Thursday, September 15th, 2011

I Object To Your Projections

Have you seen that episode of "The West Wing"? This is a Fun with Maps column, so you know which one I mean. The one where it's Big Block of Cheese Day, and CJ meets with the Organization of Cartographers for Social Equality? If that's not ringing any bells, here's what happens: the organization is promoting an alternative to the Mercator projection, that is, the version of the world map with which you're probably most familiar—it's the one that has Greenland as roughly the same size as Africa. The group would prefer the Peters projection be used, because it doesn't exaggerate the size of Europe and North America at the expense of non-Western nations, which appear proportionally a lot smaller the closer they are to the Equator.

Well, the organization might be fictional, but it’s an entirely real controversy. Arno Peters, a German historian, unveiled his eponymous map in 1973 with the goal of displaying all nations with accurate sizing. The Gall-Peters (sometimes just 'the Peters,' even though James Gall had created the same projection a full century before. Peters’ unwillingness to credit Gall’s contribution did not win him many fans in cartographic circles) preserves area all right, but the shapes of the continents are completely distorted, especially toward the poles.

Look at Russia—the area is technically correct, but the shape is elongated East-West and flattened. Closer to the equator, countries are drastically elongated North-South.

Map projections are just different ways of translating the dimensions of a globe onto a two dimensional surface. A sphere (or oblate spheriod, if you want to be fancy) can't be flattened without causing some kind of distortion, be it in scale, area, distance and/or direction. Distortions in scale mean that a linear inch from one section of the document may mean 1 mile in the middle while an inch measured from another place could be a completely different length on the ground. Distortions in area means that size is not consistent across the map, which is how you end up with Greenland and Africa looking like landmass equals. Distortions in distance are similar to scale, in a way. On a map made to conserve distance, you could place a dot on the map with a circle drawn around it, and all points on the circle would be equidistant from the original dot. The Azimuthal Equidistant is such a map. Look at it and see—the distance from the North Pole to Washington, DC is accurate. The distance from the North Pole to Warsaw, Poland is accurate. But the distance from Washington, DC to Warsaw, Poland? No way. Directional distortions have to be avoided on maps used for navigational purposes. On a map made to preserve direction, such as the Mercator, you can place your finger on Chignik, Alaska, and move it in a straight line south through the Pacific to hit Honolulu. But on something like the Goode homolosine, which doesn't preserve direction? That line would be kinda lumpy.

Projection methods fall into four classes.
• Cylindrical projections are basically what you’d get if you wrapped a flat surface around the globe. The globe being a globe and therefore round, the cylinder would only actually be tangent to the planet along a line of latitude, usually the equator. In the most direct form, longitude and latitude lines are at right angles to each other. Longitudinal meridians are all the same distance from from one another, while the latitude lines get further and further apart as you get farther from the equator. A transverse cylindrical projection switches the longitude and latitude lines, an oblique cylindrical projection uses any other line.

• Azimuthal, or planar projections use a point. Think of a flat surface held up to the planet: it would only touch at one point, right? Lat or long lines passing through the point would be represented as straight lines, all others would be circles.

• Conic projections, as you might be expecting at this point, show the spheroid surface of the planet as a cone—picture a globe in a dunce cap. Meridians will appear to converge at a point, usually the North or South Pole, while latitude lines will appear as parallel arcs.

• The fourth category is comprised of pretty much every map that doesn’t fall into any of the above three categories. This includes maps that simply aren’t projected, like hand-drawn guides or ones that don’t represent the earth, like star charts.

To figure out which map projection is right for you, channel your inner cartographer and combine a distortion you'd like to avoid with a method that will help you to avoid it.

The Mercator projection that we're familiar with was actually one of the first well-known map projections. It was developed in the 16th century by Gerardus Mercator, a Flemish cartographer and fan of floppy hats, in part to decrease shipwrecks, which can happen when you’re sailing without a trustworthy map. His Mercator projection is a straightforward cylindrical projection with the tangent at the equator, meaning that all latitude and longitude lines are at right angles to each other. The map preserves direction with rhumb lines and can be used for accurate navigation at sea. But the same grid that keeps direction standard distorts area and scale; see how, as mentioned before, Greenland and Africa look the same, right? Africa is, in actuality, 14 times larger than Greenland. The boxes formed by latitude and longitude lines are all technically the same size. But the stretching that occurs on his map closer to the poles greatly distorts the North-South distance as compared to the East-West distance, which stays the same regardless of where you are. You can see why this annoyed Arno Peters.

So you want a projection that preserves area? Good news! There are a lot of projections out there for you! And most of them have the helpful term “Equal-Area” right in the name. Let’s review some options: here are cylinder, azimuth, and conic projections that preserve area, published by the Swiss mathematician Johann Heinrich Lambert in 1772.

Lambert Equal Area Cylinder

….but look at that scale!

Lambert Equal Area Azimuth

… but look at those angles!

Lambert Equal Area Conic

… but look at those angles and that scale! Siiiiigh.

Okay, so let’s conserve distance. The Polar Azimuthal Equidistant projection, with which you may be familiar if you spend a lot of time staring at the UN emblem, uses the North Pole as the center point. Distance is correct, but you’ve thrown area out the window. I mean, is Greenland really the same size as New Zealand? But it is useful for some things, like airline routing.

What if you want to just have a general map? A nice, inoffensive map that distorts everything, but just a little bit and an equal amount across the board? In 1989, seven individual societies of professional cartographers gathered to discuss tactics for educating the public on the effects of map distortion. They put forth a resolution to “strongly urge book and map publishers, the media and government agencies to cease using rectangular world maps for general purposes or artistic displays. Such maps promote serious, erroneous conceptions by severely distorting large sections of the world, by showing the round Earth as having straight edges and sharp corners, by representing most distances and direct routes incorrectly and by portraying the circular coordinate system as a squared grid. The most widely displayed rectangular world map is the Mercator (that is, a navigational diagram devised for nautical charts), but other rectangular world maps proposed as replacements for the Mercator also display a greatly distorted image of the spherical Earth.”

So what should we use instead of the wildly distorted Mercator and Peters? The Robinson Projection was accepted by academics as an agreeable middleground map, one that distorted everything a little instead of preserving one aspect at the expense of all the others. Following the cartographer societies’ lead, the National Geographic Society used the Robinson for official maps from 1989 until 1998, when it was replaced with the Winkle Tripel, generally thought to be a more attractive and balanced view of the earth. I think it looks pinched, but my formative map years were spent gazing at the Robinson.

These aren’t the only ways to show the world, of course, there are hundreds of projections out there. Here are three notable for their nonconformity:

Buckminster Fuller’s Dymaxion map, a polyhedron that eliminates the cardinal directions.

The Peirce Quincunial, where the equator is a square.

The Werner, which, awwwww.

And finally a gem, unearthed by Strange Maps, from the woefully short-lived Globehead! Journal of Extreme Cartography: Catherine Reeves’ Equinational Projection, which allocates each nation the exact same amount of space by reducing them all, equally, to a simple square. If only it were that easy…

Victoria Johnson is a cartographer and this is her Tumblr.

Map images courtesy of Wikipedia.

29 Comments / Post A Comment

Neopythia (#353)

That was my favorite scene in the entirety of The West Wing. I'm completely fascinated by it and was disappointed to discover that it was a fictional organization. Thanks for this.

En Vague (#82)

CJ: Yeah, but you can't do that.

Cartographer: Why not?

CJ: Cause it's freaking me out.

Daniel Sargeant (#7,340)

Isn't Nick Offerman in that episode? Promoting a highway for wolves?

Ophelia (#75,576)

@Daniel Sargeant Hah, I came down here to say the same thing! "…a…wolf highway?"

statistics_lie (#14,052)

@Daniel Sargeant Different episodes. There are two that take place on Big Block of Cheese Day: "Somebody's Going to Emergency, Somebody's Going to Jail" from season 2 (with the cartographers) and "The Crackpots and These Women" from the first season (with the wolf highway, and also, SUCH A GOOD EPISODE). Ugh I am always this person with the West Wing shit Jesus God

@statistics_lie Um, can we get married?

Sarcastro (#328)

That was fun!

turd_sandwich (#5,660)

I just wanna be able to see how the continents used to fit together. I'm aware that Greenland isn't the size of Africa.

Niko Bellic (#1,312)

Just use Google Earth, damn it! I can't tell you how much (supposedly educated) people piss me off when they ask questions like: "why is my NYC – Paris flight not taking a straight line?" while looking at the damn flat map of the world. To make it worse, they satisfy themselves with an explanation that it's because it's not safe to fly over that big blue stuff, so the plane has to stick closer to the safety of the Greenland ice.

The upside-down Australia-centric map is another refreshing change, but it still has the warping problems at the poles.

Bittersweet (#765)

@John McGraw@twitter: The upside-down map is cool but always gives me a bit of vertigo.

(Love these map articles – more, please!)

Mr. B (#10,093)

O.K., but is there a way to keep the North Pole from looking like a cat's asshole?

riggssm (#760)

Toby: It's "Throw Open Our Office Doors To People Who Want To Discuss Things That We Could Care Less About Day."

It was great the way Sorkin worked random factoids into two whole episodes. (The other one was where C.J. busted up laughing about the Wolves-Only highway.)

Daniel Sargeant (#7,340)

@riggssm Thought that might be it. And now I'm headed down a TV-Links k-hole.

stephen (#14,204)

Just buy a globe already.

E (#14,552)

Oh god. I just realized, today I discussed my plans to plot a more user friendly map for public release with the GIS section after a meeting where I committed to help assist in writing road design policy to ASSIST WILDERNESS CROSSINGS. WHOA. Mind blown.

katekatekateyeah (#13,507)

Good to see I'm not the only one psyched for the new Ken Jennings book, Maphead. Or still missing The West Wing.

Alibi Jones (#82,103)

This was great, thanks!

It reminded me of something I saw a few years ago, in which a computer scientist developed a projection (he called it myriahedral projection) that cuts the sphere in different messy chaotic-looking ways to get less distortion:

thejcar (#5,689)

"Little thing called team morale, Josh… [Enters room] All right! Shut the hell up, everybody. I've fired more people than you before breakfast."

LaFabuliste (#11,706)

"So, you're probably wondering what this has to do with social equality."

"No, I'm wondering where France really is."

ladykatie (#82,550)

"Longitudinal meridians are all the same distance from from one another, while the latitude lines get further and further apart as you get farther from the equator."

This definition has me thoroughly confuzzled. Help me because I love these.

Vicky (#7,168)

@ladykatie Okay! On that map, all of the lines of longitude on that particular map are spaced exactly the same. If there is an inch of map visible between 0 degrees W and 10 degrees W, there will be an inch of map visible between 70 deg W and 80 deg W. But with longitude, while there may be an inch of map visible between 0 deg N and 10 deg N, there will be an inch and a half of map visible between 70 deg N and 80 deg N even though it covers the same amount of space on the Earth. It's stretched out a bit.

TimChuma (#9,158)

It's not really a projection, but there is also a map that has Australia and Antarctica at the "top" of the map justifying that there shouldn't really be an "up" on the map as the Earth is a globe. Also Tasmania is missing from quite a few maps or at least the ones used on the news.

Absolutely love this feature.

SeanP (#4,058)

Being in the military you learn a lot about this topic. For long distance sea voyages, you can't just draw a straight line on your Mercator chart, because you can't plot a great circle that way. But you can't navigate off a great circle chart because it doesn't conserve direction. So in practice you lay out a track on the great circle chart, then pick off segments a few days in length and transfer them to the Mercator chart. You wind up with an approximation of a great circle route that doesn't require you to continuously vary your heading.

Also, gunfire/artillery operations require very precise positioning, so the military came up with UTM – Universal Transverse Mercator. In simplified terms it's a Mercator projection with the cylinder tangent to the earth right in the center of the area of interest. So wherever you happen to be operating, you get a projection that has the least distortion right where you are.

4th and 26 (#10,443)

@SeanP I used to love telling folks about the military origins of the UTM when I was the GIS dude at a real estate development company. A lot of spatial data from the feds is in that format- wetlands, floodplains, etc., and the site plan guys hated it because it's in meters, not feet. I told them that the the data is in UTM in case the Army needed to blow up our Walmart Supercenter (that was my hope). Now I sell bicycles and can sleep at night.

hypnosifl (#9,470)

For some super map nerdery, I found this paper which uses some fancy mathematics to try to find which of various proposed projections (including some new ones invented by the authors) minimizes the error in distance between any two randomly chosen points on a globe (the winner is this guy, but I guess there are some tradeoffs because it seriously distorts the shape of Australia, and I don't know how well it does on areas as opposed to distances…they do include some equal-area projections in the paper which do well with distances though)

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