How do you take a sphere and best project it onto a flat surface so that you can carry it around on a sheet of paper, or, better yet, view it on your monitor. The answer certainly depends on what you want to use your map for. Historically, people have used the 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.