One concept or term that's really important to know and to understand is the standard line or point, and that has to do with where the developable surface touches the reference globe. So, here we have the cylinder touching our reference globe at the equator, and so the line where those two touch each other is the standard line and on our two-dimensional projected version of this map that's our standard line there. The reason the standard line is important, is that's where there was no distortion, you're literally transferring a point from the reference globe to the sheet of paper and if there are touching each other then that is the same point on both of those things and so there can't be any distortion because you actually haven't shifted it or moved it in any way. So, that's a key idea, whether there's a standard line, there's no distortion. Also, as you move away from that standard line, distortion inevitably has to increase. So, this is the diagram I showed you in another segments where we have our light bulb inside our reference globe, it's projecting the shadow on there and it's just another way of visualizing this idea of the standard line and the fact that the point on both is the same. On a conic developable surface, the standard line is also where the sheet of paper is touching the globe but it's not going to be the equator in this case, it's going to be at a mid-latitude location somewhere between, so that's the Equator that's the pole, it's touching here so that would be somewhere in the mid-latitudes. Still the same thing now, still no distortion at that standard line. On a planar also known as an azimuthal projection, there actually isn't a line, it's a point that's where it's touching, so we would refer to that as the standard point, same thing again, still no distortion at that location it's no longer a line on our map, it's just a point. So, if our reference globe has a principal scale that we've set it to in this case in my example here I've got it set to one to 100 million, this is a hypothetical three-dimensional reference globe and we can calculate the principle scale by dividing the reference globe radius to the Earth's globe radius. So, this is the calculation that gets done. I showed this in another segment that maybe you've already seen and I'm just refreshing our memory here in case you have seen it already because now we're going to apply that to quantifying the amount of distortion that we're getting. So, let's see what I'm talking about here, shall we? The way we can quantify distortion is using a thing called scale factor. If we have a reference globe and a two-dimensional map, the scale factor is when we divide the local scale by the principal scale. So, the local scale is the scale on the map, the principal scale is the scale and the reference globe. So remember, there's no distortion on the reference globe, we've taken our full-sized real earth shrunk it down to the size we need for the scale of the map that we want to create but it's hypothetical or imaginary but it's a three-dimensional globe with no distortion. That's a key idea. On our map though, there can be distortion and so the local scale is the scale on that map which may or may not be the same as on the reference globe around the principle scale. That may surprise you but the scale on a map is not always the same all over the map, it can actually change as we'll see. So, where will the local scale be the same as the principle scale? So, the scale on the globe being the same as the scale on the map, where will that happen? Can you imagine where that might be? i.e. where is there no distortion. Well, it's going to be at the standard line, it's going to be where the sheet of paper is touching the globe. So, the local scale on our flat map will be the same as the principle scale on a reference globe and so there is no distortion taking place at that location. So, if that's the case, then if this scale on our map is the same as the scale on a reference globe and we divide one scale by the other, if they're the same, then our scale factor is going to be one and the scale factor is just a way of quantifying or describing distortion and I'm just trying to put it in simple terms is that if you have the scale on your mapping the same as the scale on the reference globe, there's no distortion and the way we describe that as saying the scale factor equals one. Of course we can't have distortion in which case the scale factor will be some other number and we can use that to describe what that distortion is. So, here at the standard line the scale factor equals one but what happens to the scale factor as you move away from the standard line. So, the scale factor at the equator is one and as we've seen before with this particular projection which the Mercator projection, distortion increases as we move away from the equator. So, what's going to happen to the scale factor, it's also going to change. In fact, scale factor is going to become greater than one as we move away from the equator or from the standard line which would be a better way of putting it. And you can visualize this here, is that so this is a standard line where it's touching, you'll notice that the lines get farther and farther apart as more and more distortion takes place. So for example, we have Greenland here versus Africa there and the fact that this is so much larger than it really isn't reality, means that the scale factor is much larger than one. In fact, at this location, the scale factor would equal two. So, that means that the local scale is double the principal scale. So, if you're dividing, remember you're just dividing one by the other and so when you do that, you're going to end up with a value of two at that part of the map. Like I said, maybe kind of surprising to you is that depending on the map projection and depending on the size of the area that you're measuring, you could end up with these distortions that are quite large and the scale on the map is going to reflect that distortion and the scale factor is just a way of describing that. So, remember, on our 3D reference globe, there is no distortion and our 2D projected map there will be distortion and our scale factor is going to vary. So, we're only going to have the principle scale along our standard line. Everywhere else the scale factor is going to be different. Let me just show you how the calculation works. So, for example, if at this location, we have a scale of one to 50 million instead of the principal scale which is one to 100 million, which is what we would have at the standard line, then we have one over 50 million divided by one over 100 million and that equals a value of two. Depending on how good you are with math or how long it's been since you've done this kind of stuff, 50 million may seem like a smaller number than 100 million, but remember it's one over 50 million which is actually a larger number than one over 100 million and that's how you end up with for example this thing here is 0.00000002 is greater than 0.00000001 and so that's how you get a value of two as the scale factor. A clever way of illustrating the amount of distortion is taking place on a map is using a variable scale bar which is particularly popular with the Mercator projection because it works well for that. What this is showing is that if the standard line, this is our scale bar as you would normally see on a lot of apps it's a scale bar and it's saying that this distance at the equator is 600 statute miles, but remember the distortion increases quite dramatically on a Web Mercator map as you move towards the North Pole. So, what's happening is they're saying that if you're at this latitude that is 600 statute miles, if you're at this latitude that is 600 statute miles it's still 600 miles. Does that seem weird to you? But remember, what's happening is, you're taking the reference globe where the meridians are converging at the poll and you're stretching them out, you're actually making them parallel to each other. So, if you have a distance that's this far apart in reality and you're stretching it out on the map so it looks like this, that's actually going to distort that distance and that's what this variable scale bar is trying to illustrate is how much distortion is actually taking place and you can see that as you're moving away from the equator towards the pole, you're getting this crazy amount of distortion taking place which may not be something that a lot of you, I don't know, I didn't really tweak to it when I was first looking at maps going, why is this crazy scale bar on there, but this is the concept of the reasoning behind why that's on there and how it can help you as a map reader to better visualize distances. This is a slide that I showed in the section on scale, I just wanted to revisit it here for a minute in relation to distortion and scale factor. I think it may be more clear to you now that the amount of distortion that takes place can have an effect on the way that scale is perceived by somebody when you're looking at your map. So, one thing I wanted to point out is it's just a rule of thumb that if you have a map scale that is a smaller map scale than about one to 250,000 then it doesn't really make sense to include a bar scale on that map, because what's happening is there's so much distortion taking place. Remember, a smaller amount scale means a larger area like say a country or the world is that unless you're using a variable bar scale which is only really useful in certain situations, then a lot of times that bar scale is not going to be accurate unless it's right on the standard line. So, it's better to use a bar scale on larger scale maps where the scale is more consistent across the mapped area. What I mean by that is for example if you're making a map of a neighborhood or something like that or maybe a city even, there will be some distortion taking place from one side of the map to the other but it'll be so minimal that it really won't make a big difference to any measurements that are being made or the way that somebody who reads it. So, it's definitely a good idea then to have a bar scale on there because there won't be enough distortion for it to be misleading in any way. But if you're using a smaller map scale, so a larger area, so say you like a country or something, then the scale factor will vary more over the map. You will have more variation in scale and then a scale bar really doesn't make sense to put on a map like that. So, if you're making a map of Canada or United States or some area that size, there's no point really in putting a scale bar on there because of the amount of distortion is taking place. Really at that point, it's just useful to have a representative fraction on there, it gives people a good idea of the scale and that's good enough.
