 So today, we're going to start working through a process known as the Thesen-Polygon method. And the first step in the Thesen-Polygon method is going to be to take our data. Well, first, we're going to need some data. In this case, we have some spatially distributed data. You may have seen this before. In this case, we have a series of points, rain gauges that are giving precipitation estimates around a set of watersheds. And we're interested in sort of partitioning out the area of our map here and associating parts of this area with the different rain gauges. For example, this area 5.06 could be associated with this watershed A, but there's parts of the watershed here that are more likely to have values closer to this rain gauge value over here. And perhaps we can go around and assign different parts of the map to different rain gauges. That's the goal of the Thesen-Polygon method. Well, the first step of this method is going to be able to take this data and create a series of what are called delinear triangles or delanear triangles. I'm not sure of the pronunciation. We're going to create a series of triangles here with this map. In order to do so, you're going to need a map of spatial data. And you will want a pencil and a straight edge. Here I have a straight edge and a pencil. And there's a series of, well, there's a couple of things you need to do to start with. The first thing we want to do here, step one, is we are going to create a convex polygon around the data. If you remember, a convex polygon means that all of the angles around the outside of the polygon or around the inside of the polygon are going to be acute angles, or they're going to be less than 180 degrees. So let's go ahead and create that here. In other words, I'm going to connect points here. Here's 4.66 to the 5.37. So I continue around the outside. The 4.66 will connect to the 3.86. We continue around the grid to the 4.55, then to the 6.24, then to the 5.34. And now we have a choice here. From 5.43, it seems like we might want to go to the 4.67. But notice, if we do so, then we create an angle here. If I connect these two lines here, if I connect this point and then this point, that angle created there is going to be greater than 180 degrees. So we're now going to skip over the point here, 4.67, next to the letter D, and finish connecting all the way across from this 5.43 to the 5.37. And now we've created a polygon where all the angles here are less than or 180 degrees. Now our next step is we actually want to connect all of these points so that they're all vertices of a series of triangles. In order to do so, there's a process here. And our goal is to create these triangles with the largest minimum angles. In other words, there are going to be some triangles where we have very small, very acute angles. But we want to make those acute angles as large as possible. So the smallest angles we see, we want to be as large as possible. So it's a bit of a process for going about doing this and making sure we're doing it correctly. What I'd like to do is I'd like to identify two adjacent points along the outside. Here's 4.66 and the 3.86. We can identify two points. And a third interior point. Here's the third interior point that's close to both of them. And the question that I want to ask myself in order to make it an official De Laune triangulation is whether or not I can sketch a circle, if I lightly sketch a circle, that passes through all three of the points. And I have to roughly estimate that circle. But if I can sketch a circle that passes through all three of the points where the circumference of the circle passes, and it contains no other circle, then I can include the triangle for those three points as part of my De Laune triangulation. So it appears that a circle that would contain all these three will not contain any other points. If I had another data point that was out there, then obviously I could not include this triangle. And I would have to look for another triangulation. But in this case, I can connect the 4.66 to the 6.01 and similarly connect the 3.86 to the 6.01. Now notice that creates a fairly obvious-looking triangle here. Now there's a couple of possibilities. If I went and created a triangle with these three here, I might create, there might be the possibility of a long, skinny triangle going all the way from 4.66 to 4.67. But that seems like it's going to make very skinny angles here. I'm going to go for a simplest case and look at these three. I have two adjacent and this sort of third one here. Let's see if that, if I roughly sketch that, it does appear that any circle that's going to contain all three of those is going to be a nice big one, but off of the page and not contain any more of my data points. So I can continue by making another triangle here. And I'm going to repeat that process looking again at a couple other spots. Let's consider the 6.01 and the 3.86. And now we will consider them with this other point here. Notice again, that's going to create a circle. And I sort of roughly sketch in that circle, doing my best again to estimate how big the radius is going to be. And it does appear that it's unlikely that anything is going to fall within that circle. So once again, I can create the triangle of those three points. If I want to be a little more exact, I should probably use my straight edge. Notice I just down here around the outside, the 5.43 and the 5.37, there is this point here. And if I try to sketch something there, that's going to make a very large circle. But again, that large circle is going to be sort of off the edge of the page. And it does seem like I can include those. Notice this is still going to give me a triangle that has some very small angles. Those will sometimes have to occur. However, we're trying to minimize those values where possible. When a triangle sort of appears, we can check and see if it makes sense. If it appears to be capturing any other ones, it looks like I can sketch something here. It does not appear to be capturing any other points. So I continue with my triangulation. And sometimes it will seem fairly obvious. And other times, it may not. Now here we have a little bit of a question. We have this point 4.39. And notice there are a couple possibilities. If I actually connect, I'm going to lightly connect these two points here. I create a quadrilateral that's in this particular place. And with that quadrilateral, sometimes we can consider two possibilities, that perhaps we can connect across the quadrilateral this way. Or we might be able to connect across the quadrilateral this way, between points that are on opposite sides of the quadrilateral if we're considering something. Notice it's fairly evident in this case that the division along this line is going to create a long, very thin triangle with very small angles. Whereas opposed to dividing it this way, this is a process called flipping, where you consider a quadrilateral and you can either flip around those axes that you divided into triangles. But in this case, it seems like the most evident possibility. And if we sketch our circle within these points, it does seem like we can create a circle here that's not going to contain any other points. And similarly, although it's a little bit bigger here, now we begin to actually ask ourself, well, it's this circle here. Can we contain a circle that contains those points? There we go. So it does seem like those two triangles are appropriate. Continue here, again, sort of checking it with a bit of a sketch and up here. Now, very often, you may get to a point where you have a polygon, particularly as you start getting a little bit smaller, where you have a polygon where many of the angles are close to the same size. And it isn't evident right away. It isn't really evident whether the circle is going to contain other pieces. For example, if I start sketching the circle that contains these three here, 506, 439, and 543, it's not readily evident whether or not that's going to be big enough to contain the other two or not. If you have some question about that, one good rule of thumb is to look for the biggest angles around the outside and attempt to divide those angles. For example, this angle here appears to be less than 90 degrees. This angle here is only slightly more than 90 degrees. Whereas these other three angles tend to be little more than the other two. And if that's the case, we can usually choose to ignore the smallest ones and try to subdivide the larger ones. If you've made a mistake in this, you can also look a little more carefully and sort of compare some ideas. Perhaps if I look at this quadrilateral, and I'm not sure if this was the right division, I can consider what would happen if I did flip these two. What if this was actually a line that should have been in there? Well, then I can go back and look and see whether these two angles, and if I wanted to be much more careful I could even measure those two angles are greater than these two angles. And you want the sum of the larger two angles is the one that you want to actually split. In this case, if I did measure those, I believe I would verify that's the case. And again, if you do make a mistake in this process, it's not fatal. At some point later in the polygon method, you'll discover that things aren't lining up exactly the way you would like. And you can then go back and think about a couple of the triangles and perhaps flip a few of them to see if there is a different triangulation that actually fits our situation. So again, you start by creating a convex polygon around the data. You identify adjacent points plus an interior point and make sure you can connect with the circle that contains no other points without other points. And the third point that I like to make here is when unsure, divide largest angles. And with that process, you should have created a de launay triangulation.