What I've been showing you so far in these illustrations is that something like the cylinder that's touching the globe is only touching along one line. In other words, that cylinder is tangent to the surface of the earth. So if this is the Earth, tangent means that it is touching like that. Okay, fair enough? So, just to make sure you see that at the bottom, that means that projections that are configured that way are using what we would call a tangent case, that's just a way of describing that type of projection. So remember, the scale factor will be one at the standard line. If that's the case, then how can we reduce distortion in relation to this idea of the standard line and where the developable surface touches the globe? What could we do? Let's think about that for a minute. What could we do? Let's see. We could use what's known as a secant case. If the scale factor equals one of the standard line and the standard line is where the developable surface touches the globe, then why not make it so that that developable surface actually cuts through the globe? So, here you can see in our planar, also known as an azimuthal projection that now the sheet of paper actually cuts through the top of the globe. I don't know if you can see this well, but if that's the sheet of paper edgewise, the globe actually goes through it like that. Now, what this has the result of doing or being is that, before with the planar projection, you just had the sheet of paper touching at a point. Now that we have it with a secant case where it's cutting through, that point has become a line. What does that mean? Is that, well, if we have a line that means, that there's a larger area of our map that has no distortion because that's where the standard line is now. We've gone from a point to a line. With our cylindrical projection, our cylinder is now cutting through the globe. So again, if I just make sure that's clear, if this is our cylinder, the globe is now doing this, and so, we have a standard line here and a standard line there, that's what we're seeing there and there. So, now we have two places where we have a standard line, two places where the scale factor equals one and there is no distortion. The effect that has, of course, is that there's less distortion over a map in general, because remember, distortion increases away from the standard line, so the area between these standard lines will actually have less distortion than if you only had one standard line instead of two. The same thing is true with the conic. Secant case is now the cone is touching at two locations or our globe does this. So, now we have two standard lines here, again, same idea, scale factor equals one, two-standard lines, we've reduced the amount of distortion on our map. So, all of this is to say that it's a strategy that's used by mapmakers to reduce the amount of distortion overall on a map and to make it a higher-quality map as an end product. You'll often see the ones I'm most familiar with are maps of Canada and it's the same with US, is that they will use a conic projection hence two standard parallels which are standard lines, and they'll specify those, especially, if it's like an old-fashioned paper map or something. What they're telling you is where the secant case, where the developable surface is touching the globe where there's minimal or no distortion so that you have a sense on the map how that relates to things like the scale factor. So, let's think about scale factor a little bit in relation to the secant case. Remember with our standard line, the scale factor equals one where it touches, okay, we got that. If we think of our light bulb as being inside, this is our light bulb, I'll draw a nice little light bulb. So, if it's shining up like that, it's going to exaggerate things, so that things are going to be larger than they really are. If you think back to Greenland again, and so that makes sense that the scale factor is going to be greater than one because objects will look larger than they really are, the local scale will be larger than the principle scale, that means the scale factor will be greater than one. But what's happening here with our secant case between the standard lines is that this distance let's say is being shrunk down to here. So in this case, the scale factor will actually be less than one because you're taking objects from the reference globe and shrinking them to fit them onto our cylinder. So, now we're going to have a scale factor of less than one, still distortion, but it's a different distortion. Instead of things looking larger than they really are, they're looking smaller than they really are. A nice way of visualizing distortion in relation to projection is to use these circles that we can put on top of our reference globe. So on the reference globe, if you imagine these as being a 3D globe with circles on top of them, and in reality, it's actually meant to be infinitely small, but we'll get to that later. But you just take this idea that if you have what's on the real globe, or the reference globe is a circle, and we transfer that onto our projection, we can see what happens to those circles based on the projection that we're using. Are the circles on the globe still circles on the map, or have they been shifted or distorted in some way so that their shape has changed? The circles on the reference globe are all the same size, but have they changed in size on the projection? So, for example, this one you can see that the circles get larger as we move away from the equator. It's a way of, instead of talking about Greenland in Africa all the time, we've talked about how Greenland gets distorted so much, it's a more standardized way of trying to visualize distortion for the entire planet and say, what's happening to it over all of these different regions? Are the circles larger or smaller? Are they changing in shape? Are they changing in size? So that's just a way of visualizing this. So, when thinking about developable surfaces in relation to the reference globe and standard lines and how that relates to distortion, there's a rule of thumb that's often used when you're choosing your projection and that is that we use a planar projection when making a map of the poles. We use a cylindrical projection, when making a map of equatorial regions, and we use a conic type of projection when you're making a map of temperate areas or mid-latitude areas. So polar areas are planar, tropical are cylindrical, temperate areas are conic. This is just a guideline, is just something to work off of, but there's a useful underlying principle here to remind us about all of the things we can talk about in terms of distortion and scale factor and so on, is really this is a strategy for minimizing distortion. Is that, if you have a standard line touching near the pole, then you're going to have the least amount of distortion near the pole, so that would be good for mapping the poles. If you have standard lines between the area of the equator, then you'll have the least amount of distortion in the equatorial region, so a cylindrical projection would be good for tropical areas. If you've got a cone with two standard lines touching at the temperate areas, then you're going to have less distortion say for map of Canada or the US or something like that, in which case, you would use a conic type of projection for temperate areas. This is not to say you can't mix and match, you can move things around as we'll see, you can take this sheet of paper and you can have a touch at the equator, you can take the cylinder and you can turn it sideways. There's lots of different ways to do this, but this is a good starting point especially when you're just starting out to think about this, and they are often the more common strategies that you'll see in terms of making choices about developable surfaces and areas that are being mapped. So, it's good thing to keep in mind.
