We can observe distortion that's taking place by zooming in on one part of the world and just kind of seeing what's happening, and let's go to Greenland. It's a great example because there's a lot of distortion that takes place, especially when we look at something like the Mercator projection. So we have here what's meant to sort of look like what it would be on a reference globe. And you can see that we have the meridians that are converging towards the poles, as they should be. But then on the Mercator version of this, those same lines are not converging anymore. They're now parallel, so this is getting stretched out, right, instead of them converging like this, they become parallel. And so you're stretching, horizontally, that part of Greenland. So instead of focusing on something like Greenland or some other thing all the time. Why not use a circle as a way of being able to visualize what's happening to our distortion? And we can think of that in terms of things like scale factor as well. So we take our circles that we have on our reference globe, and we transfer those onto our projection. So on the reference globe, those circles are all the same size and the same shape. And then when we transfer them, we see what happens to them. Are they still the same size? Are they still the same shape? Something's gotta give, something's gotta be distorted. And so instead of focusing on things like Greenland, which are kind of hard to visualize. because you don't necessarily have this perfect image of what Greenland's supposed to look like all the time. You can just look at these circles and get a sense of, okay, at the equator, the circle is this size. But as we move towards the pole, it's still a circle, but it's getting larger. And that gives us an indication of what's happening with distortion. These circles that we're using are referred to as Tissot's indicatrices. So each circle is an indicatrix. These are the circles that are being drawn in the reference globe. But then the original way of imagining this, those circles are infinitely small. In other words, they have no dimensions, but of course, we have to make them larger, in terms of being able to see them on our map. So imagine though, that you're drawing these infinitely small circles on your reference globe. Which remember, has been shrunk down from full size down to the size we need for our principal scale. So on the globe, they're circles. When they're projected, they may or may not be these circles, and the sizes may have changed as well, so the size and/or the shape may be distorted. So for example, the Tissot's Indicatrix here for the cylindrical equal area map shows that we have circles at the equator. But as we move towards the poles, they're quite squashed. And they've become stretched in the horizontal direction and squished in the north south direction. So it turns out actually though that these circles have the same area as they do at the equator. So that circle in terms of it's actual coverage area is the same which is why it's referred to as a cylindrical equal area projection. As the areas are being maintained, in other words if you were measuring the areas of something on the Earth, like the size of Greenland. Now this is a bit of stretching things a little bit, because it's not exactly true, it's really only true for these small circles. But more or less, if you're measuring large areas on a map, those areas would be the same as if you were measuring them on the reference globe. So we go back to this idea of Greenland for a second and comparing it. Now I've reversed it so I've got the Mercator projection as the larger map, and the reference globe as the smaller map. And you can see that we have a scale factor that's greater than one in the horizontal direction. And that's because our meridians have been straightened out, okay? They've been, so that's been pushed over that way, that's been pushed over that way. We've stretched things. And if we stretch things, we've made them larger than they really are, the scale factor is going to be greater than one. However in the vertical direction, the opposite is happening as you can see kind if you kind of focus in here on Greenland. You can see that it's really squished vertically, and so that means that things are smaller vertically than they were on the reference globe. And so the scale factor there is actually going to be less than one. So we can have varying scale factor in the east-west direction versus the north-south direction. And we can use the Tissot's indicatrices to help us visualize that. So with the circles from the indicatrix, if we just have a look at one, what this has really meant to represent is the scale factor. So we have the scale factor in the horizontal or east-west direction, and a scale factor in the north-south direction. So we have an a and a b. And these represent scale factors or the amount of distortion that we have. So if area equals pi r squared, which is does. Which means that area equals pi times a times b, because we have a circle and a and b are both the radius. So then a equals 1, b equals 1, and a times b equals 1, as well. This is on the reference globe. Okay, all this is by the way, is just a way of quantifying our description of the distortion that's taking place. It's just kind of a way of making it more systematic or quantifiable, instead of just saying, there's more of it here, or less of it there. That's really what I'm going through right here. So this is on the original reference globe in three dimensions. Now if we take, for example, this map projection. And we see what happens to a circle at the equator versus at the poles. We can quantify the distortion that's taking place in terms of our Tissot's indicatrix. And the scale factor in the two different directions. So going back to the same idea, the same formula for area. What's happening is that our circles are being distorted, and so now, a is greater than one, b is less than one, but a times b still equals 1. So why's that important, what's the significance of that. What it means is that since a times b still equals 1, this map is equal area. In other words, the area of this yellow circle is the same as the area of the blue circle. And that indicates that the sizes of objects on a reference globe will be the same size on our projected version. The shape will be different, but the sizes will be correct. And how do we know that the shapes will be different? Because a is not equal to b anymore. And if a is not equal to b, remember on our circle, a equals b. Which makes sense because it's a circle. But as soon as a and b are different, it can't be a circle anymore. And if it's not a circle anymore, that means that the shape has been distorted. And if the shape has been distorted, then it can't be conformal. And conformal is just a word we use to say that, if a map is conformal, it means that the shapes have been maintained. And so Greenland still looks like Greenland or whatever. And so what you're seeing here is that we have, the areas are correct but the shapes are not correct. And this is just a way of saying that in a kind of a mathematical way. We can look at Tissot's Indicatrix in relation to the Mercator projection. So what we have here, as we can see that the circles are still circles, but the sizes are changing. So if we look at Greenland again, we can see that we have distortion in the east-west direction. And the scale factor is greater than one because the meridians no longer converge. And we have a scale factor greater than one in the vertical direction because we're getting the stretching taking place in the north-south direction. So with this Mercator projection, if we look at our Tissot's indicatrix again. Where we have A and B in the same formula for area, what's happening now is that the circles are actually stretching, but they're still staying as circles. So what this means is that a and b are both greater than one because they're bigger than they were on the reference globe, but they're equal to one another. So a times b equals something greater than one. So what this tells us is, since a times b does not equal one, then the area cannot be the same as what it was on the original circle. The area has to be either larger or smaller. In this case, it's larger because a times b is greater than one. What this does tell us though is that a and b are equal. And so that means that our map is conformal. That means that the shapes are being maintained. Okay, so and that's what we're seeing here is that a is still equal to b. They're bigger than they were before, but they're still equal, which means that they're still circles. And that indicates to us that generally speaking, the shapes that we have on our map projection will be the same as the shapes would have been on the globe. So Greenland still looks like Greenland or whatever. So that's usefull to know as well. This brings us to a fundamental principle about map projections is that you can preserve shape or you can preserve area, but you can't do both. When you take a three-dimensional sphere or ellipsoid, and you squish it down onto a two-dimensional sheet of paper, something's got to give. And so you can either maintain shapes or sizes but not both. So what we see here is we have Greenlands in both of these maps. And in one, the area is being maintained but the shape is wildly distorted. In the other, the shape is accurate but the size is wildly distorted. So these are two extreme versions, it's not always this dramatic but I wanted to show you something was easy to relate to, or easy to see. So you can see that we can preserve area or shape, but not both. In other words, a equals b or a times b equals 1. But you can't have both things at the same time on a two-dimensional map. And this goes back to what I showed you a minute ago, is that on the reference globe, that can be the case as we have here. But its only on a three-dimensional version of the Earth. What's this two-dimensional, you can either have a equal b, but a times b will not equal 1 or vice versa, okay? I hope that's clear. Like I said, this is just a way of being able to quantify our description of the distortion that's taking place in different parts of the world, and relating that to scale factor. And relating that to scale factor in two different directions, north-south and east-west. And so how that relates to these circles and whether they're still circles or whether they're distorted in shape or size.
