What is a map projection anyway? It's a way of transforming our 3-dimensional Earth to a 2-dimensional map. It's usually done in a mathematical way with an equation and calculations, but for us I'm really going to stick to a conceptual level for this section. But, imagine that you're trying to find ways to transfer locations from our 3-dimensional Earth to our 2-dimensional map. When you're making a map you have to choose the map scale that will be most useful for your particular map. So, it might be a large map scale like you would use for a neighborhood. Maybe it's a smaller map scale for a country or maybe for the entire world. But, whatever scale it is you have to choose that map scale and that's the beginning of our process of trying to transform our 3D Earth into our 2D map. By the way, I'll just mention here that this is not a process you have to go through manually or that you have to do the calculations. The software will do this for you, but I think it's important for you to understand conceptually what's happening as it's doing this. In this case, I've chosen a map scale of 1:100,000,000. It could be whatever scale you want, this is just an example. But, let's say that I wanted to make a map at a scale of 1:100,000,000. So, first imagine that we're going to take the full size Earth and we're going to shrink it down to the scale of the map that we want to make. So, here's our full-sized Earth. I'm going to shrink that down, so they imagined that it's still a 3-dimensional Earth. Say think of it as like just a globe you buy in a shop or whatever, but that has been sized to the same map scale that I want my finished map to look like. So, we still have a 3-dimensional version of the Earth and this is referred to as a Reference Globe. The scale that we're choosing to make our map at is the principal scale. So, for me here that scale is one to 1:100,000,000, but if it was for a neighborhood, it might be 1:10,000, or for a country, it might be say something like 1:30,000,000. So, don't get too hung up on this, it's not that this is an exact value. The fact is that we're shrinking the Earth. It's a reference globe that's set to the scale that we want to make our map out. That's referred to as the principal scale. By the way, that Earth that we're starting from could be the Earth that we're treating as a perfect sphere or it could be the Earth treated as an ellipsoid. It will have an effect on how the calculations are done for the projection. Again, this isn't something that you have to do yourself, but it's something to be aware of. So, I'm just letting you know that either one of those is an option. So, now we have our hypothetical 3-dimensional reference globe that's been shrunk down to our principal scale. So, the principal scale is a ratio or the way we calculate this is it's the reference globe radius over the Earth's globe radius. So, the Earth's globe radius in this case for the scale that I'm interested in would be 6.378 centimeters. This is the actual radius of the full, real-sized earth, and so when you divide those two, this is the number you get, which is equivalent to the scale that I want my map to be at, which is the principal scale. So, this is just a way of explaining that idea of principal scale. So, then the software, whoever is doing this takes the principal scale, looks at the points on the map that you want to create or points on the globe, sorry, and then transfers them to the 2-dimensional map. So, every single location that you want to map will go from this 3-dimensional version down to the 2-dimensional version. Probably already you're starting to realize that there's going to be some distortion that takes place. Here's Greenland again you can see that it's quite large in relation to Africa, which is kind of crazy. If I really want to blow your mind, if you think of the North Pole as being a point on this particular projection that's being stretched out as a line all the way across the top of that map. Crazy isn't it? So, all I'm trying to point out there is that already we're seeing that some distortion is taking place by doing this process of transfer from 3D to 2D. So, now we have our reference globe at the principal scale of 1:100,000,000 and we have our flat 2-dimensional version of our map that the same scale. Another way to think about distortion is these distances here. So, the Equator on our globe is the same distance as the equator on the map. But, this distance up here is now stretched out to be the same distance as the equator. So, obviously, on our globe, this distance is less than this distance. But, with this particular map projection these distances are now the same length. So, this version or this line that's closer to the North Pole has been distorted, it's been stretched out to match the distance at the equator. This isn't the same for every map projection, this is just for this particular map projection. Again, I'm just trying to show you how these distortions can take place, and to start you to visualize what's happening. When we flatten our 3D globe onto our 2D surface, something's going to give. If we imagine this as a cross-section of a 3-dimensional globe, or a reference globe, and then we've got our flat sheet of paper that were trying to transfer things onto. You'll notice that this point crosses the two and this point crosses the two. But, look what happens here is you have an actual distance that's over there the curved part of the Earth that then has to fit onto the map distance. In this case, that's going to actually be shrunk down or distorted in some way. So, just to make sure that's clear, this distance is longer than this distance, but it's the same point A and point B that you're measuring distances between. So, our real distance is now in this case being shrunk down to a shorter distance in order to fit it on the map. There's lots of different ways to do this. There isn't necessarily one way, but again I'm just trying to give you ways of imagining what's happening here. There's a great quote from a book called How to Lie with Maps, ''Not only is it easy to lie with maps, it's essential. To portray meaningful relationships for a complex three-dimensional world on a flat sheet of paper or video screen, a map must distort reality.'' I love the way that he puts that because what he's really emphasizing there is that anytime you make a 2-dimensional map from a 3-dimensional world, there will be distortion. How to Lie with Maps is tongue and cheek. That's the point he's trying to make is that every map is a lie and that's another quote from him, I think, is that you have to distort things. It's not to say that it's necessarily wrong, it's that it can never be a true representation of the entire planet because of the distortion and the squashing and the plotting that's taking place.
