In everyday conversation, or when you're just thinking about the world, if you do, you probably imagine it as being perfectly spherical. Well, it's not exactly perfectly spherical. So, let's talk about that, and what an ellipsoid is, and why we would want to know about that in terms of describing the locations of things and being able to make measurements correctly. I have some shocking news for you, perhaps even earth-shattering, is the earth is not round. Well, it's almost completely round, but not totally perfectly round. It's actually flattened a little bit. So, why is the earth flattened a little bit? Well, it has to do with the fact that the earth is spinning. There's a really long complicated answers to why this is, and there's a short simple answer. I think I'll just stick with the short simple one here. Essentially, just imagine if the earth was like a big water balloon that was spinning, and that as it's spinning, there's a centrifugal force that starts to push or pull the center of the earth out a little bit, and if it's pulling that mass out at the center, it's actually stretching that out and pushing it in at the top and the bottom. So, maybe this helps to visualize this, is that because it's spinning, you've got this effect where it's stretching at the equator and flattening a little bit at the poles. Now, this is wildly exaggerated. It's not something that you would notice. If you've ever seen images from space, you're like, "How come it doesn't look flat?" Well, okay, it's a very small amount. The point here though is that, sometimes, we want to take that flattening into account. If we treat the earth strictly as a sphere, our measurements will actually be off if we're describing distances, say, from one location to another. They will be off by a lot. But if we're trying to be more exact and more accurate with the measurements that we're making, after all, we're GIS professionals, then we may want to try and reduce that error and make our measurements better. So, that's what we're doing here, is we're going to try and take into consideration the fact that the earth is spinning, that there's this flattening that's taken place, and can we measure that amount of flattening and take it into account. So, the earth can be modeled using this thing called an ellipsoid, which is really just a mathematical version of a sphere that's been stretched out. If you want, you can think of it as called an oblate ellipsoid, and it has a semi-major axis and a semi-minor axis. So, if this was perfectly spherical, those would be equal. But the fact that they're not, because it's been stretched along the equator, this dimension, the major-axis is longer, which is why it's called the major. This one is a little bit shorter, which is why it's called the minor. So, the a is a little bit bigger than b. All this does is, it gives us a way of being able to quantify our description of the amount of flattening that's taken place. That's really just a difference, so we're subtracting one from the other. Remember, if it was perfectly spherical, the difference between the two would be zero because they'd be equal. But they're not, so all we're trying to do is express the fact that there is a difference, and then we just standardize it by dividing by a. So, we're just going to say a minus b divided by a is the flattening value that we have for the earth. It turns out that, generally speaking, that is equivalent to about one over 300. So, why am I prevaricating there a little bit and saying, "Well, it's one over 300? Don't we know?" Well, actually, there's been many different measurements of flattening over a very long period of time, literally over thousands of years. So, the amount of flattening that's taken place, or the way that that flattening has been measured, has come up with different numbers. You might think, "Well, okay, but isn't there just one number now?" Well, sort of, but there's actually several versions of this flattening that are still in popular use today, that she will still come across when you're getting data or making maps, and you need to be aware of the fact that these different versions of it, but we'll get into that more in a minute. So, basically, just remember for now that we've got the stretching, we've got one axis longer than the other, we're going to use a little bit of math to describe what that is or to quantify it, and that that equals around one over 300. So, here are some examples of ellipsoids that have been created or generated over time. There's lots of them out there. These are just some of the more popular ones I wanted to show you is an example. So, for example, this one is Airy 1830, and yes, that means it was developed in the year 1830 by Sir George Airy. At that time, he determined that the semi-major axis was 6,377,563.396 meters, and that the flattening was, remember, it was about one over 300, so he actually determined it to be one over 299.3249646. Great. So, then there's the Australian National. There's Clarke 1866. That's a very popular one in North America. You'll notice that, for example, that one, the numbers are a little bit different than they are, say, with the 1830, and this is different as well. One of the most popular or common ones that you'll see used today is GRS 80. So, yes, that means it was adopted in the year 1980. These things don't change that often. So, again, the values are a little bit different. So, now, instead of one over 300 or one over 299, it's 298.257, blah, blah, blah. So, why am I telling you this? Because, as I said, these are different versions of flattening for the earth. So, what they've done is, if you think of it this way, the earth is not perfectly spherical, and the earth is not a perfect ellipsoid either, but this ellipsoid is a mathematical approximation of the amount of flattening that we can use to describe the earth. So, there's been different descriptions over time that have been used to then map locations. So, if you, I'm time-jumping ahead a little bit here, but if you map locations of something using one of these ellipsoids, and then try to take those coordinates and use them with a different ellipsoid, they won't be in exactly the same place, because they're different versions of the earth. So, we have to know which version of the earth was used to create that data, and then use that same version of the earth to map that data, then we can change it if we want, or adjust it, or transform it in some way, but we need to know what was used to begin with. So, just if you're wondering where I'm going with this, it's really important to know all of this in order to be able to map locations correctly. Just to illustrate why this matters in particular, if the earth was a sphere, and we are measuring angles from the center of the earth, and so if these are equal angles, so this angle is the same as that angle, and all of these angles are the same, then all of these distances would be the same as well. They'd all be equal. So, this distance will be the same as that one, that one, that one, right? Because if we have equal angles, we're going to have equal distances. Everything is consistent, easy to work with, no big problems. Unfortunately, that's not how things actually work. Now, I should mention, you can treat the earth like this if you want to, and for some purposes, that's perfectly fine. It's simpler. It's easier. If you don't care about getting that little bit extra accuracy, that's okay. There are situations where that's fine. But if you do need more accuracy, then it's not going to be so great. Now, if the earth is an ellipsoid, and again, this is wildly exaggerated just so we can see what's going on, if we try to draw those points in the same way we just did before, that distance is not going to be the same as that distance. If you think of it as though the earth has been stretched along the equator, then this distance has also been stretched, and so that distance here is a little bit longer than that distance there. But we're trying to use angles to measure these distances or to describe our locations, and so now there's a bit of inconsistency if we try to treat the earth like a sphere when it's not. So, that's what we're trying to talk about here and trying to figure out. A little bit of terminology for you is that if you do treat the earth as though it's a sphere, and if you measure a latitude from that mathematical spherical version of the earth, what you're using is known as geocentric latitude. How this works is if you're on the surface of the earth, we draw a line that's at a tangent to the surface of the earth and then draw another line that's perpendicular to that tangent. So, this is the one that's tangent here. We draw a line that's perpendicular to that, and that, if the earth was perfectly spherical, would go right through the center of the earth. So, the center of the earth is geocentric. So, geocentric. So, if it goes through the center of the earth, it's geocentric latitude. More terminology for you to remember. Like I said, you will find data that's stored in geocentric latitude. It's not that you should never use it. It's just not going to be as accurate as if you were using one that takes into account the flattening in the ellipsoid. If you do that, if you use an ellipsoid, and you do exactly the same thing, you draw a line that's tangent to the surface of the earth, and then one that's perpendicular to it, it will not go through the center of the earth. That will only happen at the poles or at the equator. So, just in terms of geometry, I guess, it will go right through. I didn't draw that quite right, but it'd go through like that. But anywhere else on the surface of the earth, it will pass through the equatorial plane, but not through the center of the earth. So, you can still measure an angle from that, in this case, 45 degrees, and what this is referred to is as geodetic latitude. So, if you see geodetic latitude, that means that the coordinates that you're using have taken the flattening of the earth into consideration and are using an ellipsoid. So, we have geocentric versus geodetic. Geocentric is not as accurate. Geodetic is more accurate. Geocentric is using the earth as a sphere. Geodetic is using the earth as an ellipsoid. So, why does this all matter? I've mentioned a few times that the drawings that I'm using of the flattening are wildly exaggerated. It turns out that, really, if you treat the earth as a sphere to measure distances, you'll be off by about one kilometer for every 110 kilometers, or just under one percent. You may be thinking, "Less than one percent? That's what this guy is going on about here, is less than one percent? Who cares?" Well, like I said, sometimes it matters. Imagine if it was a football game, and someone was throwing the football for a touchdown, and you're off by, say, just under a yard out of the entire football field, which is 100 yards. Then, is it a touchdown, or is it not a touchdown? You could say, "Oh, who cares? It's close enough." Well, in some situations, it really does matter. Or imagine if you're an architect or somebody building a building and looking for the cornerstone, where you're going to start the building from. If you're off by a meter, that's a big deal. Or if it's a rail line, and you have two railways that are meeting, and you're off by a meter, that would be a big problem. So, of course, there are situations where it doesn't matter. You might be in a board meeting, you've got a map of the world on the wall, and you're talking about flying from Japan to New York or something. In that situation, it doesn't really matter if you're off by less than one percent. So, if you're not taking measurements, certainly go ahead and use the geocentric. But if that level of accuracy is important to you, then you'll want to use geodetic. The rule of thumb is that if you're making a map at a map scale of one to five million or smaller, which means more of the world, remember, map scale is the opposite of what you may think, so smaller map scale means it's mapping more of the world, so if you're at a scale of one to five million or smaller, then that difference of less than one percent probably isn't going to really matter very much. But if you're making a map at one to one million or larger, then it may be noticeable, and you may want to use geodetic. So, one to five million or less, you can use geocentric. One to one million or larger, you can use geodetic. You may say, "Well, why wouldn't I just use geodetic all the time?" Not all data comes in geodetic. So, it's a matter of, whether you're making that decision or deciding from scratch, then you can decide. Maybe you always want to use geodetic if you have the choice. I would. But this is just a rule of thumb or a guideline that you can use if you're working with different coordinates you're not sure which would be the best way to approach it.