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 (either that of the Earth or the celestial sphere). It was first proposed in 1975 by Chan and O'Neill for the Naval Environmental Prediction Research Facility.^{ [1] }

This scheme is also often called the **COBE sky cube**,^{ [2] } because it was designed to hold data from the Cosmic Background Explorer (COBE) project.^{ [3] }

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. The vertices of the cube correspond to the cartesian 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 and one face centered on the Greenwich meridian (although any definition of pole and meridian could be used). The faces of the cube are divided into a geodesic grid of square bins, where the number of bins along each edge is a power of 2, selected to produce the desired bin size. Thus the number of bins on each face is 2^{2N}, where *N* is the binning depth, for a total of 6 × 2^{2N}. For example, a binning depth of 10 gives 1024 × 1024 bins on each face or 6291456 (6 × 2^{20}) in all, each bin covering an area of 23.6 square arcminutes (2.00 microsteradians).

The second key feature is that the bins are numbered serially, rather than being rastered as for an image. The total number of bits 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, which is referred to as the * quad-tree *. 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 schemes are separable: the map projection 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.

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.

In geometry, a **cube** is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

In geography, **latitude** is a geographic coordinate that specifies the north–south position of a point on the Earth's surface. Latitude is an angle which ranges from 0° at the Equator to 90° at the poles. Lines of constant latitude, or *parallels*, run east–west as circles parallel to the equator. Latitude is used together with longitude to specify the precise location of features on the surface of the Earth. On its own, the term latitude should be taken to be the *geodetic latitude* as defined below. Briefly, geodetic latitude at a point is the angle formed by the vector perpendicular to the ellipsoidal surface from that point, and the equatorial plane. Also defined are six *auxiliary latitudes* which are used in special applications.

The **Mercator projection** is a cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because of its unique property of representing any course of constant bearing as a straight segment. Such a course, known as a rhumb or, mathematically, a loxodrome, is preferred by navigators because the ship can sail in a constant compass direction to reach its destination, eliminating difficult and error-prone course corrections. Linear scale is constant on the Mercator in every direction around any point, thus preserving the angles and the shapes of small objects and fulfilling the conditions of a conformal map projection. As a side effect, the Mercator projection inflates the size of objects away from the equator. This inflation is very small near the equator, but accelerates with latitude to become infinite at the poles. So, for example, landmasses such as Greenland and Antarctica appear far larger than they actually are relative to landmasses near the equator, such as Central Africa.

A **geographic coordinate system** is a coordinate system that enables every location on Earth to be specified by a set of numbers, letters or symbols. The coordinates are often chosen such that one of the numbers represents a vertical position and two or three of the numbers represent a horizontal position; alternatively, a geographic position may be expressed in a combined three-dimensional Cartesian vector. A common choice of coordinates is latitude, longitude and elevation. To specify a location on a plane requires a map projection.

In cartography, a **map projection** is a way to flatten a globe's surface into a plane in order to make a map. This requires a systematic transformation of the latitudes and longitudes of locations from the surface of the globe into locations on a plane. All projections of a sphere on a plane necessarily distort the surface in some way and to some extent. Depending on the purpose of the map, some distortions are acceptable and others are not; therefore, different map projections exist in order to preserve some properties of the sphere-like body at the expense of other properties. Every distinct map projection distorts in a distinct way, by definition. The study of map projections is the characterization of these distortions. There is no limit to the number of possible map projections. Projections are a subject of several pure mathematical fields, including differential geometry, projective geometry, and manifolds. However, "map projection" refers specifically to a cartographic projection.

In geometry, the **stereographic projection** is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles at which curves meet. It is neither isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures.

In geometry, the **snub cube**, or **snub cuboctahedron**, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices.

The **great-circle distance** or **orthodromic distance** is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere. The distance between two points in Euclidean space is the length of a straight line between them, but on the sphere there are no straight lines. In spaces with curvature, straight lines are replaced by geodesics. Geodesics on the sphere are circles on the sphere whose centers coincide with the center of the sphere, and are called *great circles*.

A **gnomonic map projection** displays all great circles as straight lines, resulting in any straight line segment on a gnomonic map showing a geodesic, the shortest route between the segment's two endpoints. This is achieved by casting surface points of the sphere onto a tangent plane, each landing where a ray from the center of the sphere passes through the point on the surface and then on to the plane. No distortion occurs at the tangent point, but distortion increases rapidly away from it. Less than half of the sphere can be projected onto a finite map. Consequently, a rectilinear photographic lens, which is based on the gnomonic principle, cannot image more than 180 degrees.

The **scale** of a map is the ratio of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the Earth's surface, which forces scale to vary across a map. Because of this variation, the concept of scale becomes meaningful in two distinct ways.

In computer graphics, **environment mapping**, or **reflection mapping**, is an efficient image-based lighting technique for approximating the appearance of a reflective surface by means of a precomputed texture image. The texture is used to store the image of the distant environment surrounding the rendered object.

The **equirectangular projection** is a simple map projection attributed to Marinus of Tyre, who Ptolemy claims invented the projection about AD 100. The projection maps meridians to vertical straight lines of constant spacing, and circles of latitude to horizontal straight lines of constant spacing. The projection is neither equal area nor conformal. Because of the distortions introduced by this projection, it has little use in navigation or cadastral mapping and finds its main use in thematic mapping. In particular, the plate carrée has become a standard for global raster datasets, such as Celestia and NASA World Wind, because of the particularly simple relationship between the position of an image pixel on the map and its corresponding geographic location on Earth.

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 single texture. The cube map is generated by first rendering the scene six times from a viewpoint, with the views defined by a 90 degree view frustum representing each cube face.

The **oblique Mercator** map projection is an adaptation of the standard Mercator projection. The oblique version is sometimes used in national mapping systems. When paired with a suitable geodetic datum, the oblique Mercator delivers high accuracy in zones less than a few degrees in arbitary directional extent.

The **Lambert azimuthal equal-area projection** is a particular mapping from a sphere to a disk. It accurately represents area in all regions of the sphere, but it does not accurately represent angles. It is named for the Swiss mathematician Johann Heinrich Lambert, who announced it in 1772. "Zenithal" being synonymous with "azimuthal", the projection is also known as the **Lambert zenithal equal-area projection**.

A **geodesic grid** is a spatial grid based on a geodesic polyhedron or Goldberg polyhedron.

**HEALPix**, an acronym for **H**ierarchical **E**qual **A**rea iso**L**atitude **Pix**elisation of a 2-sphere, refers to either an algorithm for pixelisation of the 2-sphere or to the associated class of map projections. The pixelisation algorithm was devised in 1997 by Krzysztof M. Górski at the Theoretical Astrophysics Center in Copenhagen, Denmark, and first published as a preprint in 1998.

**Web Mercator**, **Google Web Mercator**, **Spherical Mercator**, **WGS 84 Web Mercator** or **WGS 84/Pseudo-Mercator** is a variant of the Mercator projection and is the de facto standard for Web mapping applications. It rose to prominence when Google Maps adopted it in 2005. It is used by virtually all major online map providers, including Google Maps, Mapbox, Bing Maps, OpenStreetMap, Mapquest, Esri, and many others. Its official EPSG identifier is EPSG:3857, although others have been used historically.

A **Discrete Global Grid** (**DGG**) is a mosaic which 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 strategy, to simplify position calculations, each region is represented by a point, abstracting the grid as a set of region-points. Each region or region-point in the grid is called a **cell**.

The **Goldberg–Coxeter construction** or **Goldberg–Coxeter operation** is a graph operation defined on regular polyhedral graphs with degree 3 or 4. It also applies to the dual graph of these graphs, i.e. graphs with triangular or quadrilateral "faces". The GC construction can be thought of as subdividing the faces of a polyhedron with a lattice of triangular, square, or hexagonal polygons, possibly skewed with regards to the original face: it is an extension of concepts introduced by the Goldberg polyhedra and geodesic polyhedra. The GC construction is primarily studied in organic chemistry for its application to fullerenes, but it has been applied to nanoparticles, computer-aided design, basket weaving, and the general study of graph theory and polyhedra.

- ↑ Chan, F.K.; O'Neill, E. M. (1975).
*Feasibility Study of a Quadrilateralized Spherical Cube Earth Data Base (CSC - Computer Sciences Corporation, EPRF Technical Report 2-75)*. Monterey, California: Environmental Prediction Research Facility. - ↑ "COBE Quadrilateralized Spherical Cube".
- ↑ Max Tegmark. "What is the best way to pixelize a sphere?".

- O'Neill (1976) "Extended Studies of a Quadrilateralized Spherical Cube Earth Data Base", original scanned document of May 1976. (Naval Environmental Prediction Research Facility, Naval Postgraduate School Monterey, California 93940).

- Fred Patt. "SphereCube". Information from the Sternberg Astronomical Institute

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