Quadrilateralized spherical cube
Encyclopedia
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.
This scheme is also often called the COBE sky cube,
because it was designed to hold data from the Cosmic Background Explorer (COBE) project.
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 the unit sphere. 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 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 22N, where N is the binning depth, for a total of 6 × 22N. 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 microsteradian
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
s required for the bin numbers at level N is 2N+ 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 0–1048577, face 1 has numbers 1048576–2097151, 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 and results in a bin size of 0.0922 square arcminutes (7.80 nanosteradians).
In principle, the mapping and numbering scheme
s 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 susceptible to partitioning
into a set of square arrays. Used together, they make a flexible and efficient system for storing map data.
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.
Cartography
Cartography is the study and practice of making maps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively.The fundamental problems of traditional cartography are to:*Set the map's...
, a quadrilateralized spherical cube, or quad sphere for short, is an equal-area mapping and binning
Data binning
Data binning is a data pre-processing technique used to reduce the effects of minor observation errors. The original data values which fall in a given small interval, a bin, are replaced by a value representative of that interval, often the central value...
scheme for data collected on a spherical
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...
surface (either that of the Earth or the celestial sphere
Celestial sphere
In astronomy and navigation, the celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with the Earth and rotating upon the same axis. All objects in the sky can be thought of as projected upon the celestial sphere. Projected upward from Earth's equator and poles are the...
). 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 (COBE) project.
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 projectionCurvilinear perspective
Curvilinear perspective is a graphical projection used to draw 3D objects on 2D surfaces. It was formally codified in 1968 by the artists and art historians André Barre and Albert Flocon in the book La Perspective curviligne, which was translated into English in 1987 as Curvilinear Perspective:...
that preserves area. The sphere is divided into six equal regions, which correspond to the faces of the cube
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. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
. The vertices
Vertex (geometry)
In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
of the cube correspond to the cartesian
Cartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...
coordinates defined by |x|=|y|=|z| on the unit sphere. For an Earth projection, the cube is usually oriented with one face normal to the North Pole
North Pole
The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is, subject to the caveats explained below, defined as the point in the northern hemisphere where the Earth's axis of rotation meets its surface...
and one face centered on the Greenwich meridian
Prime Meridian
The Prime Meridian is the meridian at which the longitude is defined to be 0°.The Prime Meridian and its opposite the 180th meridian , which the International Date Line generally follows, form a great circle that divides the Earth into the Eastern and Western Hemispheres.An international...
(although any definition of pole and meridian could be used). The faces of the cube are divided into square bins
Bin (computational geometry)
In computational geometry, the bin data structure allows efficient region queries, i.e., if there are some axis-aligned rectangles on a 2D plane, answer the question Given a query rectangle, return all rectangles intersecting it. kd-tree is another data structure that can answer this question...
, where the number of bins along each edge is a power of 2
Power of two
In mathematics, a power of two means a number of the form 2n where n is an integer, i.e. the result of exponentiation with as base the number two and as exponent the integer n....
, selected to produce the desired bin size
Histogram
In statistics, a histogram is a graphical representation showing a visual impression of the distribution of data. It is an estimate of the probability distribution of a continuous variable and was first introduced by Karl Pearson...
. Thus the number of bins on each face is 22N, where N is the binning depth, for a total of 6 × 22N. 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
Minute of arc
A minute of arc, arcminute, or minute of angle , is a unit of angular measurement equal to one sixtieth of one degree. In turn, a second of arc or arcsecond is one sixtieth of one minute of arc....
(2.00 microsteradian
Steradian
The steradian is the SI unit of solid angle. It is used to describe two-dimensional angular spans in three-dimensional space, analogous to the way in which the radian describes angles in a plane...
s).
The second key feature is that the bins are numbered serially, rather than being rastered
Rasterisation
Rasterisation is the task of taking an image described in a vector graphics format and converting it into a raster image for output on a video display or printer, or for storage in a bitmap file format....
as for an image. The total number of bit
Bit
A bit is the basic unit of information in computing and telecommunications; it is the amount of information stored by a digital device or other physical system that exists in one of two possible distinct states...
s required for the bin numbers at level N is 2N+ 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 0–1048577, face 1 has numbers 1048576–2097151, 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
Binary tree
In computer science, a binary tree is a tree data structure in which each node has at most two child nodes, usually distinguished as "left" and "right". Nodes with children are parent nodes, and child nodes may contain references to their parents. Outside the tree, there is often a reference to...
, 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 and results in a bin size of 0.0922 square arcminutes (7.80 nanosteradians).
In principle, the mapping and numbering scheme
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 system table, whose table definitions require a...
s 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 susceptible to partitioning
Partition of a set
In mathematics, a partition of a set X is a division of X into non-overlapping and non-empty "parts" or "blocks" or "cells" that cover all of X...
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 singularitiesSingularity
Singularity or Singular points may refer to:-Mathematics:* Mathematical singularity, a point at which a given mathematical object is not defined or not well-behaved, for example infinite or not differentiable-Geometry:...
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.
External links
- http://www.sai.msu.su/~megera/wiki/SphereCube Information from the Sternberg Astronomical InstituteSternberg Astronomical InstituteThe Sternberg Astronomical Institute , also known as GAISh , is a research institution in Moscow, Russia, a division of Moscow State University. The institute is named after astronomer Pavel Karlovich Shternberg.The asteroid 14789 GAISH was named in its honour.-External links:*...