Cropping the Mercator Projection

Mercator_projection_SW

The Mercator projection is the map people love to hate. The West Wing humorously laid out the case in one of their “big block of cheese” episodes, and the complaints generally fall on the issue of distortion: things far from the equator are stretched more than things near the equator. Greenland appears to be the same size as Africa, but is actually about 14 times smaller.

One thing that people generally don’t comment on is the fact that a Mercator world map never actually shows the entire world. It’s not uncommon, like the image above, to cut off the northern latitudes at 80 or 82 degrees, leaving off the northernmost parts of Canada, Russia, and Greenland, and to slice Antarctica at a point that just barely leaves in all the coastline. Even Google Maps – yes, Google Maps, like most online mapping services, uses a form of Mercator – cuts off at right around 85 degrees (although the precise choice is for a secondary cool reason I’ll mention below).

If you stop and think about it, the cutoff makes sense. On the Earth, the lines of longitude get closer and closer as you move away from the equator until they finally meet at the poles. This might lead you to think that the reasoning is simply that the distortion would be so extreme if you showed the poles, that it’s not worth including on a map. Indeed, the distortion will get worse and worse as you keep moving north or south. David Swart made an image to help you get an idea of what it might look like if you kept going south (see his paper on the subject here).

W0YuyC6

You get to the point where you can make out individual arms of a snowflake resting on the pole. The Amundsen-Scott South Pole Station complex, just a few hundred meters across, is as big as Asia. But even this image doesn’t make it to the pole itself, and the reality is pretty interesting: a Mercator map can never show the whole world. In fact, an uncropped Mercator map would extend infinitely long.

Explanation 1

globecylinder

Looking at the layman’s description of Mercator makes this clear. Take the globe and wrap it in a piece of paper to form a cylinder that touches at the equator. Shine a light in the middle of the sphere, trace what appears on the paper, and then unroll it. (Or, put another way, draw a line from the center of the Earth, through a point on the globe, and extend it until it hits the paper.) The reason the areas near the equator aren’t very distorted is pretty clear: they’re relatively close to the paper to begin with. But as you move away from the equator, the light has a lot of time to spread out between passing through the globe and finally reaching the paper.

As you get very close to the poles, though, the light has to travel very far, requiring quite a long piece of paper to pick them up. And the poles themselves? Their beams of light keep shooting up or down parallel to the paper, and never actually reach it.

Explanation 2: The Math

With Mercator, as with many projections, you can use two formulas – one for latitude, one for longitude – to go from geographic coordinates to x and y coordinates on your piece of paper: plug in your lat and long, and it spits out coordinates on an x and y graph for where to put it.

Mercator’s longitude formula is incredibly simple. If you use the Prime Meridian as your x-axis, the x coordinate is the same as the geographic longitude in radians. If you want to find the x coordinate on your map where 90 degrees east belongs, just convert it to radians (π/2 = 1.571). Longitudes west of the Prime Meridian are treated as negative numbers. This will give you a world map that spans from -π to π, which you can then adjust by multiplying by a constant to have a more convenient scale – just be sure you use the same constant for the y coordinates.

The y coordinate formula is a little more complicated, but not too bad:

latform

where θ is the latitude. Plug in 60 degrees north, and you get ln (tan (75)), which is 1.317. Similar to west longitude, south latitude is treated as negative. (Note: this is the formula if you assume the Earth is a sphere, which it isn’t, but it’s pretty darn close. Assuming the Earth is an ellipsoid, which is standard, adjusts the θ/2 term by a factor that works out to be less than 1%. It’s not super important for this discussion.)

One thing to point out is that latitude isn’t included in the x-coordinate formula, and longitude isn’t included in the y-coordinate formula. This is why all lines of latitude in Mercator are parallel to each other and all lines of longitude are parallel to each other – no matter what latitude you have, a longitude of 45 degrees will give you the same answer. That is not universal among all map projections.

But let’s break this formula down a little bit. On the outside is the natural logarithm function, which when graphed, looks like this:

natlog

 

The limit as x goes to zero approaches negative infinity, so a zero value would make our formula blow up. To figure out where that happens, we have to step inside the natural log, and look at the tangent function, which graphs like this:

tangraph

 

Tangent has a value of zero at multiples of 180 degrees (including 0), so this is where the natural log function would break. However, it is also undefined at 90 degrees – approaching it from the left goes to positive infinity, so a value there would also break our y coordinate function.

Going back to the full formula, then, and plugging in 90 degrees (North Pole) and -90 degrees (South Pole) confirms the first explanation. The North Pole gives us ln (tan (90)), and the tangent of 90 degrees is undefined, and the South Pole gives us ln (tan (0)), which is the same as ln (0), which is also undefined.

Additionally, we can keep plugging in numbers closer and closer to 90 or -90, and we will get increasing  and decreasing y coordinate values, approaching positive infinity in the north and negative infinity in the south. This is why something like David Swart’s image above is possible, and could even be extended as far down as we wanted to go.

Playing with the Formula

Since we can go as far up or down a map as we wanted to go, how big of a map would we need to see something very very tiny? How about an electron centered on the North Pole? To avoid a discussion of how to measure an electron’s radius, I’m just going to use the classical definition I found on Wikipedia: 2.8 x 10-15 m. Next, we need to compare that to the length of a degree of latitude. On a spherical assumption of the Earth, this is constant, and even in reality, it doesn’t vary a ton, unlike longitude. For a rough estimate, we’ll use 1.1 x 105 m. If we divide our electron radius by our latitude length, we can get the fraction of a degree that the edge of our election would be sitting at: 2.5 x 10-20 of a degree. Put another way, we’re looking at a longitude of 89.999999999999999999975 degrees north.

Plugging this into the y-coordinate formula gives us a value of just about 50. If we wanted a map that was one meter wide that cut off the map at electrons at the North and South Pole, then, we’d need a map that was about 16 meters high. The Swart map above gets just over halfway there in the southern direction.

One last note: I mentioned that Google uses a cutoff very close to 85 degrees for its online mapping; a more precise figure is about 85.0511. Why that? Let’s try plugging it in: ln (tan ( 87.5256 ) ) = 3.1416, or in other words, π. In order to make life easier for themselves, they went with a cutoff that makes their world map perfectly square.

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