[MUSIC] Hello everyone and welcome back. In this lecture we'll dive into the details of how projections work to give you an understanding of how projections preserve properties of maps and a way to conceptualize and visualize their construction. We'll cover the types of distortions and projections and the mechanics of translating a sphere into a flat surface. And in the process, you'll build an imaginary device to project the earth onto flat surfaces. As part of this, we'll discuss planar, cylindrical and conic projections as well as normal, oblique and transverse variants. If that all sounded like scary jargon don't worry it will all make sense soon. Projections can be dull and frustrating but I'm hoping this lecture will give some of you an, aha moment where you understand them at a higher level and even think they're somewhat fun. So to begin with, remember that regardless of spatial data, it must be projected in order to be displayed on a flat surface. Projections aren't free, though. That is, they have consequences in the form of distortions to our spatial relationships. We can distort shape, area, distance, direction, and scale. Even if your data isn't stored in a projected coordinate system, it still needs to be projected to display on your screen. GIS software chooses a default projection to use in this case often one called the equirectangular projection. It won't show up as that on your data frame but that's what it's using to translate latitude and longitude to your screen if you didn't specify it by projecting your data. Projections take the coordinates of the projected coordinate system often in meters, feet, etc. And a sign or map each one to the corresponding decimal degree coordinates on a globe through the use of mathematical transformations. For this reason, projections are always built on a geographic coordinate system or GCS. So data stored in a projected coordinate system has both the projected coordinate system and an underlying geographic coordinate system. ArcMap will show you the GCS when you look at the projected coordinate system as well. Since projections are in linear units and the globe necessitate angular units, this transformation is required, but the math varies by projection and its properties. Don't worry. I won't make you learn any of it. We're focused on the practical side, and you can learn the projection map elsewhere if it interests you. Thinking back to distortions, some projections protect against specific types of distortion by using different techniques and mathematical transformations to preserve some properties at the expense of others. Now, this distortion is not constant on a map. Some areas on a map are more accurate than others based on how their projection is set up. This is similar to what we did with datums where they can be optimized for different parts of the earth based upon how you construct the data. The same thing happens with projections. So now, I need to get the features of a globe on to a flat sheet of paper. To do so, I need to turn this spherical representation of the Earth, a globe, into a projector that projects my globe onto a screen or a sheet of paper. Now, just to prove that we need that projector instead of just flattening the globe, let's see what happens if we try to flatten this right now. So now we have a flat globe. But it's not truly flat. Notice all these bumps and ridges everywhere. That's not just the air that remains inside it, that's the inability of this round item to fully flatten out. If I flatten one area, I have to move the distortion to another area. Projections work similarly, but we still get distortion of some amount inside the areas that we flatten out. The mechanics of how we do this though are by turning the Earth as a sphere into a projector. So now I'm going to blow this back up. Now let's turn this globe into a projector. And to do so I'll need the place that our map will ultimately end up, our projection surface, like a sheet of paper. Imagine that there is a bright light inside of this globe and the light passes through the Earth from the center to the outside. But as it does so it takes on the characteristics of the part of the Earth it passes through. Like the color and it carries them to the paper where it prints onto the paper. We can do this a few different ways. This is the core of projections here. Is how we handle projecting onto this sheet of paper. If I put the sheet of paper on top of the Earth, I can do a planar projection, often called a polar or azimuthal projection. Imagine that light coming out of the Earth and putting the features of the half of the Earth that is visible to the sheet of paper as features on the paper. This is one way to get features onto the paper. Note where accuracy varies here though. At the point that the paper contacts the globe it's perfectly accurate but everywhere else the angles involved in the projection distort the features on the globe. If we look at features near the equator they're condensed if we look down the top or the bottom of the globe. Effectively flattening it. And we can't see much of what's on the side. If we have a light bulb inside and a huge planner sheet of paper. We could end up with distortions near the edges where everything is much bigger because the light angles to pass from the center of the globe out through it and to a sheet of paper. Require the paper to grow much larger, linearly, for smaller increases in angles of the Earth that we can capture. To illustrate, imagine a single light ray inside of the Earth here, coming out of the globe. If it comes straight up through the pole it doesn't widen the features out. But if we got a handful of degrees and measured the distance on the surface of the globe, and the distance on the sheet of paper, they would differ. This is much more apparent as we go even further out where the features here would come from somewhere down here on the globe. But what if we need to get most of the planet. Not just this top half. Say all of the continents on to a flat sheet of paper like most maps. Well we need to get more creative then. To demonstrate this next projection. I'm going to switch globes to something I can manipulate more easily. To get most of the globe we need what's called a cylindrical projection. If holding the sheet of paper flat against the Earth was a planar projection what do you think a cylindrical projection will look like on our globe projector? Cylindrical projections are probably the easiest to understand and the results are commonly seen. They're the source of the Mercator projection that's used in most web maps and that we talked about before. To get that cylindrical projection, we need to wrap our sheet of paper around the globe. So rotate it here so you could see it. And now I take this and put it under this Earth here and wrap it around like a cylinder with the sheet of paper. We can make a cylindrical projector. Now imagine that the sheet of paper also comes back this way, so you can capture this end of the globe. If we imagine the paper like photo paper again, it will retain the image of the Earth that we've projected onto it, after we unwrap it from the globe, into a flat sheet of paper again. Now, with the planer projection we had one accurate point at the top of the globe. In this case we now have the whole line going around the globe that has now distortion. Wherever the sheet of paper makes direct contact with the globe. This line where the globe contacts the sheet of paper is called a standard parallel. The further we get away from the standard parallel out towards you. The more error we accumulate due to the angle of projection. But close to the standard parallel we have nearly no distortion and along the standard parallel we have zero distortion. If we relate this back to the Mercator Map since Mercator is built on a cylindrical projection. The areas out here are the more distorted areas up in the north or the south. And the area in here on the standard parallel is equivalent to the equator on a Mercator map. In this case it's rotated or is transversed. But it's still a very similar type of projection, and the mechanics are the same. The standard parallel in the Mercator map is the equator and has no distortion. And the further away you get, closer toward you, is much more distorted like in the north on the Mercator map. Despite the growing errors on Mercator maps they illustrate why we have so many different map projections because they preserve bearings or relative directions. So they're still useful for navigation. They're poor for areas but great for navigation. Other projections make different compromises and enable them to be used for other purposes. Still, we have the same issue we had with the planar projection. As we get further away from the standard parallel, we accumulate more error on our map, because the light rate travels at angles that force the spread of information. This is really evident in the Mercator projection. Greenland is pretty small but in the Mercator projection, since it's out here, it's as big as the continent of Africa, which is over here along the standard parallel. In reality Africa is more than seventeen times the size of Greenland. But again, since Africa is positioned at the more accurate location along the standard parallel, in that case, the equator, which is not demonstrated here. So it's size is kept mostly correct on the standard parallel while Greenland is projected to a large size by being very far from the standard parallel. We can create different variants of these cylindrical projections. We can do, quote, normal projections that follow a known pattern north to south and where the standard parallel is along the equator like in the Mercator projection. Or we can do transverse projections, where we rotate the paper 90 degrees relative to the Earth, just like I have here. And oblique projections where we rotate it to any angle to capture and put the standard parallel in a more accurate spot for a given use. These rotations help us align that standard parallel to the region of interest. Many states, nations, and regions have standardized projections that do this rotation and optimize for a specific area. So one thing you should do is go find out what projection is important in your are that's optimized for it and then use that projection. So, what if we wanted to make our map have accuracies similar to being on the standard parallel across a larger area of the map? We can do this. We can achieve this with conical projections and a little bit of imagination. Conical projections are the last major type of projections with planer, cylindrical and conical projections. If instead of a cylinder we wrap the sheet of paper as a cone. And place it on the globe. We can preserve a different set of properties and don't get the same distortion near the poles that we did with the Mercator or cylindrical projections. Now all the projections I've shown you so far have been tangent to the globe. Just like this one is right now. It is that they touch the outside of the globe here and run tangent to it. But, and I can't do this with the sheet of paper, if we slice through the globe as a secant, where this paper goes under the surface of the globe for a moment and then comes back out. We can greatly increase the accuracy over a larger area because that gives us two standard parallels where it contacts the globe in two separate spots. You can imagine this almost like sinking the cone inside the globe a bit. The end result is increased accuracy. The area between the two standard parallels curves back toward the other standard parallel, so it doesn't get too far from the projection surface. The rest of the globe curves away under here as usual. But it's still usable near the standard parallel. Conic projections like this are a great way to achieve higher accuracy further from the equator and across large areas to the two standard parallels. And they're commonly used at mid latitudes on the Earth. To wrap this all up, in this lecture we made an imaginary globe projection device. That we used to understand the fundamental concepts of translating a spherical surface such as the Earth. To a flat surface like a sheet of paper or your screen. We also learned about planar projections, cylindrical projections, and conical projections. As well as how we can increase accuracy in certain regions by varying the angle of the projection. And using a secant method to obtain two accurate standard parallels on our maps. That's it for this lecture on how projections are constructed. In the next lecture, we'll put them together into some common projections that are in use around the world so that this becomes practical again. See you there.
