Quadrilateralized Spherical Cube

In mapmaking, a quadrilaterlized spherical cube, or quad sphere for short, is an equal-area mapping and binning scheme for data collected on a spherical surface (either Earth data or the celestial sphere). It was first proposed by Chan and O'Neill in 1975 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 (COBE) project.

Elements
There are two key elements to the quad sphere:


 * 1) The mapping consists of projecting the sphere onto the faces of an inscribed cube using a curvilinear projection which preserves area. The sphere is divided into six equal areas which correspond to the faces of the cube. The vertices of the cube correspond to the cartesian coordinates defined by |x|=|y|=|z| on the unit sphere. For an Earth projection, the cube is normally 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 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**(2*N), where N is the binning level, and the total number of bins is 6*2**(2*N). For example, a level of 10 gives 1024x1024 bins on each face and 6291456 (6*2**20) total bins, which are 23.605 square arcminutes (1.99737E-6 steradians) in size.
 * 2) The bins are numbered serially, rather than being rastered as for an image. The bin numbers are determined as follows. The total number of bits required for the bin numbers at level N is 2*N+3, where the 3 most significant bits (msb) are used for the face numbers and the remaining bits are used to number the bins within each face. The faces are numbered 0-5 with 0 being the North face, 1 through 4 being equatorial with 1 corresponding to Greenwich, and 5 being South. Thus at level 10, face 0 has bin numbers 0-1048577, face 1 has numbers 1048576-2097151, etc. Within each face the bins are numbered serially from one corner (the convention is to start at the "lower left") to the opposite corner, with the ordering such that each pair of bits corresponds to a level of bin resolution. This ordering in effect is a two-dimensional binary tree, which is referred to as the quad-tree. The conversion between bin numbers and coordinates is straightforward. If 4-byte integers are used for the bin numbers the maximum practical bin level is 14, which uses 31 of the 32 bits and results in a bin size of 0.0922 square arcminutes (7.80223E-9 steradians). The advantages of this numbering approach are stated below.

In principle, the mapping and numbering schemes are separable; the 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 which can be partitioned as a set of square arrays. Used together, they comprise 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. The distortion is moderate over the entire sphere, so that at no point are shapes distorted beyond recognition.