So, this map is an equidistant projection, but what about direction? Well, it turns out that this projection type is also known, not only as a planar projection but as an azimuthal projection. Azimuths are related to direction, and so that gives us something we can talk about in terms of, like why is it called an azimuthal projection. What does that mean? How does that relate to direction? Well, let's have a look. So, azimuths relate to direction. If I try to tell somebody what direction they need to go in to get from point A to point B, that has to be in reference to some standard reference line. I don't want to call it a standard line because that's something different. So, that's a line connecting A to B, but what's the direction of that line? Well, typically what we would do is, talk about the angle of that line in relation to a line that goes true north. So, in this case, that's a 45-degree angle. We would say that that is an azimuth of 45 degrees. In other words, if I'm at point A and I want to get to point B, I could take a direction or azimuth of 45 degrees and that would get me there. Now, this is generally true for small areas. I mean, if you're taking a hike or something like that, then that would be correct. But what happens when we talk about azimuths in directions on a map projection for a larger area. This brings us to a discussion of things like the great circle routes and navigation. So, let's just talk about that for a minute and how this relates to projections and directions. So, the great circle route, between any two points, is a line that goes along a plane and that plane cuts through the center of the earth. Really, that's a complicated way of saying, it's the shortest distance between two points over the surface of the earth following the curvature of the earth, if you want. So, if I wanted to get from New York to London, whether it's by plane or by boat, we would follow the great circle route in order to get there as the shortest distance. So, this line is actually a curved line over the curvature of the earth, and that's actually the shortest distance between these two points, the great circle route. Just to visualize, this a bit better. This is the same line from a different angle. So, this is New York to London. So, that curved line, I'm hoping you're seeing this, is that's the great circle route is that's actually following the curvature of the earth. That's the most direct route you can take to get from one to another. Okay? Now, that maybe the shortest route, but it can actually be a little more complicated in terms of navigation to follow that great circle route. If we zoom in a little bit here and look at the same line, so this is from New York to London. What you'll notice is that, if you were to describe the azimuth or in other words, the direction that you have to take, remember the definition of an azimuth is that, it's the angle in relation to a line that goes directly north. So, these red lines, all point directly North. Isn't that funny. It looks weird. Doesn't it? But if you remember, these are meridians. Meridians all travel towards the North Pole. So, if you're looking at these lines, you're seeing that that's an angle, that's an azimuth between the route that we're taking and true north. This is an angle between the route we're taking in true north and so on. But the thing is, that these are all different angles, because the lines are converging. That's not a mistake. That's not something that's wrong. It's just a fact of life that these meridians will all curve. So, the angles that we're using to navigate, if I was to say sailing ship and trying to figure out what bearing to go on to what azimuth to take to navigate. I'd have to keep shifting that or adjusting it, because these angles, these azimuths are all different. Okay. So why am I telling you all this? Well, it has to do with azimuths, directions, and projections as you'll see. So, along comes this guy named Gerardus Mercator, and he invented a long time ago a map projection that solved this problem, that made navigation a lot easier. How that worked was, is that he made it so that the azimuths were constant. But I'm getting ahead of myself. Let's just have a look at this. On the Mercator projection, this is the great circle route. It is not, on this projection it's not a straight line, but that is actually the shortest distance between these two points. It's still the same line I was showing you a minute ago, but it just looks different. That's because the projection is actually curved out. Let me show you what I mean. So, here's that line, as a straight-line distance that's the great circle route, between the two points using what is meant to look like a reference globe, with no distortion and it just watch. If we take that same line and use the Mercator projection, remember that it's stretching the lines out. There no longer meeting at the poles they're becoming parallel to one another. So, that line is being bent or curved upwards in the way that it's being shown. The actual distance on the ground isn't changing. That still the great circle route. It's still the shortest distance, but the way it's being shown in the projection has made it look like it's a curved line. Okay. So, if that's what's happening with the Mercator projection which I'm showing here, what if you do draw a straight line between two points? Your brain wants to think that that's the shortest distance. But it's actually not. It's a rhumb line, as it's called. So that's the term down here, R-H-U-M-B. That's a line of constant bearing that you can draw on the curvature of the earth, or on a projection. So, you'll see here well, actually I've got it, like this. But now, the azimuths along this rhumb line are constant. So, if you wanted to sail a ship from New York to London, theoretically you could set it to that one direction, that one azimuth and follow that and you would get where you want it to go. You wouldn't have to worry about adjusting it along the great circle route. It would be a much simpler way of navigating. So, that's why the Mercator projection became so popular with sailors at the time, is that this hadn't been done before and it simplified navigation quite a bit. You can just set it and forget it if you will. So, the price that's paid for that though is that the rhumb line is longer than the great circle routes. So, yes it's simpler, but it means you have to travel a bit farther to get where you want to go. So, there's a price there. So, there's the great circle route, that's the shortest distance, believe it or not, and then the rhumb line is the simplest route in terms of the direction but it's a longer distance. If you don't believe me, let's have a look at it. Here's the great circle route in red on our globe and the rhumb line is the one of constant direction or azimuth in yellow. Then, when we put those on the Mercator projection. Are straight red line becomes a curved line here, and I actually did this with the software. It was calculated correctly. There's nothing that's wrong. It just looks different depending on what projection you're looking at. So, it's important when you're looking at these things. One of the reasons I'm telling you what this is, I think it's interesting and it's useful in terms of just understanding how the navigation has evolved and how maps are made. But it also I think, really impresses upon you. I'm hoping that if you understand what's happening here and why these lines look the way they do, then you also understand how projections work, distortion works, and how the reference globe works. It's all tied together. These concepts are all related to one another. To me, it's a nice way of encapsulating these ideas. If you can explain this to somebody, if you it makes sense to you, then it means that you've understood a lot of the stuff that we're talking about in this whole section on projections. One note about the Mercator projection and navigation is, I said earlier that if you follow the rhumb line it will be simpler, but it was a longer distance. So, a strategy that sailors would actually use is to approximate the great circle distance by breaking it into smaller rhumb lines. So, you can see here that this is a rhumb line, this is a rhumb line, and this is a rhumb line. So, they would only have to set their campass to an azimuth three times. So, 1,2,3, and so you'd get the benefits of not having to constantly shift the azimuth, and you'd still be able to cut down on your distance. Now, whether they did it three times or five or whatever, that's up to them. But you get the idea that this is a strategy to be able to combine a bit of the best of both. You're getting the navigational simplicity, but still being able to approximate a great circle route.
