Map projections for wide-field surveys

I developed an appreciation of cartography over the question:

Which map projection(s) should we choose to truthfully represent data from the celestial sphere in 2D?

Note: truthfully, and celestial.

For what follows I will assume a survey of up to 20,000 sq. degrees with predominant East-West extension. This will do just fine for DES and LSST.

How to choose a map projection of the sky

When you look at widely used projections, they (can) have four main characteristics (in brackets: the feature they preserve when you move around the map):

  1. equidistant (distance)
  2. conformal (angle)
  3. equal-area (size)
  4. perspective (relation between distance and size)

It is unfortunate but inevitable that any mapping from a curved onto a flat surface can only preserve some of those features, but definitely not all, and in principle maybe none. So which ones matter most?

This is where truthfully comes in. When studying visual perception, it turns out that humans can distinguish some features better than others (see the seminal work by Cleveland & McGill, 1984). The order of the list above reflects that: distances (along a common scale, as in a bar chart) can most precisely be distinguished, followed by objects with different angles (think: trend lines or model predictions in a scatter plot), sizes, and volumes (for which perspective is relevant). Only after that will saturation and color be incorporated to distinguish objects.

So, we should be using equidistant maps, right? No.

The questions we face is not how precisely or rapidly can someone perceive features of the map. It’s how well can we show what we want to show.

Let’s take an equidistant map. If I found myself on an airplane with a finite amount of fuel, that’s exactly the map I’d want to have available to me. In astronomical terms, it’ll tell us how long it takes to slew a telescope, but most viewers won’t care about that. Use only if distances are what really matters.

Conformal maps face a similar problem. Outside of navigation, it’s rare that angles are of great importance over large areas—and on small areas most of what we discuss here is irrelevant. One can alter the angles of the stellar constellations quite a bit, and they will remain recognizable. For interpreting e.g. the effects of the moon illumination, a conformal map might be useful, but again: not necessarily the main focus here.

Equal-area maps are interesting because size matters, specifically for wide-field survey maps. The reason is that we often want to show particularly large features. Think of large cosmic voids, which we may want to compare visually to the size of the full moon. It would be odd if the size of those features change if we move them around on the map. Hence, I argue that most wide-field maps should be equal-area (or at least close to).

Brings us to the perspective projections. They feel natural because they mimic how we would look upon a sphere from some distance. Remember: celestial. We view that sphere from the inside, so the classical perspective does not help here. I would still maintain that maps of the sky should maintain as much of the curved appearance that we feel to be natural. We can create this perception with a graticule with at least one set of curved lines (either meridian or parallels, or both). This aesthetic preference rules out cylindrical maps, which map the sphere onto a rectangle.

These considerations leave only a few projections for hemisphere-sized maps:

  1. Lambert Azimuthal Equal-Area
  2. Albers Equal-Area Conic
  3. Lambert Conformal Conic*
  4. Equidistant Conic*

The latter two are not equal-area, but can be useful if angles or distances are most relevant for the purpose at hand.

The azimuthal projection is great for an area that extends equally in North-South and in East-West direction, while both of the conic projections work best when the extension of the data is mainly East-West. For surveys observed from one fixed location on the ground, the latter is the more typical situation because the range of visible declinations is limited, whereas RA is not. The conics have this property for all reasonably moderate latitudes, while the cylindrical ones work best only along the equator (and don’t have curved appearance).

Leaves Albers Equal-Area, Lambert Conformal, and Equidistant conics. All three of them are implemented in skymapper, a frontend to matplotlib I created to codify best practices. Albers is strictly equal-area at the expense of rather strong distortions towards the poles. Lambert preserves angles, sizes only along the standard parallels. In particular, it shows the pole as a point, unlike the Albers, where it’s an arc (see figure below). The Equidistant conic is a compromise between both, which will work well for smaller areas.

Albers Equal-Area vs Lambert Conformal around South Pole

Summary

In essence, if you stay in moderate latitudes the equal-area character of the Albers projection is preferable. If you need to approach or cover the poles, Lambert is a better choice. If distances really matter, use the Equidistant conic.

P.S.: Equal-area all-sky projections

For all-sky maps, we essentially have to throw out the preference of natural perspective altogether, which leave those equal-area maps.

  1. Lambert Cylindrical Equal-Area and variants (like Behrmann)
  2. Mollweide
  3. Eckert IV
  4. Healpix and others

I will leave the discussion of those to a later post…

© 2017 Peter Melchior ~ Email