 So in engineering, we're often interested in talking about how to express information about a three-dimensional object in only two dimensions. Since screens and pieces of paper are sort of our more common ways of sharing information. We're going to do this by talking a little bit about orthographic projection. And we'll start by projecting simple geometric objects like points, lines, and planes, and talking how we can represent those using orthographic projection. So let's start by looking at, well, here's a three-dimensional object. It's a very simple three-dimensional object. It's a paperclip that's been straightened out. And in essence, it's been reduced from really having three dimensions to having one dimension in the sense that it's very long in one dimension. And the other dimensions are so small that they could almost be ignored if we're thinking about this. And mathematically, we could shrink it so that it's basically a line in geometry class is effectively just one-dimensional because the thickness is not important. Well, notice if you take this line and we're trying to draw it in space, depending on how you look at it, it might look differently. Right now, you're looking down on it from the top. But if I take and rotate it 90 degrees, it still looks the same if I rotate it 90 degrees. But if I now rotate it 90 degrees in this other dimension by turning it and spinning it, if I hold it exactly 90 degrees so it's pointing straight up at you, it almost disappears. All you see is the teeny, tiny point, and the rest of the line all disappears in behind there. So basically, it's represented by a point. This is kind of a key concept in our orthographic projection. This idea that any line will be visible from two of the dimensions, from the top, and from looking at it from the front, but it will become nearly invisible and fade to a point when we look at it from one of the sides. Now, that changes if you reorient your line. If I take my line and make a diagonal, well, then it looks like one form of diagonal from one side. If you rotate it 90 degrees, it might look a little different from another side. And if you rotate it 90 degrees, let's see here, looking from the top, looking at it from the front like this. If you rotate it 90 degrees like that, it'll look even different. And in none of those cases, does it completely disappear because you're not orienting your axis at all with parallel to that line. Notice this doesn't just apply for lines. This also applies for lines that make up the edges of something. The line that makes the edge here of this index card, similarly, has a different view if you look at it from 90 degrees. Okay, and again, we're just thinking about this line. You can still see it from 90 degrees, but if I turn and rotate it here, the line almost disappears if the card's straight. Okay? So we're gonna talk about how to sort of create those lines. Let me bring out one more example. Three dimensions. Again, here's a line. You can see the line fairly clearly from this dimension, but if I rotate in this dimension, that same line disappears from your view as long as it's parallel to one of your, to your viewpoint. So let's talk about how we can represent some of these things in space. Here we have a series of examples where we have a picture of something and we're providing two of the three orthographic views. Two of the three orthographic views. In this case, we have an orthographic view of something like our little example here, running from point one to point two. And when we look at it from the front, okay, so let's see if I can rotate this down really quickly. If we look at it from the front, it looks something like this. Hey, you can see all the screens. If you look at it from the top, actually, I'm sorry. If you look at it from the front, it actually has a slant to it. There we go. Whereas if you look at it from the top, it's straight, even though there is a slant that's there, what does it look like from the side if we walk over here and look at it from this side? Well, let's see if we can figure that out and learn how to do some orthographic projections at the same time. Notice if we've taken two views, if we've taken two views of our object, that's enough to give us most of the information we want. All right, but I wanna go ahead. I've got a front view. This is the front. And I have a top view or a plan view. And I wanna create a view of the right side. Let's see how we do that. Let's start by creating some axes. Okay, on this thing here, I'm gonna go ahead and create a point here. We're gonna go ahead and create an axis. This is going to be our x-axis in the x-direction, okay? Notice that x-direction works for both the front view. It's the side to side in the front view as well as the side to side in the plan view, okay? The front view also goes up. So I'll create a z-dimensional up on the front view. And that same z-dimension, I can repeat it here if I would like to, that same z-dimension applies to the right side. Just like the same x-dimension. You don't have to redraw it, but just to be clear, the x is repeated on both of those. And then our last piece here is our y-dimension. In other words, our back to front. Well, when I look from the plan view, there is a y-dimension on the plan view. Let me go ahead and draw it here. Y, okay? But it's not the same if it's, this is the y's going from the back to the front, okay? But when we look at it from the side, when we rotate and look at back to the front from the side, it ends up being a different coordinate here. And this is where, as we've seen before, we're going to need our miter line. Let me draw this 45 degree angle, which allows us to line up. If I take this y-axis, I can line up my y-axis so that it travels horizontally in the right view, but lines up with the vertical version of it in the plan view. Here it's up and down, here it's side to side. So if we want to recreate this line segment here in the right side view, all we have to do is start by recreating points. Let's recreate point one. I'm gonna mark point one in blue right here. And I notice that the x-axis of point one, if I draw straight up, that point one also has, is located directly above it. They both have the same x-coordinate. And in fact, any time you draw a point in orthographic projection, you're going to create a box. This is one side of the box. The second side of the box looks like this. I'm going to go directly sideways from the front view where point one is into the right side view. That's gonna establish the z-axis. So we know that in the right side, point one is somewhere along this line. It has the same z-coordinate as the front view. Where is it? Well now, we do the same thing. We go horizontally from point one until we hit our miter line. When we hit our miter line, we turn 90 degrees. And what we've established is a location, a y-axis. Here's y, we'll say that's y equals zero. And notice we'll line it up with y equals zero here on our right side axis. So we've created one point, two points, a place where it bounces off the miter line. And then the fourth point is the view on our right side. And that becomes, fill it in, there is point one. We can now do the same process with point two. I'll use a slightly different color here. Let's use a red. I'll start with point two here. I'll notice that point two lines up with the point two in the plan view. I'm gonna move across from point two here somewhere. But where do I stop? Well, let's figure it out. If I go across from point two, I hit the miter line. At the same point as point one did because they have the same y-coordinate, drop down from that point, and there is my point two in my right side view. And if I connect those two points, there's my right side view. So when I had this thing that was angled, from the top view it looked angled, but I'm gonna lift my right hand up here so that when you looked at it from the side, there's the angle we saw from the side. And if you look at it from, let's see if I can rotate it this way. You'll notice that you see pretty much a straight line that way, but a shorter straight line. My arm is kind of in the way there. Let's see if we can think about that from another perspective. Let's go ahead and find the actual coordinates of our points here. Point one has coordinates. Let's see, our point one has an x-coordinate, a y-coordinate, and a z-coordinate. Okay, well, let's see here. Point one has an x, y, and z. We'll go ahead and say that it's located at the beginning of the axis. So it's an x-component of zero. It has a y-component of zero. And we'll say it has a z-component of zero. So it's at zero, zero, zero. Okay. For our y-component, we can look, for our point two, we can see that it has, it's different x. Let's say that's one, two, three, four, five units. It has a five units. Notice it has five units in both of these views. On the z-axis, it goes up two units. So it has a z-coordinate of two units. And a y-coordinate of, well, let's see here, y-coordinates there. A y-coordinate looks like is at zero. Yup. Is where the y-coordinate starts. So it has coordinates of five, zero, two. Okay. So in each of those, we sort of have a position. We know where each point is, and then the two points connect to each other. Let me see if I can show you that in sort of a three-dimensional drawing here. We're using that logo in a three-dimensional drawing for just a second. Gonna put in some coordinates. Where'd it go? There we go. Okay, and I'm gonna put in some values here. I'm actually going to make the points just a little, I'm gonna add one so that they're actually a little bit more visible. So this was zero, zero, zero, and the other one is going to be, I will say, 51, one, and 21. So I'm gonna draw a very similar picture here. Okay, I'm actually using 10 units to be one unit for our, so it's a little easier to see it. And I'm shifting it by just a little bit so it's easier to see. Okay, but if you look at our viewpoint here, this is our x-axis. There's our y-axis going back into the page, and here's our z-axis going up and down. You can see the viewpoint from the front is indeed that angle that we see. Let me show you the angle, is indeed that first angle. Okay. If I then rotate and look down on it from the top, there's the top view, which is the flat line that we saw in the original. And then if I rotate it and look at it from the side, there you can see that we, again, have a vertical line. I haven't kind of oriented right there, but there's a vertical line that's much shorter. It just represents our changes in the z dimension. Okay, but there's that vertical line that we recreated, so we apparently did a correct job of it there. Let's go ahead and continue with another segment example. We'll do a simpler segment example. We'll do this one immediately to the right here. Okay, let's see if we can do that. Now, this changes things just a little bit because now we have our front view and our side view, but we do not have our top view. Well, how do we handle that? Let's pick a point. Here's point three. Here's point three. Notice three and four are located right on top of each other, which means there's some line extending out toward us when we're looking at it from this side. But if we take point three and point three, that establishes the bottom of our box for point three. Point three moves up from here, and it also moves up from here until it hits the miter line. Well, now I've got to create a miter line, make a 45 degree line here, and I bounce that blue target. There I hit the miter line and I turn 90 degrees. And where those two lines intersect, that is our location for point three on our top view. How about for point four? See if we can do it in red here as well. Point four connects across here. They both have the same z-coordinate. Here's the x-coordinate in the top, okay? And if we look, point four and point three both have the same y-coordinate. So that means point four in the top view is located at that fourth point in the rectangle. Fill that in, draw a line across, and there is our view. And notice that view is pretty straightforward. That's just a straight line, rotated 90. We still see the straight line, rotated 90. We see straight down and see just the point. So that's a, because it's lined up with one of our dimensions, it disappears into a point. Last example, using just segments. I'll walk through this without talking. See if you can stay a little bit ahead of me as you create point five and six this time from a front view. Notice it's a little bit easier. Why was it easier here? Well, we didn't actually need the miter line. The relationship between the top view and the side view wasn't really necessary. I can go back and put that miter line into play and you will notice that point six does indeed line up when bounced off the miter line as does point five. But we didn't actually need those in this case because we already had the two views that are related through the miter line. Let's check this one more time and see if we can see what it looks like in 3D. Let me go ahead and get for point one, the coordinates and for point, I'm sorry for point five, the coordinates and point six, the coordinates. Point five, let's go ahead and assume that this point down here will start, will let this base point BRZ axis and you'll notice that point five has a Z coordinate of zero. If point five has a Z coordinate of zero, point six has a Z coordinate one, two, three, four of four units. Point five has an X coordinate if we're going in that direction, an X coordinate of zero and that means that point six has coordinates one, two, three, four units to the right of point five. So point six has an X coordinate of four, four units to the right. And then finally, if we're talking about our Y axis, we're going to define our Y axis up here. We're going to start our Y axis at the point six. That's the furthest to the front. So we'll let six have a value of zero, okay, and then five has a value of two. So the coordinates of point five are zero on the X axis, two on the Y axis and zero on the Z axis, zero on the Z axis. And for point six, it's over four on the X axis, zero on the Y axis and up four on the Z axis. Let's see if I can put those into net logo. I'm going to translate them into this net logo for me to see if we can look at it in the three dimensional view here. Okay, so the points I'd like to see here are going to be, we'll say zero one, 21 and zero one, just making each unit mean 10 units. And then here we'll have 41, one and 41. And when I update that, let's compare our views, okay? Here's our XY view. In other words, I mean, I'm sorry, here's our XZ view, looking at it from the front. Let's compare, okay? Here's our front view. See, it's slanted from the front view. If I rotate down and look at it from the top, there's the top view. And finally, if I rotate it over here and look at it from the side, whoops, there's an extra line there, there is our side view. So hopefully from that, you get an idea how we can create a line segment or make line segments connect using orthographic viewpoint.