Tissot Indicatrix in order to provide guidance. The method is simple: Imagine painting small circles on the earth (or any other planet -- above, we project the surface of Jupiter), and project the surface of the earth (and the circles) onto the map. Some applications will require that all of the circles still be circular (conformal projections). Some will require that all circles be of equal area. No map projection can meet both criteria.
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.
Decide for yourself. Wes Colley has prepared an Eckert IV projection of the WMAP 3rd year results:
preprint, you may want to look at this stuff for yourself. We are therefore making our IDL code available to all who want it.
There are two main programs
In each case, we made our measures uniform on the globe. The specific metrics are discussed in our paper.