science

HEALPix

HEALPix is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand HEALPix rather than just read about it. In short: HEALPix (sometimes written as Healpix), an acronym for Hierarchical Equal Area isoLatitude Pixelisation, is an algorithm for pixelisation of the 2-sphere and the associated class of map projections. The pixelisation algorithm was devised in 1997 by Krzysztof M.

Key takeaways

HEALPix belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect HEALPix to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of HEALPix from memory before moving on to harder problems.

Reference excerpt

HEALPix (sometimes written as Healpix), an acronym for Hierarchical Equal Area isoLatitude Pixelisation, is an algorithm for pixelisation of the 2-sphere and 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.

Projection and pixelisation The HEALPix projection is a general class of spherical projections, sharing several key properties, which map the 2-sphere to the Euclidean plane. 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 for maps of the cosmic microwave background. 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 standard for writing astronomical data files. It was approved as part of the official FITS World Coordinate System (WCS) by the International Astronomical Union FITS Working Group on April 26, 2006. The spherical projection combines a cylindrical 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 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 cosmic microwave background. Gaia mission 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. Unlike many other spherical grid systems, HEALPix is not based on mapping a sphere to a polyhedron. In particular, the standard H=4, K=3 HEALPix projection has 12 quatrilateral faces, similar to a rhombic dodecahedron, but its vertex configuration is different and in fact incompatible with any genus 0 polyhedron. The H=6 HEALPix has similarities to another alternative grid based on the icosahedron.

See also List of map projections Spatial grid Geodesic grid

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

Illustrations

Worked examples

Example 1 — a first encounter with HEALPix

Start with the simplest possible case. Write down what HEALPix claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to HEALPix before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about HEALPix ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of HEALPix

In research: HEALPix appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses HEALPix in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: HEALPix is common in secondary-school and first-year university syllabi. It links to neighbouring topics Map projections, so understanding it makes those chapters shorter.
In everyday life: Look for HEALPix outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study HEALPix in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what HEALPix means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain HEALPix out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is HEALPix in simple terms?

Why does HEALPix matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study HEALPix?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on HEALPix.