![HEALPix projection SW](https://upload.wikimedia.org/wikipedia/commons/1/12/HEALPix_projection_SW.svg)
HEALPix (sometimes written as Healpix), an acronym for Hierarchical Equal Area isoLatitude Pixelisation of a 2-
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
, is an
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
for
pixel
In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a raster image, or the smallest point in an all points addressable display device.
In most digital display devices, pixels are the smal ...
isation of the 2-sphere and the associated class of
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 ...
s.
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.
Projection and pixelisation
The HEALPix
projection
Projection, projections or projective may refer to:
Physics
* Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction
* The display of images by a projector
Optics, graphic ...
is a general class of spherical projections, sharing several key properties, which map the 2-
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
to the
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
.
Any of these can be followed by partitioning (pixelising) the resulting region of the 2-plane. In particular, when one of these projections (the H=4, K=3 HEALPix projection) is followed by a pixelisation of the 2-plane, the result is generally known as the HEALPix pixelisation,
which is widely used in
physical cosmology
Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of f ...
for maps of the
cosmic microwave background
In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all spac ...
. This pixelisation can be thought of as mapping the sphere to twelve square facets (diamonds) on the plane followed by the binary division of these facets into pixels,
though it can be derived without using the projection.
The associated software package HEALPix implements the algorithm.
The HEALPix projection (as a general class of spherical projections) is represented by the keyword ''HPX'' in the
FITS
Flexible Image Transport System (FITS) is an open standard defining a digital file format useful for storage, transmission and processing of data: formatted as multi-dimensional arrays (for example a 2D image), or tables. FITS is the most com ...
standard for writing astronomical data files. It was approved as part of the official FITS World Coordinate System (WCS) by the
International Astronomical Union
The International Astronomical Union (IAU; french: link=yes, Union astronomique internationale, UAI) is a nongovernmental organisation with the objective of advancing astronomy in all aspects, including promoting astronomical research, outreac ...
FITS Working Group on April 26, 2006.
The spherical projection combines a
cylindrical
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infini ...
equal area projection, the
Lambert cylindrical equal-area projection
In cartography, the Lambert cylindrical equal-area projection, or Lambert cylindrical projection, is a
cylindrical equal-area projection. This projection is undistorted along the equator, which is its standard parallel, but distortion increase ...
, for the equatorial regions of the sphere and a
pseudocylindrical
In cartography, map projection is the term used to describe a broad set of Transformation (function) , transformations employed to represent the two-dimensional curved Surface (mathematics), surface of a globe on a Plane (mathematics), plane. In ...
equal area projection, an interrupted
Collignon projection
The Collignon projection is an equal-area pseudocylindrical map projection first known to be published by Édouard Collignon in 1865 and subsequently cited by A. Tissot in 1881.
For the smallest choices of the parameters chosen for this project ...
, for the polar regions.
At a given level in the hierarchy the pixels are of equal area (which is done by bisecting the
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
in the case of the H=4, K=3 projection) and their centers lie on a discrete number of circles of latitude, with equal spacing on each circle. The scheme has a number of mathematical properties which make it efficient for certain computations, e.g.
spherical harmonic
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
Since the spherical harmonics form a ...
transform
Transform may refer to:
Arts and entertainment
* Transform (scratch), a type of scratch used by turntablists
* ''Transform'' (Alva Noto album), 2001
* ''Transform'' (Howard Jones album) or the title song, 2019
* ''Transform'' (Powerman 5000 album ...
s. In the case of the H=4, K=3 projection, the pixels are squares in the plane (which can be inversely projected back to quadrilaterals with non-geodesic sides on the 2-sphere) and every vertex joins four pixels, with the exception of eight vertices which each join only three pixels.
The latitude of transition between equatorial-orthogonal and polar-convergent longitude lines has been selected to allow the folding of the projection into a perfect cube — "cubing the sphere"; indeed in this way the Arctic Circle becomes a square.
Usage and alternatives
The pixelisation related to the H=4, K=3 projection has become widely used in cosmology for storing and manipulating maps of the
cosmic microwave background
In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all spac ...
.
Gaia mission
''Gaia'' is a space observatory of the European Space Agency (ESA), launched in 2013 and expected to operate until 2025. The spacecraft is designed for astrometry: measuring the positions, distances and motions of stars with unprecedented preci ...
uses HEALPix as the basis for source identification.
An alternative hierarchical grid is the
Hierarchical Triangular Mesh
Hierarchical Triangular Mesh (HTM) is a kind of quad tree used for mesh generation in 3-D computer graphics and geometric data structures
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with prope ...
(HTM).
The pixels at a given level in the hierarchy are of similar but not identical size. The scheme is good at representing complex shapes because the boundaries are all segments of
circles of the sphere. Another alternative hierarchical grid is the
Quadrilateralized Spherical Cube
In mapmaking, a quadrilateralized spherical cube, or quad sphere for short, is an equal-area polyhedral map projection and discrete global grid scheme for data collected on a spherical surface (either that of the Earth or the celestial sphere). It ...
.
The 12 "base resolution pixels" of H=4, K=3 HEALPix projection may be thought of as the facets of a
rhombic dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.
Properties
The rhombic dodecahedro ...
.
The H=6 HEALPix has similarities to another alternative grid based on the icosahedron.
What is the best way to pixelize a sphere?
/ref>
See also
* 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 ...
* Spatial grid
In the context of a spatial index, a grid or mesh is a regular tessellation of a manifold or 2-D surface that divides it into a series of contiguous cells, which can then be assigned unique identifiers and used for spatial indexing purposes. A ...
References
External links
Official implementation
with many languages support (C, C++, Fortran90, IDL, Java and Python) for resolutions up to 0.4 mas (milliarcsecond
A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of one degree. Since one degree is of a turn (or complete rotation), one minute of arc is of a turn. The na ...
)
Java port
of original Fortran code by Nikolay Kuropatkin, supporting resolutions up to 0.3 arcsec
Java port
optimized to use RangeSet, very good for high resolutions
healpy
Python wrapper
astropy-healpix
BSD-licensed HEALPix for Astropy
healpix.cxx
c++ code to transform healpix coordinates
JavaScript port
convert between lonlat and HEALPix coordinates in JavaScript
Typescript healpix
An implementation of HEALPix in JavaScript / TypeScript
Healpix.jl
Healpix library written in Julia
{{DEFAULTSORT:Healpix
Map projections