HEALPix (sometimes written as Healpix), an acronym for Hierarchical Equal Area isoLatitude Pixelisation of a 2-

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, 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 Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...

for

pixel In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a Raster graphics, raster image, or the smallest addressable element in a dot matrix display device. In most digital display devices, p ...

isation of the 2-sphere based on subdivision of a distorted

rhombic dodecahedron In geometry, the rhombic dodecahedron is a Polyhedron#Convex_polyhedra, convex polyhedron with 12 congruence (geometry), congruent rhombus, rhombic face (geometry), faces. It has 24 edge (geometry), edges, and 14 vertex (geometry), vertices of 2 ...

, and the associated class of

map projection In cartography, a map projection is any of a broad set of Transformation (function) , transformations employed to represent the curved two-dimensional Surface (mathematics), surface of a globe on a Plane (mathematics), plane. In a map projection, ...

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 or projections 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, graphics, and carto ...

is a general class of spherical projections, sharing several key properties, which map the 2-

to the

Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...

. 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 fu ...

for maps of the

cosmic microwave background The cosmic microwave background (CMB, CMBR), or relic radiation, is microwave radiation that fills all space in the observable universe. With a standard optical telescope, the background space between stars and galaxies is almost completely dar ...

. 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 used for storage, transmission and processing of data: formatted as multi-dimensional arrays (for example a 2D image), or tables. FITS is the most commonl ...

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; , UAI) is an international non-governmental organization (INGO) with the objective of advancing astronomy in all aspects, including promoting astronomical research, outreach, education, and developmen ...

FITS Working Group on April 26, 2006. The spherical projection combines a

cylindrical A cylinder () has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a Prism (geometry), prism with a circle as its base. A cylinder may ...

equal area projection, the Lambert cylindrical equal-area projection, for the equatorial regions of the sphere and a pseudocylindrical equal area projection, an interrupted Collignon projection, 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 geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

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 transforms. 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

Gaia mission ''Gaia'' was a space observatory of the European Space Agency (ESA) that was launched in 2013 and operated until March 2025. The spacecraft was designed for astrometry: measuring the positions, distances and motions of stars with unprecedented ...

uses HEALPix as the basis for source identification. An alternative hierarchical grid is the Hierarchical Triangular Mesh (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. The 12 "base resolution pixels" of H=4, K=3 HEALPix projection may be thought of as the facets of a

. The H=6 HEALPix has similarities to another alternative grid based on the

icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical tha ...

References

External links

Official implementation
with many languages support (C, C++, Fortran90, IDL, Java and Python) for resolutions up to 0.4 mas ( milliarcsecond)
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

Projection and pixelisation

Usage and alternatives

See also

References

External links