Get to Know a Projection: Lambert Conformal Conic

What the heck are projections, anyway? First of all, projections aren’t maps, even though most maps have projections. It’s a little weird, but think about it like this: If every point on a globe has a coordinate, then the projection is the formula that tells all those points where they will move when that globe is flattened – or projected. The operation never goes perfectly, and the final map is always a bit stretched and distorted.
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Mike Bostock.

The America you see above is a proud country. Its northwestern border is arched like a subtly confident grin. The scrawl from Minnesota to Maine looks like a doctor's confident signature. East and west masses aren't so much symmetrical as they are complimentary – California's barrel chested coast feels like a good match to Chesapeake's washboard bays. Like most maps, this one has a projection, and Lambert's conformal conic is the name of the equation behind, below, and between every flattened shape on this beautiful America. It's not perfect – no projection ever is – but for a place like America, it's ideal.

What the heck are projections, anyway? First of all, projections aren't maps, even though most maps have projections. It's a little weird, but think about it like this: If every point on a globe has a coordinate, then the projection is the formula that tells all those points where they will move when that globe is flattened – or projected. The operation never goes perfectly, and the final map is always a bit stretched and distorted.

Johann Heinrich Lambert, the brains behind the projection that bears his name, was a cherubic, Swiss polymath born in 1728. Drawn to self-improvement at an early age, he wrote a manifesto when he was 14 in which he swore to use scientific reasoning to make the right life choices. Unlike most teenage proclamations in history, Lambert's stuck, and it even worked. He chose to study light, and discovered how it was absorbed by matter. He chose to study astronomy, and made accurate predictions about how star systems behave. He chose to study mathematics, and wrote the first exhaustive proof that pi was both infinite and non-repeating. Near the end of his life, he chose to study geography, and in 1772 published a set of seven new map projections.

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He gave his tome the gasp-inducing title "Notes and Comments on the Composition of Terrestrial and Celestial Maps" (it's even more impressive in the original German: Anmerkungen und Zusätze zur Entwerfung der Land- und Himmelscharten). Of the seven, two were interesting, two were ugly, and three revolutionized cartography. Prior to Lambert, projections were designed with the whole globe in mind. We can't know for sure if Lambert was thinking of different scales, but those three projections were optimized for displaying the three basic shapes that areas take. The conformal conic works best for areas, like the United States, that are wider than they are tall. The other two -- the transverse Mercator and the azimuthal equal-area -- work best on areas taller than they are wide, and polar regions.

A conic projection, like the one above, is what you'd get if you rolled a piece of paper up like a party hat, placed it on the planet's brow, and projected up into it (There is no evidence Lambert actually did this, and it's debatable whether he was much of a partier).

Regional maps are large enough to have some distortions, but small enough to keep those distortions contained. The word 'conformal' means this the Lambert preserves the shapes of objects at the expense of their relative sizes (an equal-area map does the opposite). On the conformal conic, the north pole is pinched while the southern hemisphere extends infinitely, and usually gets cut off somewhere around Rio de Janiero.

It's an interesting way to look at the world (see below), but not very useful unless you tighten the focus. This is what makes it the best projection for the United States.

Actually, it's not the only thing. As I mentioned above, projections are geometric formulas. In the Lambert's case, that means the cone doesn't need to just rest on the globe's surface, it can pass through it. This means it can have two standard parallels, two lines where the map is perfect. Since all distortions radiate away from standard parallels, if you are smart about where you put your standard parallels, you can drastically minimize your distortions. For a U.S. map, these lines are 33 and 45 degrees North.

This makes for a map that is closer to 3-D reality, and ultimately a map that is more beautiful. Look, for example, at how Maine sticks out like a mako shark's fin on the map above. It looks like the highest point in the country, but did you know that Maine's northern point is actually over 100 miles lower than Minnesota's border with Canada? Compare it with a Mercator projection, and the entire eastern U.S. seems to shrivel and sag in contrast to the beefy west. Perhaps most uniquely, on a Lambert the entire country rests on an unlikely fulcrum of balance: Texas.

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