Curvature in Map Projections

Links

Map Distortions
All about Maps Dave Goldberg-Related Stuff

Intro

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.

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:

Eckert IV

Mollweide

Our Code

While you should certainly feel free to simply peruse our results and look at our maps in our 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

  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?

Incidentally, Rich has designed a very nice Conformal map of the universe. You should check it out!

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

In our paper, we measure a number of different qualities for maps: How well area is preserved (A), how much ellipticity distortion there is (E), how much flexion (F) and skewness (S) there is, the variation in distance errors (D), and how many boundaries a randomly selected pair of points on the globe will be separated by.

In each case, we made our measures uniform on the globe. The specific metrics are discussed in our paper.

Projection EAFSDB
Non Interrupted Projections
Azimuthal Equidistant 0.87 0.60 1.0 0.57 0.356 0
Gott-Mugnolo 1.2 0.20 1.0 0.59 0.341 0
Lambert Azimuthal 1.4 0 1.0 2.1 0.343 0
Stereographic 0 2.0 1.0 1.0 0.714 0
1 180 deg. Boundary Cut
Breisemeister 0.79 0 0.81 0.42 0.372 0.25
Eckert IV 0.70 0 0.75 0.55 0.390 0.25
Eckert VI 0.73 0 0.82 0.61 0.385 0.25
Equirectangular 0.51 0.41 0.64 0.60 0.449 0.25
Gall-Peters 0.82 0 0.76 0.69 0.390 0.25
Gall Stereographic 0.28 0.54 0.67 0.52 0.420 0.25
Gott Elliptical 0.86 0 0.85 0.44 0.365 0.25
Gott-Mugnolo Elliptical 0.90 0 0.82 0.43 0.348 0.25
Hammer 0.81 0 0.82 0.46 0.388 0.25
Kavrayskiy VII 0.45 0.31 0.69 0.41 0.405 0.25
Lagrange 0 0.73 0.53 0.53 0.432 0.25
Lambert Conic 0 1.0 0.67 0.67 0.460 0.25
Mercator 0 0.84 0.64 0.64 0.440 0.25
Miller 0.25 0.61 0.62 0.60 0.439 0.25
Mollweide 0.76 0 0.81 0.54 0.390 0.25
Polyconic 0.79 0.49 0.92 0.44 0.364 0.25
Sinusoidal 0.94 0 0.84 0.68 0.407 0.25
Winkel-Tripel 0.49 0.22 0.74 0.34 0.374 0.25
Winkel-Tripel (Times) 0.48 0.24 0.71 0.373 0.39 0.25
You can find a Romanian translation of this page here.