Perfecting the World Map?
/On Ideas and Measures of "Distortion" and the Recent Claims Made for a Two-Sided Hemispheric Projection
To continue the criticism of the physplaining Princeton astrophysicists’ take on map projections, I have looked at the two previous papers in Cartographica by Richard Gott and his colleagues, especially Goldberg and Gott (2007) in which they introduce the metric on which their claims rest. I am not concerned with the mathematics — I’m not qualified — but with why they think that paying attention to distance errors and what they call “boundary cuts” is important. These are the two factors, in addition to the general metric, that underpin their claim to have created the most accurate map to date. (See my previous post.)
1. The Unobjectionable Treatment of Map Projection Distortions by Goldberg and Gott (2007)
A persistent element of any general text on map projections is that each projection is defined in part by how it “distorts” the surface of the earth in transforming it from a curved surface in three dimensions to a plane in two. One set of projections do not distort angles and shapes; they are variably called as eumorphic, conformal, or isotropic projections. Another set do not alter the relative areas of defined entities; they are generally called equal-area. A further set is often defined, but not always, in which the scale factor along many meridians or parallels, or other great or lesser circles, is constant; this is not to say that scale factor is consistent along any and ever circle on such “equidistant” map projections, just on a defined subset. Another set are those projections designed in the nineteenth and twentieth centuries to balance or in some other way minimize shape and areal distortions; these are called “compromise” or “minimum-error” projections. And then there are a great many other map projections that have some other special properties, or none at all.
(Important: Goldberg and Gott [2007] quite properly refer to lines across the ellipsoid as geodesics; given that for most purposes the difference in mapping the entire earth approximated as an ellipsoid or approximated as a sphere is negligible, I’ll continue using standard geographical terms like “great circle” intended for the earth approximated as a sphere.)
Goldberg and Gott (2007) extended the usual discussion of conformality and equal-areaness to also consider further kinds of distortion that are evident “on continental scales” and on world maps. Specifically, they defined and gave examples of “flexion” (or the degree of bending of great circles from the straight line of the great circle across the earth’s surface) and of “skewness” which they gloss as “lopsidedness” (i.e., scale factors are not equal to either side of a point, so that the weight of distortion at a point leans one way or another). Flexion and skewness are interesting concepts and are mathematically defined. So far, so good. And, actually, the authors’ application of these factors to modify the usual Tissot indictrix generates revealing nuances and precision in the pattern of distortion on a map projection.
2. Problems with Goldberg and Gott (2007)
Things start to get weird when Goldberg and Gott (2007, §6) pursue a numerical analysis to define the “overall quality” of any given projection. Mercator’s projection makes a good fit close the equator (or central meridian in transverse aspect, which is one reason it is used for the multiply projected zones of UTM), so that the fact that it goes off to infinity at the poles does not make it “infinitely bad” (p. 315). What then, the author’s ask is the overall quality of the projection as compared to others?
They create a single metric in two steps. First, they randomly selected 30,000 points across the globe and calculated the root mean square (RMS) of all their isotropic and areal distortions (I and A) and also the indices for flexion and skewness (F and S). But then they also throw in two other undefined factors:
D, “corresponding to distance errors,” which is to say the RMS of the ratio of map to globe distances along the great circles between the pairs of points;
B, “corresponding to the average number of map boundary cuts crossed by the shortest geodesic connecting a random pair of points” which is calculated as B = Lb/4π where “Lb is the total length (in radians) of the boundary cuts” It is not explained how this calculation gives the specified correspondence.
They presented these metrics in a big table for a number of common map projections:
Science, amirite?
Finally, Goldberg and Gott (2007, 317) combine all the summary metrics for the different kinds of distortion, I, A, F, S, D, B to create a single metric, what the recent pre-pub paper called the “fidelity metric” (Gott, Goldberg, Vanderbei 2021). How did they weight these different values in combining them? Equally: each kind of error is as important as the rest.
So, the problems:
It is unexplained why “distance error” is important to consider. Although, Gott, Mugnolo, and Colley (2007) state as axiomatic that “Maps convey important information about distances between pairs of points. It is therefore desirable to minimize the errors made in representing distances between pairs of points on maps.”
It is unexplained why “boundary cuts” are a significant kind of error.
Nor is it explained how the provided formula matches the verbal definition. As stated, B seems to require the calculation of intersections cutting boundaries/edges along many great circles and then taking the RMS.
And the big one: the equal weighting of the contributory factors is neither explained nor justified. At all. Just a brief statement that someone else made such a single metric, so we can too. No analysis of how that previous scholar had constructed his metric, how it succeeded, and how it failed (because why else do they present their own?).
If flexion and skewness are derivatives of isotropy and areal distortions, as the authors state right up front, then they are dependent variables and have no place in the (un)weighted metric alongside the independent variables I and A.
Problems 1–3, at minimum, should in my book have been occasion for revisions. Problem 4 is such a basic flaw that the paper should be rejected, or the entire discussion and conclusions (together §6) should be cropped in its entirety. And, if I am correct, problem 5 is just bad and a sign the journal editors should have just run away from the paper. Screaming.
All of this is to say:
the metric of quality
that these authors present
on which they have determined the absolute quality (from 6 to 0 [perfect]) of any map projection and
by which they have exalted their own new map as the best thing since sliced bread,
features six parameters
two of which are of undefined relevancy (D and B)
two more of which are probably dependent and as such have no place in the metric (F and S)
that are combined by weighting that
is simply unjustified and unexplained and
has been made up at the drop of a hat to give equal weight to the spurious parameters.
To base any comparison of the quality or value of different map projections through this metric is a load of fetid dingoes’ kidneys. (To quote the late, great Douglas Adams.) It simply cannot be taken seriously as a mathematical exercise.
To then base claims of the amazingly quality of one’s own map projection by relying primarily on the disputable parameters D and B, especially when B is dropped to zero by simply declaring the interruption to be a fold, is mathematically sloppy, utterly self-serving, and borders on the disreputable.
3. Why Bother?
And we haven’t even started on issues of why world map projections should be subjected to a competition for “the best.” If you want the best world map, when “best” is defined strictly by geometrical parameters, then buy a globe (and squish it a bit to make it an ellipsoid). If you want a flat world map, then accept that you are engaging in one of several different spatial discourses in which geometrical accuracy and mathematical principles are largely irrelevant, beyond the basic issue of visual propriety (using equal-area projections in analytical mapping) and thoroughly idiosyncratic reactions to shapes. World mapping is ineluctably a social and cultural act and social and cultural considerations should take precedence.
A map projection manifests in two ways: as the graphic network of meridians and parallels, and as a set of two or more mathematical formulae. Under the modern idealization of cartography, discussion of the projections of world maps means discussion of their formulae and their geometry and their “accuracy.” These are indeed matters of absolute importance to certain communities of mapping, notably engineers seeking to modify the earth’s surface contours, artillerymen seeking to lob shells against an enemy, or hikers seeking to know how far they have yet to walk to get to food, drink, and rest. But they are not important at all in world mapping. World maps do not just denote the earth’s features, they connote “the world.” Treating world maps as anything else is a fools errand.
References
Goldberg, David M., and J. Richard Gott, III. 2007. “Flexion and Skewness in Map Projections of the Earth.” Cartographica 42, no. 4: 297–318.
Gott, J. Richard, III, David M. Goldberg, and Robert J. Vanderbei. 2021. “Flat Maps that Improve on the Winkel Tripel.” Instrumentation and Methods in Astrophysics. Pre-publication submission, 15 February 2021. https://arxiv.org/abs/2102.08176v1.
Gott, J. Richard, III, Charles Mugnolo, and Wesley N. Colley. 2007. “Map Projections Minimizing Distance Errors.” Cartographica 42, no. 3: 219–34.
