 So let's complete our discussion about orthographic projection of points, lines, and planes by discussing how we would extend a series of points that create a plane and see what it looks like from different orthographic projection perspectives. So for example, let's say I have something like an index card. Well, if I have an index card, this is made out of a series of 1, 2, 3, 4 lines. And they're all sort of connected. And in this case, it's a nice rectangular shape. Keep it nice and clean here. But notice if I take that and I start changing its orientation, it still looks like a rectangle, more or less. Notice in reality, we see a bit of perspective, that things tend to get a little smaller as they get further away from us. But in orthographic projection, we don't presume that. We usually draw it so that this side and this side end up having the same lengths. That's one of the things that makes an orthographic projection where everything's perpendicular, a little bit different than reality. If we continue moving, you'll look straight down one side. You'll notice that the plane will disappear and simply become a line. Similarly, if you rotate it the other 90 degrees, you'll see it disappears and becomes a line in another dimension. So when we're trying to draw this from different perspectives, it depends on whether or not we're parallel. If we're looking at it parallel to some faces, we're going to just see a line. If we're looking at it at some angle, we're going to see some variation of the plane itself. So let's see if we can apply our orthographic principles here to recreating something. Well, here's a perspective. We have here, we have something that looks like a plane from the front view, but has faded into a line from the top view. Can you picture what that looks like in your head? What do you think it's going to look like from the right side? Let's see if we can go ahead and create that in the same way that we've done with our other points. We'll start with each point. Here's point one. With point one, we've already established its x-coordinate lined up with point one in the top view. If we come horizontally from both of those, we're going to need our miter line to establish what the y-coordinate is. But if we bounce off the miter line to get the y-coordinate and have it meet the z-coordinate, then we've established point one. Let's repeat that process with a different color here for point two. Notice point two has the same x-coordinate as point one, but a different z-coordinate. So we'll bring that z-coordinate there. We'll notice that it also has the same y-coordinate. One and two are on top of each other. So they have the same y-coordinate. So if we come across here, we're going to hit the miter line at the same location, drop down, and there is point two on the corner of that box after we've reflected it to the miter line. So there's point two. And because it's a straight line segment between one and two here, and as far as we can tell, it's just a dot there, must be disappeared in the line segment. We can connect points one and two at that point there. Now let's move to point three. Point three, we establish its z-coordinate. We notice that it lines up along the x-coordinate. We create the y-coordinate, bounce it off the miter line to establish the y-coordinate down here in the right side view. And there we've established point three. Now we have to look carefully here. It looks like point one is connected to point three here in the top view, but it's pretty clear that point one is not collected to point three here in this view. In fact, it's point two that is connected to point three in both of those views, so we can connect point two to point three in this right side view. And then finally, we'll take point four, noticing that they already are aligned along the x-coordinate. We create our z-coordinate, and we see that the y-coordinate for three and four are the same. So we create our little orange box there, and that establishes point four. There's point four, and you'll notice that point four is connected both to point one and to point three. So we connect it to point one, and we connect it to point three. And there is the face. There's the plane that's defined we see from the right side. Notice it's smaller than what we see from the front. Does that make sense? Well, if I have an index card and I slant that index card so part of it's a little further away from you, you'll see that from one perspective, it actually seems to be fairly long. But depending on how much the slant is, if I rotate this down, and there's another perspective, you can actually see that it seems to be a little bit shorter from that perspective. Let's see if we can see that on our net logo picture. I've recreated it here. Here's our viewpoint. If we're looking at it from the front, you can sort of see this little window. But as I rotate down and look at it from the top, there's the angle that we had. But if I rotate it and look at it from the side, you can see that we still see the rectangle, but it's a little bit shorter in this dimension. And that's based on the difference between the angles here. Notice it extends more in this dimension than it does in that dimension, more in the x direction than it does in the y direction. So there's an example of a plane. Let's do one final example here. We'll combine our two-dimensional plane with a curve. On section 4 or 5 here, we have this little thing that looks like, well, we'll say it's a quarter pizza. If you had a piece of pizza, you sliced through it, and you can see sort of a quarter of this piece of pizza. And you looked at it from the top. I don't have the actual thing, but let's see if we can go ahead and draw something like that. Here's my curve. If I look at it from the top, I see that. But when I'm looking at it from the side, I'm seeing some sort of angle. Well, what does it look like from the right side? Looks like that from the top. Looks like this from the angle. Let's see if we can recreate that. Once again, start with the point. Create a miter line in this case. We will need the miter line. Make sure that the points lined up on the x-coordinate, they share an x-coordinate, find the y-coordinate off of the miter line, and find the z-coordinate off of the front view. And there we've established point 1. Move on to point 2. There's the y-coordinate bouncing off the miter line. Here's the x-coordinate, which is lined up. And here's the z-coordinate, which is on the same place as point 1. But further out on the right side. So there's point 2. And notice, 1 and 2 are connected in a straight line here, and they disappear to a point there. So we can feel good about there not being any curve between 1 and 2. It's a straight line between 1 and 2. For point 3, seems like they have the same x-coordinate. Get the y-coordinate by bouncing off the miter line, and get the z-coordinate from the front view. And there's point 3. And again, the line between 1 and 3 seems to be a line. We'll fill in the line. There's point 3. Now the question is, what does the curve look like? We can see that the line between 2 and 3 is curved here, but it's straight there. So what does it look like here? How much of a curve is visible from there? To do that, we go back to our curve idea. We pick a location on one of the curves. We figure out what the x-coordinate is. We line up the x-coordinate. So there's my other location. So my intermediate point between 2 and 3. Come across to the miter line and bounce. Come across from this point, and that recreates for us an intermediate point on the curve between 2 and 3. And I do my best to sketch the curve so it goes between those two. Notice it seems to be maybe a little flatter curve, maybe not much, perhaps a little flatter than the original one in the top view. That actually should be about the same curve because that's a 45 degree angle. But now we've recreated the view of a curved piece. So again, if I take this piece, let's see here, and show it to you from here. And you know it's tilted. If I then take it, look at it from there, you can see how we've replicated. Going to encourage you, there are a number of different exercises here that we haven't actually practiced with. In order to get comfortable with this, you're going to need some practice. So I'm going to encourage you to look at the other examples here. There are four full pages of them. Practice them on your own so you're comfortable with recreating lines, points, and planes in orthographic projection view.