Let's have a look at properties of map projections and in particular distance and direction. Let's start with a fairly straightforward map projection. This is an equidistant projection known as the Plateau Cutaway, it's French. The distinctive factor or thing about this projection is that you'll notice that we have squares that are created by the grata cual. All of these are squares no matter where you are on the map, the're all squares of the same size. Okay? So, you might think well that must be a good thing then because that means maybe there's not as much distortion. Well, there is or there isn't a depends like any map rejection there's going to be distortion. If we look at the standard line, it's touching the developable surface at the equator. So, that's where we have a scale factor of one. We also have a scale factor of one along each of the meridians. That's because these meridians are the same distance on the map as they are on the reference globe. So, there's no distortion taking place along those lines. So, if you see here, you'll see that there's a meridian here. They're all the same length on the reference globe, and they're all the same length on our projected version. So, that gives us an indication about what's happening with distortion at least in that vertical dimension. However, when we look at distortion in East to West direction because these lines are not converging at the poles where they're supposed to on the reference globe, there being stretched out and so that indicates to us that the scale factor has to be greater than one. So, we're not maintaining our distances correctly on this map. Another way of looking at this is that if you had a line of latitude say somewhere around here, or actually let's do one down here since it's easier to draw on the South Pole or towards the South Pole here, is that this line is shorter than the equator, right? But on this map, here's the equator, let's say that's the same line here that we have up here, okay? So, now, this line is the same distance same length sorry as that line. So, from here to here, that's not the same distance as the equator. That's another way of just kind of visualizing the fact that that line has been stretched out. All of the lines of latitude which gets shorter and shorter as you go towards the poles, have all been stretched out so that they're all the same length. So, that gives us a good indication that something's not quite right here, at least in terms of distortion. What we can see in terms of things like distance, is that the distances are correct. In other words if I measure a distance on this map, it's correct where the scale factor equals one. So, what happens a lot with projections is that you can measure distances or directions correctly, but only in certain situations, only under certain conditions, only in certain parts of the map and that's what's happening here is that we have a situation where the scale factor is one along the standard line. So, if you want to measure something along the equator, that distance will be correct. The scale factor equals one along each of the meridians. So, if you measure a distance along any of the meridians, that distance will be correct. But because we're having this scale factor greater than one in the horizontal direction, any other line that you draw on that map is going to have distortion in East-West direction and so the distances will not be correct for anything that's not on the standard line or is not on a meridian. In other words, where the scale factor is not one. So, part of what I'm doing here is trying to tie together this idea of how distortion is taking place, how we measure scale factor, and how we can relate that to measuring distances. This is another equidistant projection. In other words, distances can be measured accurately but only in certain circumstances or in certain conditions. Here, this is a planar as immutable projection and it's planar because if you think of the sheet of paper as a plane that's touching the globe at the north pole, that's one reason why it gets the name as it's planar and in this case. So, here's where it's touching this is the standard point in this case. Any of the lines that are meridians that are radiating out from here are all the same length and they all have a scale factor of one. So, if you measure a distance in any direction from the North pole, then the scale factor equals one and those distances will be correct. But if you try to measure a distance along here, you'll see that these circles they indicate that the seas had been stretched and that tells us that the scale factor is greater than one in that direction and we can't measure distances in any direction other than along those meridians. What's interesting is that this line also has a scale factor of one and both of these purple lines point North-South, really? Yeah if you take a look at it, remember these are lines along meridians, meridians run North-South. So, all of these lines run North-South all of them. I know it's a little crazy at first if you haven't seen something like this before, but that's what's actually happening. So, this is just to show that the scale factor is greater than one again because you can see these circles have been distorted, we're getting distortion in the East to West direction. So, now we're getting distortion this way. So, what that means in the context of what we're talking about here is that even though this is an equidistant projection, it's only equidistant in certain situations. Equidistant meaning that the distances are correct. So, if you measure the distance from here to here, that will not be correct because of the distortion taking place along that dimension the scale factor being greater than one.
