• Intro
• Who we are
• Our IDL Codes
• Rogues Gallery of maps
• Quality Scores
• astro-ph/0608501.

• My big Photo Map of Philadelphia
• Tissot Indicatrices
• Various Map Projections

• My Main Page

Intro

J. Richard Gott and I wanted to take this analysis another step forward. What happens when you project large bodies: the US-Canada Border, for example, or Australia, or the bands of Jupiter, onto a particular map projection? Are the features faithfully reproduced? To that end, we have introduced a formal measure of the flexion and skewness of a map.

This can be visualized simply via the Goldberg-Gott Indicatrices. Pick a point on the earth and drive north 12 degrees (about 1300 kilometers). Even if you hit the north pole, don't turn your steering wheel. Follow a geodesic. Do the same thing but heading west, east, and south. From your perspective, you've drawn a big plus sign on the ground. When projected onto a map, this indicatrix immediately reveals the curvature of the projection. Moreover, if you connect the dots and close the indicatrix, you get an ellipse -- the Tissot ellipse.

Rich and I have evaluated about 20 different projections and measured them with respect to area preservation, ellipticity of the Tissot, flexion (the bending of geodesics), skewness (the rate of change of speed on a map), boundary cuts, and interruptions between random points. All in all, we found that the best two overall projections are (respectively), the Winkel-Tripel (left), and the Kavrayskiy VII (right). See if you agree with our numerical assessment.

While you're at it, stop by Wes Colley's homepage. He has also worked with Rich on map projections, and was good enough to provide his map reprojection software. You can also get a paper by Gott, Mugnolo and Colley on optimal map distance measures here.

Now, for each take a look at the "Goldberg-Gott" indicatrices:

But the best overall map isn't necessarily the best for every purpose. For example, the Kavrayskiy VII has the nice property that lines of longitude are horizontal. So, for a projection, of, say, Jupiter, we find:

For many applications, (such as the CMB, Large-Scale Structure, etc), all-sky maps are required. Since anything other than an area-preserving projection would grossly distort structure, and thus should be immediately discounted, only those should be used. But which one? The WMAP team, for example, has used the Mollweide projection. However, we show that the all-around best area-preserving map is the Eckert IV projection. It beats the Hammer (Aitoff) on shapes by a large margin (the most noticable effect in this case), ties it on flexion, and barely loses on distances. On skewness, however, the Hammer wins by a fair margin.

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Decide for yourself. Wes Colley has prepared an Eckert IV projection of the WMAP 3rd year results:

Eckert IV

Mollweide

Our Code

There are two main programs

1. map_flexion.pro, which creates a projection with Goldberg-Gott indicatrices. The calling procedure is:
   map_flexion,projection (string), [outfile=filename]
   The optional outfile argument creates a postscript file rather than printing to the screen. There are also optional arguments which can specify the range of latitudes and longitudes to be plotted. A list of available projections can be found in deproject.pro. If you wish to add a projection to any of the programs, simply use the existing projections as a template.
2. map_random.pro, which throws random points on the globe and computes the statistics found in the Goldberg & Gott paper. The calling procedure is:
   map_random,projection[,npts=npts][,latmax=latmax][,latmin=latmin]
   By default, we use 30000 points and select points randomly over the whole globe.

Who Are We?

• Dave Goldberg, Assistant Professor of Physics at Drexel University
• J. Richard Gott III, Professor of Astrophysics at Princeton University

Rogues Gallery

Eckert IV [ps of lat/lon grid]	Equirectangular [ps of lat/lon grid]	Gall-Peters [ps of lat/lon grid]	Gnomonic Cube
Gott [ps of lat/lon grid]	Gott-Mugnolo Elliptical	Gott-Mugnolo [ps of lat/lon grid]	Hammer [ps of lat/lon grid]
Kavrayskiy VII [ps of lat/lon grid]	Mercator [ps of lat/lon grid]	Mollweide [ps of lat/lon grid]	Winkel-Tripel [ps of lat/lon grid]

Quality Scores