related projection called the S2 projection was created at
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for the purpose of defining a discrete global grid scheme. It is similar to the quad sphere but is not equal-area.


Elements

The quad sphere has two principal characteristic features. The first is that the mapping consists of projecting the sphere onto the faces of an inscribed cube using a curvilinear projection that preserves area. The sphere is divided into six equal regions, which correspond to the faces of the
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...
. The vertices of the cube correspond to the
cartesian Cartesian means of or relating to the French philosopher René Descartes—from his Latinized name ''Cartesius''. It may refer to: Mathematics *Cartesian closed category, a closed category in category theory *Cartesian coordinate system, modern ...
coordinates defined by , ''x'', =, ''y'', =, ''z'', on a sphere centred at the origin. For an Earth projection, the cube is usually oriented with one face normal to the
North Pole The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is the point in the Northern Hemisphere where the Earth's axis of rotation meets its surface. It is called the True North Pole to distinguish from the Mag ...
and one face centered on the
Greenwich meridian The historic prime meridian or Greenwich meridian is a geographical reference line that passes through the Royal Observatory, Greenwich, Royal Observatory, Greenwich, in London, England. The modern IERS Reference Meridian widely used today ...
(although any definition of pole and meridian could be used). The faces of the cube are divided into a grid of square bins, where the number of bins along each edge is a
power of 2 A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer  as the exponent. In a context where only integers are considered, is restricted to non-negative ...
, selected to produce the desired bin size. Thus the number of bins on each face is 22''N'', where ''N'' is the binning depth, for a total of 6 × 22''N''. For example, a binning depth of 10 gives 1024 × 1024 bins on each face or 6291456 (6 × 220) in all, each bin covering an area of 23.6 square  arcminutes (2.00 micro
steradian The steradian (symbol: sr) or square radian is the unit of solid angle in the International System of Units (SI). It is used in three-dimensional geometry, and is analogous to the radian, which quantifies planar angles. Whereas an angle in radian ...
s). The second key feature is that the bins are numbered serially, rather than being rastered as for an image. The total number of
bit The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represente ...
s required for the bin numbers at level ''N'' is 2''N'' + 3, where the three most significant bits are used for the face numbers and the remaining bits are used to number the bins within each face. The faces are numbered from 0 to 5: 0 for the north face, 1 through 4 for the equatorial faces (1 being on the meridian), and 5 for the south. Thus at a binning depth of 10, face 0 has bin numbers 0–1,048,575, face 1 has numbers 1,048,576–2,097,151, and so on. Within each face the bins are numbered serially from one corner (the convention is to start at the "lower left") to the opposite corner, ordered in such a way that each pair of bits corresponds to a level of bin resolution. This ordering is in effect a two-dimensional
binary tree In computer science, a binary tree is a k-ary k = 2 tree data structure in which each node has at most two children, which are referred to as the ' and the '. A recursive definition using just set theory notions is that a (non-empty) binary t ...
, which is referred to as the ''
quad-tree A quadtree is a tree data structure in which each internal node has exactly four children. Quadtrees are the two-dimensional analog of octrees and are most often used to partition a two-dimensional space by recursively subdividing it into four ...
''. The conversion between bin numbers and coordinates is straightforward. If four-byte integers are used for the bin numbers the maximum practical depth, which uses 31 of the 32 bits, results in a bin size of 0.0922 square arcminutes (7.80 nanosteradians). In principle, the mapping and
numbering scheme There are many different numbering schemes for assigning nominal numbers to entities. These generally require an agreed set of rules, or a central coordinator. The schemes can be considered to be examples of a primary key of a database management ...
s are separable: the
map projection In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitud ...
onto the cube could be used with another bin-numbering scheme, and the numbering scheme itself could be used with any arrangement of bins susceptible to partitioning into a set of square arrays. Used together, they make a flexible and efficient system for storing map data.


Advantages

The quad sphere projection does not produce singularities at the poles or elsewhere, as do some other equal-area mapping schemes. Distortion is moderate over the entire sphere, so that at no point are shapes altered beyond recognition.


See also

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List of map projections This is a summary of map projections that have articles of their own on Wikipedia or that are otherwise notable Notability is the property of being worthy of notice, having fame, or being considered to be of a high degree of interest, signif ...
*
Cube mapping In computer graphics, cube mapping is a method of environment mapping that uses the six faces of a cube as the map shape. The environment is projected onto the sides of a cube and stored as six square textures, or unfolded into six regions of a s ...
*
Geodesic grid A geodesic grid is a spatial grid based on a geodesic polyhedron or Goldberg polyhedron. Construction A geodesic grid is a global Earth reference that uses triangular tiles based on the subdivision of a polyhedron (usually the icosahedron, and ...


References

* * {{DEFAULTSORT:Quadrilateralized Spherical Cube Cartography Map projections