 OK, so we're going to continue our discussion of orthographic projection of points, lines, and planes. But we're going to talk about a different kind of shape or a different kind of line here. We're going to get a little more complicated by considering a curve, throwing you a curve ball if I didn't did it. All right, so we're going to take a curve and notice that if we have something that's sort of curved, a curved line, all right, that when we rotate it, if we rotate it in the right set, you can actually get it to the point that you can no longer even notice the curve. If you look at it from a particular angle, if the curve is flat in one plane, it will eventually flatten out and become a line. And in fact, if we rotate it again another 90 degrees, you'll still see a line. So it's possible that something that's curved may only look like a line from some other perspectives. You notice that's very true if we have something like a cup. Here I have a cup that I'm holding all my pencils in, okay? Notice if we look at the top of the cup, we have a round curve, but if we turn and look at it from the side of the cup, the curve disappears into a line and from just that perspective, you don't necessarily know that it's a curve without additional information. So let's see how we can represent a curve in our orthographic space. How do we go about taking information about a curve and translating it into three pictures? Well, it's not gonna be much different than a segment. In fact, the first thing we're gonna do here is exactly what we do for a segment. We're going to note the end points of the curve and we're gonna treat them exactly the same way. So I wanna go ahead and recreate this curve on the right side. So to do so, I'm going to take point one. Notice that it lines up with point one in my top view. Create my miter line so that I can move horizontally from point one in the front view, horizontally from point one in the plan. When we hit the miter line, we drop straight down and find a location for point one on the right side. Here's a location for point one on the right side. Do the same thing for point two. We'll notice that point two has a common X coordinate that we'll connect vertically here. We'll define the Z coordinate by coming straight across from there, and then we'll define the Y coordinate by coming across from the plan view, hitting the miter line, and dropping down to finish our box. And the location where they meet is point two. Now, the question is, how do we connect point one and point two? Because in the front view, they connect with a straight line. In the top view, they connect with the curve. Do I use a curve here, or do I use a straight line? What does it look like? Well, this is where it gets a little more complex, but what we're gonna do is we're gonna need to sort of make a point in the middle. We can pick any point along the curve. In fact, I'm gonna pick a point right here that's about two thirds of the way between point one and point two, okay? And by simply picking that point on the curve, I can figure out by going straight up what all three coordinates it has are, okay? It has a point right here. Notice this point seems to be in the middle of that curve, but it's about two thirds of the way over from the side. Does that make sense? Let's take a look. If I'm looking at something that's curved and I grab it right here in the middle, okay, so that it's halfway between, sort of halfway around the curve, and then I rotate it up for you to look at it, you'll notice there's a lot more, about two thirds of the curve is here and there's less of it on the part that's coming to you because it's basically turning and going vertical. So that does make some sense. So let's continue with this intermediate point we have here. It has one point here on the curve. It has one point here on the line. They have matched X coordinates. Let's find the Z coordinate, which is still on this same line as everything else, and let's find the Y coordinate, which if we move along here, we find a bouncing point, we bounce down here, and it turns out it ends up on that straight line between one and two. Well, I could do another one just to check, but it seems pretty clear to me that everything's going to end up on this line, so there is what our curve looks like from the side. Let's see if we can repeat that process. Go to another page here, page four, four. Okay, repeat that process for another curve, okay? And this one's a little bit easier. We don't have to use the miter line because we're making a front projection, but let's see how we do that. We'll start with point one. Establish our Z axis by coming horizontally from point one. Establish our X coordinate by coming vertically from point one in the plan view, and fill it in, there's point one, okay? And we'll do the same thing for point two. Establish the Z coordinate. Establish the X coordinate. There's where point two meets point two. However, we don't know what the curve looks like. Is it a line? Is it a curve? Before it ended up being a line, let's see if we can verify it. I pick any point along my curve. Let's do what we did last time. Let's go ahead and pick this point that's right. Let's, well, actually, let's pick a point right here in the middle of the curve. I'll pick this point here, some intermediate point between one and two. We could call it one and a half. If I drop straight down from one and a half, okay? Drop straight down from one and a half. Well, that establishes where the X position is, but I don't know where the Y position is. Looks like I might need my miter line after all, okay? But here's what's important. I can't just draw any miter line. I've gotta make sure my miter line is lined up. When I don't have something, I can draw the miter line where I would like. But now there is a miter line. It's definitely defined by these two things. So I gotta make sure that my miter line matches with my previous part. So I'm gonna go up from point two and over from point two, up from point one and over from point one to make sure that these meet, okay, and make sure my miter line runs between those two points because I could accidentally put my miter line shifted over someplace and that would mess things up. Okay, but notice that completes the box there. So now with my green intermediate point, I can come straight across. It appears to be, actually, it's a little bit below that point there. So I'm gonna come just a little below that line till it hits my miter line and then come straight down. Here we go and it seems to hit it right there on the, or maybe a little bit off, but on the one third point or so, okay? Well notice if I line those up, it's not real prominent here, but if we look carefully, you can see that that point on the front is just a little bit off. It's not directly between one and two. It's a little bit off of the curve, okay? So I would need to do my best to run my curve through that point. If I wanted more accuracy, I could pick more intermediate points and see where they reflect, but you see there's a very minor curve associated with this piece. Does that make sense? Well, let's see if we can take a look at it, okay? If you're looking at a plan view of something curved, so here's my curved piece, but I'm going to recognize that the side view has the top up a little bit higher. So here's the plan view that you see. If I rotate it 90 degrees, oops, I actually did it the wrong way. It's gonna be rotated this way. Here's the plan view that you see. If I rotate it 90 degrees so you can see the side, you'll see there's the angle. Now, trust me that I'm rotating at the right amount. And then if I rotate it so that you're seeing the front, you'll notice that there is, oops, too far, I did too far. You'll notice there is just a little bit of a curve. There's some, it's a combination of that diagonal view that we saw and the curve view that you saw from the top. So hopefully from that, you can see how you can approximate curves or get a sense for how to draw curves as long as you know the end points and have some sense for the curve and at least one of the view points.