WEBVTT 00:00:00.666 --> 00:00:03.816 A:middle >> This lecture is on the first part of descriptive geometry 00:00:04.106 --> 00:00:06.036 A:middle and you remember this figure here. 00:00:06.066 --> 00:00:08.406 A:middle This was the figure that we did for one of our labs 00:00:08.406 --> 00:00:10.126 A:middle on secondary auxiliary views. 00:00:11.266 --> 00:00:14.566 A:middle We were given the front here and the right side view 00:00:14.656 --> 00:00:18.156 A:middle and you were given this primary auxiliary reference plane here, 00:00:18.156 --> 00:00:20.606 A:middle the secondary auxiliary reference plane here. 00:00:21.036 --> 00:00:24.266 A:middle The first reference plane gave us this auxiliary, 00:00:24.346 --> 00:00:25.526 A:middle primary auxiliary, view. 00:00:25.526 --> 00:00:28.606 A:middle And then the secondary auxiliary view is here. 00:00:28.836 --> 00:00:31.896 A:middle It turns out to be the true size of plane ABC. 00:00:32.526 --> 00:00:34.786 A:middle So the question is, "How did we know how 00:00:34.786 --> 00:00:38.776 A:middle to position these two auxiliary reference planes in order 00:00:38.776 --> 00:00:41.086 A:middle to get the true size of plane ABC?" 00:00:41.136 --> 00:00:44.806 A:middle And that's one of the questions that we're going to be answering 00:00:44.806 --> 00:00:47.286 A:middle in this topic called descriptive geometry. 00:00:48.656 --> 00:00:51.796 A:middle It involves the projection of three dimensional figures 00:00:51.796 --> 00:00:55.046 A:middle on a two dimensional plane of the paper in a manner 00:00:55.046 --> 00:00:56.776 A:middle that allows geometric manipulation 00:00:56.776 --> 00:00:59.096 A:middle to determine lengths, angles, shapes, 00:00:59.096 --> 00:01:02.436 A:middle areas and other quantitative information by means of graphic. 00:01:02.436 --> 00:01:05.726 A:middle So we're going to use both the graphic projections specifically 00:01:05.726 --> 00:01:08.886 A:middle in order to extract useful quantitative information 00:01:09.176 --> 00:01:09.946 A:middle from the drawings. 00:01:10.006 --> 00:01:15.386 A:middle The auxiliary views are used to show things like true length 00:01:15.386 --> 00:01:18.316 A:middle and point of views of lines, true sizes and edge views 00:01:18.316 --> 00:01:21.686 A:middle of planes, angles between two lines, between two planes, 00:01:21.946 --> 00:01:23.236 A:middle and between the line and the plane. 00:01:24.676 --> 00:01:27.636 A:middle This is based -- The method we're going to use is based 00:01:27.636 --> 00:01:31.266 A:middle on the Folding-Line Model which we can use in order 00:01:31.266 --> 00:01:33.916 A:middle to visualize the resulting auxiliary views. 00:01:33.916 --> 00:01:38.366 A:middle Starting from the simplest entity, we can look at in 3D, 00:01:38.886 --> 00:01:41.606 A:middle a point is nothing but a location in space. 00:01:41.606 --> 00:01:43.946 A:middle Of course no dimension, just a location. 00:01:44.316 --> 00:01:46.816 A:middle And to specify the location you're going 00:01:46.816 --> 00:01:51.396 A:middle to need three coordinates, X, Y, Z, to define a point in space. 00:01:51.786 --> 00:01:55.906 A:middle So for instance this could be the height H below the 00:01:55.976 --> 00:01:56.626 A:middle top plane. 00:01:56.626 --> 00:01:59.176 A:middle The width, number of units to the left 00:01:59.176 --> 00:02:02.136 A:middle of the right profile plane P. And also the depth 00:02:02.136 --> 00:02:05.446 A:middle which is how far behind you are from the frontal plane. 00:02:06.036 --> 00:02:11.876 A:middle In order to define all three of the dimensions, height, 00:02:12.416 --> 00:02:16.406 A:middle width and depth, we're going to need at least two views. 00:02:16.706 --> 00:02:18.426 A:middle Okay. Here's an illustration of -- 00:02:18.806 --> 00:02:24.246 A:middle This seems to be an oblique perspective of the glass box 00:02:24.916 --> 00:02:27.806 A:middle that we're trying to illustrate using the folding line method. 00:02:28.466 --> 00:02:32.976 A:middle Here's point two, okay, and here's the projection 00:02:33.016 --> 00:02:36.216 A:middle of point two in the front view, top view 00:02:36.216 --> 00:02:37.406 A:middle and the right side view. 00:02:38.116 --> 00:02:41.716 A:middle And it's illustrating the three dimensions in height 00:02:42.206 --> 00:02:46.516 A:middle which is measured in how many units you are below the top 00:02:46.576 --> 00:02:47.546 A:middle horizontal plane. 00:02:48.956 --> 00:02:52.116 A:middle There's the dimension of width, how many units you are 00:02:52.116 --> 00:02:54.386 A:middle to the left of the profile plane here. 00:02:55.236 --> 00:02:59.506 A:middle And this depth here, how many units you are behind the 00:03:00.056 --> 00:03:00.706 A:middle frontal plane. 00:03:00.706 --> 00:03:03.426 A:middle And take note that for each fold in the -- 00:03:03.706 --> 00:03:05.846 A:middle corresponding to the projection planes, 00:03:06.636 --> 00:03:09.876 A:middle you have your planes labeled horizontal here 00:03:10.056 --> 00:03:14.706 A:middle for the top view, F for frontal, P for profile. 00:03:15.766 --> 00:03:18.906 A:middle Going to lines now which are nothing but straight paths 00:03:18.906 --> 00:03:23.846 A:middle between two points, a line can appear in a view 00:03:23.846 --> 00:03:27.576 A:middle as either foreshortened, true length or point view. 00:03:27.796 --> 00:03:31.916 A:middle The first set of top and front views here, one, two, 00:03:31.916 --> 00:03:35.666 A:middle is shown both in the front view here and top view 00:03:35.666 --> 00:03:38.876 A:middle as foreshortened, meaning it appears as a line, 00:03:38.876 --> 00:03:41.286 A:middle but it's shorter than its actual true length. 00:03:41.506 --> 00:03:45.976 A:middle And the second example here, line three, four, is horizontal, 00:03:45.976 --> 00:03:50.916 A:middle HL, because it's parallel to the fold line that corresponds 00:03:50.966 --> 00:03:51.966 A:middle to the top plane here. 00:03:52.986 --> 00:03:55.096 A:middle So this line here, three, four, 00:03:55.636 --> 00:03:57.096 A:middle three and four have the same height. 00:03:57.426 --> 00:04:00.846 A:middle Therefore it's horizontal and a horizontal line when viewed 00:04:00.846 --> 00:04:04.566 A:middle from the top like this is true length or TL in this case. 00:04:05.256 --> 00:04:07.236 A:middle The last combination of front 00:04:07.236 --> 00:04:11.426 A:middle and top views illustrates line five, six, which happens 00:04:11.456 --> 00:04:13.526 A:middle to be perpendicular to the frontal plane. 00:04:13.786 --> 00:04:17.766 A:middle And, as a result, this line five, six becomes true length 00:04:17.966 --> 00:04:22.026 A:middle in the top view, but a point view in the frontal plane, 00:04:22.026 --> 00:04:24.906 A:middle meaning five and six both correspond to the same point. 00:04:25.996 --> 00:04:29.216 A:middle Classification of lines according to their orientations 00:04:29.216 --> 00:04:32.716 A:middle with respect to the principal projection planes of horizontal, 00:04:32.716 --> 00:04:34.476 A:middle frontal and profile planes. 00:04:35.066 --> 00:04:39.436 A:middle We have a line classified as principal, principal line. 00:04:39.436 --> 00:04:42.686 A:middle It was parallel to at least one of the principal planes. 00:04:43.206 --> 00:04:45.986 A:middle An oblique line is one that's neither parallel nor 00:04:46.216 --> 00:04:48.836 A:middle perpendicular to any of the principal planes. 00:04:50.146 --> 00:04:54.356 A:middle Horizontal line is a principal line. 00:04:54.706 --> 00:04:58.176 A:middle It's parallel to the top horizontal plane and, 00:04:58.176 --> 00:05:01.326 A:middle as a result, appears true length in the top view. 00:05:01.726 --> 00:05:05.546 A:middle A frontal line is parallel to the frontal plane 00:05:05.876 --> 00:05:07.836 A:middle and appears true length in the front view. 00:05:07.836 --> 00:05:11.876 A:middle Similarly, a profile line is parallel to the profile plane 00:05:12.136 --> 00:05:15.966 A:middle and appears true length in the side views. 00:05:16.476 --> 00:05:18.996 A:middle Here is an illustration of the horizontal line. 00:05:19.506 --> 00:05:20.866 A:middle One, two is horizontal. 00:05:20.866 --> 00:05:22.996 A:middle They have -- One and two have the same height. 00:05:23.456 --> 00:05:25.956 A:middle As a result, it's true length in the top view. 00:05:26.616 --> 00:05:28.006 A:middle Three, four is frontal. 00:05:28.506 --> 00:05:31.336 A:middle Why? Because three and four have the same depth 00:05:32.216 --> 00:05:36.596 A:middle or distance behind the frontal plane either viewed from the top 00:05:36.736 --> 00:05:38.396 A:middle or here at the right side view. 00:05:38.396 --> 00:05:41.656 A:middle As a result, the frontal line three, 00:05:41.656 --> 00:05:44.386 A:middle four appears true length in the front view. 00:05:44.386 --> 00:05:49.226 A:middle And five, six is an example of a profile line. 00:05:49.506 --> 00:05:52.876 A:middle It's parallel to the profile plane, the right profile plane. 00:05:53.566 --> 00:05:57.306 A:middle As a result, it's true length in the right side view. 00:05:58.496 --> 00:06:02.776 A:middle One of the things we can find using descriptive geometry is 00:06:03.076 --> 00:06:04.666 A:middle the true length of a given line. 00:06:05.276 --> 00:06:09.456 A:middle We can either use Pythagorean Theorem using true length 00:06:09.456 --> 00:06:12.126 A:middle diagram or by using an auxiliary view 00:06:12.426 --> 00:06:13.736 A:middle to find the true length of the lines. 00:06:14.336 --> 00:06:15.986 A:middle Using Pythagorean Theorem, 00:06:16.056 --> 00:06:18.006 A:middle really Pythagorean Theorem in 3D. 00:06:18.006 --> 00:06:22.476 A:middle Those of you who have taken physics and Calc 3, you know 00:06:22.476 --> 00:06:25.266 A:middle that this line here, that it 00:06:25.266 --> 00:06:28.566 A:middle in 3D generally can have three components, 00:06:28.626 --> 00:06:31.156 A:middle one along X which you call the width, 00:06:31.516 --> 00:06:33.166 A:middle one along Z which you call the depth 00:06:33.246 --> 00:06:35.256 A:middle and Y which we call height. 00:06:35.766 --> 00:06:41.966 A:middle If you look at the projection of the plane in the XY plane 00:06:42.516 --> 00:06:47.466 A:middle of the -- in the XY plane, this length here is equal 00:06:47.466 --> 00:06:49.846 A:middle to the square root of X squared plus Y squared 00:06:49.846 --> 00:06:50.746 A:middle by Pythagorean Theorem. 00:06:50.746 --> 00:06:54.676 A:middle And now if you look at this triangle over here now, 00:06:55.366 --> 00:06:58.676 A:middle it is also a right triangle where this is one of the legs 00:06:59.296 --> 00:07:01.056 A:middle and this is the other leg, is Z -- 00:07:01.176 --> 00:07:02.766 A:middle If we use the Pythagorean Theorem, 00:07:02.766 --> 00:07:05.216 A:middle we see that the true length of the line here 00:07:05.486 --> 00:07:07.846 A:middle from point one here to point two 00:07:07.896 --> 00:07:10.526 A:middle in 3D space is just the square root of the sum 00:07:10.526 --> 00:07:14.246 A:middle of the squares of the XYZ components. 00:07:14.626 --> 00:07:17.016 A:middle Here is a graphic where you're finding the true length. 00:07:17.276 --> 00:07:19.696 A:middle Here's the top view and here's the front view. 00:07:20.276 --> 00:07:23.196 A:middle What you do is you create your answers here 00:07:23.716 --> 00:07:27.656 A:middle to create your true line diagram, 00:07:27.976 --> 00:07:30.036 A:middle a vertical axis here and the horizontal. 00:07:30.616 --> 00:07:33.436 A:middle And what you do is you measure the height of the line. 00:07:33.486 --> 00:07:37.096 A:middle That is the vertical distance from point one to point two. 00:07:37.856 --> 00:07:39.826 A:middle And plot it on the vertical side here. 00:07:39.826 --> 00:07:42.316 A:middle And then measure the horizontal extent 00:07:42.756 --> 00:07:44.206 A:middle or the horizontal projection 00:07:44.836 --> 00:07:46.496 A:middle which can be seen from the top view. 00:07:46.626 --> 00:07:49.146 A:middle That's H in the horizontal. 00:07:49.916 --> 00:07:54.476 A:middle And just connect this to points here to find the true length 00:07:54.476 --> 00:07:57.376 A:middle of the line, measure the true length of the line. 00:07:57.376 --> 00:08:00.436 A:middle Another way of doing -- finding the true length of the line, 00:08:00.906 --> 00:08:03.446 A:middle of a line, is by using an auxiliary view. 00:08:03.446 --> 00:08:06.316 A:middle Given, for instance, say the top view of AB here 00:08:06.316 --> 00:08:09.926 A:middle and the front view of AB, we're asked to find an auxiliary view 00:08:09.926 --> 00:08:12.756 A:middle that would show the true length of line AB. 00:08:12.756 --> 00:08:14.926 A:middle Now for you to see the true length 00:08:14.926 --> 00:08:16.856 A:middle of a line, visualize this. 00:08:16.856 --> 00:08:18.556 A:middle If you have a line in 3D space, 00:08:18.886 --> 00:08:21.726 A:middle the only way you can see the length of that line is 00:08:21.726 --> 00:08:23.576 A:middle if your line of sight is perpendicular 00:08:24.036 --> 00:08:25.156 A:middle to the line itself. 00:08:25.156 --> 00:08:28.196 A:middle So the line of sight needs to be perpendicular to the line, 00:08:28.656 --> 00:08:31.486 A:middle but we know that the line of sight is always perpendicular 00:08:31.486 --> 00:08:34.676 A:middle to the projection planes or the reference plane which means 00:08:34.766 --> 00:08:38.616 A:middle that if two things are perpendicular to the line 00:08:38.616 --> 00:08:43.056 A:middle of sight, okay, those two things must be parallel to each other. 00:08:43.056 --> 00:08:46.456 A:middle Therefore the reference plane must be parallel to the line 00:08:46.456 --> 00:08:48.486 A:middle that you're trying to find the true length of. 00:08:49.246 --> 00:08:51.526 A:middle What you need to do is, as shown here -- 00:08:52.166 --> 00:08:56.296 A:middle You take a reference plane here, parallel to line AB. 00:08:56.496 --> 00:08:58.036 A:middle I did that on the top view. 00:08:59.036 --> 00:09:02.746 A:middle And then construct these green lines 00:09:02.846 --> 00:09:07.356 A:middle which are projection lines projected perpendicular 00:09:07.436 --> 00:09:09.476 A:middle to the reference plane here. 00:09:09.476 --> 00:09:12.096 A:middle So we have labeled the fold line here. 00:09:12.666 --> 00:09:14.816 A:middle It's between the top view and the front view, 00:09:14.816 --> 00:09:17.006 A:middle so H and F, H and F here. 00:09:17.396 --> 00:09:21.916 A:middle The fold line here is between the horizontal or the top view 00:09:21.916 --> 00:09:23.786 A:middle and the primary auxiliary view. 00:09:24.286 --> 00:09:26.636 A:middle So this is the auxiliary reference plane number one. 00:09:27.146 --> 00:09:28.766 A:middle This is the fold line between the two. 00:09:29.286 --> 00:09:32.996 A:middle And what we do is to locate the position 00:09:32.996 --> 00:09:37.596 A:middle of A along this construction line, the depth of point A 00:09:37.596 --> 00:09:40.026 A:middle which represents the perpendicular distance 00:09:40.386 --> 00:09:45.406 A:middle from H here, the top plane, is the same as the depth 00:09:45.816 --> 00:09:47.446 A:middle as seen in the front view. 00:09:47.856 --> 00:09:51.146 A:middle So our current central view is the top view corresponding 00:09:51.146 --> 00:09:52.286 A:middle to the horizontal plane. 00:09:52.706 --> 00:09:55.756 A:middle Perpendicular distance away from H as seen 00:09:56.086 --> 00:09:58.496 A:middle from the front view should be exactly the same 00:09:58.496 --> 00:10:01.266 A:middle as the perpendicular distance away from H 00:10:01.266 --> 00:10:02.736 A:middle as seen in auxiliary view. 00:10:03.286 --> 00:10:05.676 A:middle So, as we can see here, A is at the height 00:10:05.676 --> 00:10:08.586 A:middle of 1.5 below the top plane 00:10:08.586 --> 00:10:14.886 A:middle or H. Therefore it's also 1.5 units away from the H here. 00:10:14.886 --> 00:10:18.406 A:middle Similarly, B is anywhere along this green projection line, 00:10:18.816 --> 00:10:25.116 A:middle but it's -- a height of 0.5 below H. Therefore it is going 00:10:25.116 --> 00:10:30.236 A:middle to be 0.5 away from H. So the procedure is, just to review, 00:10:30.566 --> 00:10:32.836 A:middle take this reference plane parallel to AB, 00:10:32.836 --> 00:10:36.346 A:middle create these construction lines perpendicular 00:10:36.346 --> 00:10:39.336 A:middle to that reference plane that was created parallel to AB -- 00:10:39.336 --> 00:10:44.546 A:middle By the way, the position of this auxiliary reference plane is 00:10:44.616 --> 00:10:47.506 A:middle arbitrary as long as it's parallel to AB. 00:10:47.506 --> 00:10:51.696 A:middle It could be slightly farther here or slightly closer to AB. 00:10:51.696 --> 00:10:52.976 A:middle It wouldn't make a difference. 00:10:53.556 --> 00:10:57.436 A:middle And then to find the locations of A and B we need to measure 00:10:57.436 --> 00:11:00.446 A:middle or borrow the heights or distances away 00:11:00.446 --> 00:11:02.966 A:middle from H here shown in the front view. 00:11:04.516 --> 00:11:06.646 A:middle Direction of a line. 00:11:06.646 --> 00:11:07.626 A:middle When you're looking at a map, 00:11:07.896 --> 00:11:10.346 A:middle the map is really the top view of the terrain. 00:11:10.406 --> 00:11:13.516 A:middle So direction of a line is always shown in the top view. 00:11:14.116 --> 00:11:16.326 A:middle And there's two common ways in engineering 00:11:17.126 --> 00:11:19.156 A:middle to give the direction of a line. 00:11:19.456 --> 00:11:22.796 A:middle One is using the compass bearing which is nothing 00:11:22.796 --> 00:11:25.816 A:middle but an angle measured from the north or the south, 00:11:25.816 --> 00:11:29.476 A:middle depending on which one's closer, measured towards the east 00:11:29.476 --> 00:11:31.846 A:middle or the west between 0 and 90 degrees. 00:11:32.386 --> 00:11:36.646 A:middle Or as the azimuth usually used by civil engineers. 00:11:37.356 --> 00:11:40.096 A:middle It's the angle measured always from the north. 00:11:40.236 --> 00:11:42.516 A:middle At least in the northern hemisphere we use north 00:11:42.516 --> 00:11:43.326 A:middle as the reference. 00:11:43.436 --> 00:11:44.466 A:middle In the southern hemisphere, 00:11:44.466 --> 00:11:46.976 A:middle their azimuths are measured from the south. 00:11:47.406 --> 00:11:49.506 A:middle And it's an angle that's measured from 0 00:11:49.506 --> 00:11:52.336 A:middle to 360 degrees clockwise. 00:11:52.956 --> 00:11:57.826 A:middle So, for instance, line AB -- So with A as the reference, 00:11:57.826 --> 00:12:05.086 A:middle I have four lines, AB, AC, over here AD and AE. 00:12:05.086 --> 00:12:07.106 A:middle And we are given these angles here. 00:12:07.706 --> 00:12:11.246 A:middle Based on these angles we can find the compass bearing 00:12:11.386 --> 00:12:12.036 A:middle and azimuth. 00:12:12.036 --> 00:12:13.696 A:middle So AB, here's AB. 00:12:13.756 --> 00:12:16.496 A:middle It's closer to the north. 00:12:16.496 --> 00:12:18.706 A:middle So we start measuring from the north. 00:12:18.706 --> 00:12:20.366 A:middle And it's towards the east so we -- 00:12:20.366 --> 00:12:22.806 A:middle We're going to need this angle here which happens 00:12:22.846 --> 00:12:24.476 A:middle to be the complement of 65. 00:12:24.976 --> 00:12:30.826 A:middle Therefore the compass bearing of AB is north 25 degrees. 00:12:31.286 --> 00:12:34.226 A:middle 90 minus 65 east. 00:12:34.576 --> 00:12:36.836 A:middle North 25 degrees, east. 00:12:37.796 --> 00:12:40.106 A:middle Whereas its azimuth is right here 00:12:40.106 --> 00:12:41.416 A:middle from the north and clockwise. 00:12:41.796 --> 00:12:43.736 A:middle It just says north 25 degrees. 00:12:44.366 --> 00:12:46.866 A:middle No east or west because it's always measured clockwise. 00:12:47.466 --> 00:12:50.966 A:middle AC, on the other hand, is closer again to the north, 00:12:51.066 --> 00:12:55.266 A:middle but it goes towards the west by 35 degrees. 00:12:55.266 --> 00:12:59.546 A:middle So it's north, 35 degrees west which means that the azimuth -- 00:12:59.546 --> 00:13:02.266 A:middle We need to find this angle measured 00:13:02.266 --> 00:13:05.946 A:middle from the north all the way to where AC is. 00:13:06.746 --> 00:13:10.606 A:middle That gives you 360 minus 35 which 325. 00:13:10.606 --> 00:13:12.826 A:middle So it's north 325 degrees. 00:13:13.546 --> 00:13:16.236 A:middle Same thing for AD and AE. 00:13:16.496 --> 00:13:19.756 A:middle So hopefully you'll see why for AD it's south, 00:13:19.866 --> 00:13:23.056 A:middle because it's closer to the south, we need the angle here 00:13:23.826 --> 00:13:25.986 A:middle from the south towards the east. 00:13:26.506 --> 00:13:30.086 A:middle And that's a complement of 60 which is 30 whereas 00:13:30.086 --> 00:13:34.426 A:middle for E it's also measured from the south, starts closer 00:13:34.426 --> 00:13:36.986 A:middle to the south, but we need this angle here towards the west. 00:13:37.916 --> 00:13:39.856 A:middle So it's going to be the complement of 38 00:13:39.996 --> 00:13:43.236 A:middle which is 52 south, 52 degrees west, 00:13:43.626 --> 00:13:46.046 A:middle whereas for the azimuth it's always measured from the north, 00:13:46.136 --> 00:13:48.726 A:middle clockwise towards AE here. 00:13:49.616 --> 00:13:50.806 A:middle That's why it's 132. 00:13:51.326 --> 00:13:56.776 A:middle If a line is going directly east, we don't say it's north, 00:13:56.776 --> 00:13:59.226 A:middle 90 degrees east or south 90 degrees. 00:13:59.226 --> 00:14:02.816 A:middle We simply say it's due north -- due east, rather. 00:14:02.976 --> 00:14:03.286 A:middle We don't .... 00:14:04.256 --> 00:14:05.816 A:middle Here's an illustration of a line. 00:14:06.356 --> 00:14:10.136 A:middle If a line is not perfectly horizontal, such as this one is, 00:14:10.566 --> 00:14:13.736 A:middle what you need to do is two appears to be higher than three. 00:14:14.046 --> 00:14:17.576 A:middle What you need to do is project two and three 00:14:17.616 --> 00:14:21.106 A:middle in to a horizontal plane which basically is your top view. 00:14:21.106 --> 00:14:24.116 A:middle That's why we look for directions in the top view. 00:14:25.086 --> 00:14:27.436 A:middle And the way you give directions, 00:14:28.106 --> 00:14:31.816 A:middle you always use the higher point as your reference. 00:14:31.816 --> 00:14:34.256 A:middle So in giving the direction of line two, three, 00:14:34.606 --> 00:14:37.996 A:middle you need to start from point two and move towards three 00:14:38.086 --> 00:14:40.216 A:middle because two is the higher end of the length. 00:14:40.216 --> 00:14:44.086 A:middle In this case, directional line two, three is north, 00:14:44.496 --> 00:14:48.946 A:middle 45 degrees east or azimuth of north 45 degrees. 00:14:49.356 --> 00:14:51.836 A:middle Slope of a line is the measure of the steepness 00:14:51.886 --> 00:14:53.916 A:middle or inclination with respect to the horizontal horizontally, 00:14:53.916 --> 00:14:56.376 A:middle can be specified using a slope which -- 00:14:56.506 --> 00:15:00.906 A:middle The slope which is rise over run, you see that in algebra. 00:15:01.446 --> 00:15:03.346 A:middle So the rise of the vertical distance 00:15:03.346 --> 00:15:05.406 A:middle and the run would be what we call "H" here, 00:15:05.406 --> 00:15:07.576 A:middle or the horizontal projection of the line. 00:15:08.296 --> 00:15:11.706 A:middle Which -- It can also be expressed in grade which is -- 00:15:11.916 --> 00:15:13.536 A:middle percent grade which is the slope -- 00:15:13.536 --> 00:15:17.146 A:middle numerical value of the slope times 100 percent. 00:15:17.416 --> 00:15:20.946 A:middle Or it's also common to give you the actual angle. 00:15:21.056 --> 00:15:25.886 A:middle So, as an example, a slope of 3 00:15:25.886 --> 00:15:28.626 A:middle over 4, (I know a 3-4-5 triangle). 00:15:29.076 --> 00:15:33.656 A:middle So 4 is along the horizontal and three's the vertical. 00:15:34.596 --> 00:15:36.196 A:middle The slope would be 3 over 4. 00:15:36.776 --> 00:15:39.886 A:middle The percent grade of that would be 75 percent 00:15:40.426 --> 00:15:43.636 A:middle and the slope angle would be equal to the arctangent of 3 00:15:43.636 --> 00:15:47.476 A:middle over 4 which is about 36.87 degrees. 00:15:48.286 --> 00:15:51.106 A:middle So the slope angle is the angle between -- 00:15:51.336 --> 00:15:54.706 A:middle You have to be measuring the true length of the line relative 00:15:54.706 --> 00:15:55.856 A:middle to the horizontal plane. 00:15:56.676 --> 00:15:59.616 A:middle And to do this graphically, first you need 00:15:59.656 --> 00:16:01.976 A:middle to find the true length of the top view 00:16:02.636 --> 00:16:05.936 A:middle and then the resulting view will show the angle 00:16:05.936 --> 00:16:07.846 A:middle with the true length and the edge view 00:16:07.846 --> 00:16:09.926 A:middle of the horizontal plane which corresponds 00:16:09.986 --> 00:16:13.896 A:middle to the fold line between H and ARP. 00:16:13.986 --> 00:16:15.616 A:middle Here's an illustration here. 00:16:15.926 --> 00:16:17.986 A:middle It's the same exact line we looked at earlier, 00:16:17.986 --> 00:16:20.096 A:middle AB in the front, AB in the top. 00:16:20.526 --> 00:16:23.596 A:middle What we did was we looked for the true length of the line 00:16:23.596 --> 00:16:27.836 A:middle by taking this reference plane parallel to AB in the top view. 00:16:27.876 --> 00:16:29.976 A:middle If you want to find the slope angle, 00:16:30.446 --> 00:16:33.516 A:middle you have to do the reference plane, 00:16:33.866 --> 00:16:37.286 A:middle auxiliary reference plane number one, parallel to the top view. 00:16:37.916 --> 00:16:40.756 A:middle Here is the true length of the line AB, 00:16:41.766 --> 00:16:44.516 A:middle which means that if this is true length, 00:16:45.196 --> 00:16:51.206 A:middle this line here is parallel to the horizontal plane 00:16:51.446 --> 00:16:52.906 A:middle because it's parallel to H 00:16:53.816 --> 00:16:56.466 A:middle which is the fold line folded from the top. 00:16:56.906 --> 00:16:59.456 A:middle So what that means is this angle here measured 00:16:59.456 --> 00:17:01.976 A:middle at 17 degrees is the slope angle of the line 00:17:01.976 --> 00:17:04.456 A:middle because it's the angle meeting the true length of AB 00:17:04.456 --> 00:17:08.026 A:middle and this line here which happens to be horizontal 00:17:08.026 --> 00:17:12.956 A:middle because it's parallel to H. Point view of a line. 00:17:13.656 --> 00:17:19.036 A:middle So to find the point view of a line we need to make our line 00:17:19.036 --> 00:17:21.626 A:middle of sight parallel to the true length of the line itself. 00:17:21.626 --> 00:17:24.426 A:middle And the procedure is first find the true length of the line. 00:17:25.006 --> 00:17:27.116 A:middle The way you find the true length of a line, 00:17:27.116 --> 00:17:29.006 A:middle as we just discussed this earlier, 00:17:29.296 --> 00:17:31.056 A:middle take a reference plane parallel to the line. 00:17:32.196 --> 00:17:34.286 A:middle Then project the end point perpendicular 00:17:34.286 --> 00:17:35.326 A:middle to the reference plane. 00:17:35.326 --> 00:17:37.216 A:middle Label all the reference planes. 00:17:37.286 --> 00:17:39.996 A:middle This distance away from the previous corresponding 00:17:39.996 --> 00:17:40.716 A:middle reference plane. 00:17:41.256 --> 00:17:45.156 A:middle Then take the reference plane -- the next reference plane, 00:17:45.156 --> 00:17:48.966 A:middle perpendicular to the true length project, and transfer. 00:17:49.046 --> 00:17:51.776 A:middle That should be the point view of that. 00:17:51.776 --> 00:17:55.446 A:middle I will illustrate this in the next lecture. 00:17:56.856 --> 00:18:00.996 A:middle Edge view of a plane involves finding the point view 00:18:00.996 --> 00:18:02.326 A:middle of any line in the plane. 00:18:02.636 --> 00:18:04.686 A:middle So imagine a plane. 00:18:06.276 --> 00:18:11.256 A:middle If you can find any line in the plane and find the point view 00:18:11.256 --> 00:18:14.696 A:middle of that line, you will get the view 00:18:15.236 --> 00:18:16.416 A:middle that is edge view of the plane. 00:18:16.416 --> 00:18:19.996 A:middle So the procedure is find a line in the plane 00:18:19.996 --> 00:18:21.596 A:middle that is already shown as true length. 00:18:22.286 --> 00:18:26.096 A:middle And the easiest way to do that is find a line that's horizontal 00:18:26.806 --> 00:18:30.216 A:middle because if it's horizontal it will be true length 00:18:30.216 --> 00:18:31.036 A:middle in the top view. 00:18:32.156 --> 00:18:35.006 A:middle Then take a reference plane perpendicular 00:18:35.006 --> 00:18:36.586 A:middle to the true length in the top view. 00:18:37.636 --> 00:18:40.896 A:middle Project the points of the plane. 00:18:41.776 --> 00:18:44.516 A:middle Take measurements of height from the front view. 00:18:45.796 --> 00:18:48.846 A:middle And then if you did your procedure correctly, 00:18:49.516 --> 00:18:52.006 A:middle the points will line up into an edge view. 00:18:52.316 --> 00:18:53.416 A:middle Here's an illustration. 00:18:54.856 --> 00:18:55.786 A:middle Here's step one. 00:18:56.136 --> 00:19:02.686 A:middle We are given the front view here of 1, 2, 3, the top view of 1, 00:19:02.686 --> 00:19:06.986 A:middle 2, 3, and we're asked to find the edge view of the plane, 00:19:07.586 --> 00:19:09.736 A:middle a view that will show line 1, 2, 00:19:09.736 --> 00:19:12.556 A:middle 3 with an edge or a straight line. 00:19:12.556 --> 00:19:15.666 A:middle So you need to have 1, 2, and 3 lined up perfectly. 00:19:16.176 --> 00:19:18.086 A:middle The way you do that is step one. 00:19:18.316 --> 00:19:24.066 A:middle Find a horizontal line, 1, 0. 00:19:24.366 --> 00:19:25.746 A:middle How did I find 0? 00:19:25.786 --> 00:19:28.316 A:middle I created this line here parallel 00:19:28.426 --> 00:19:31.006 A:middle to the horizontal plane here. 00:19:31.866 --> 00:19:34.976 A:middle Wherever this horizontal line intersects 2, 3, 00:19:34.976 --> 00:19:36.206 A:middle I will call that point 0. 00:19:37.086 --> 00:19:40.276 A:middle Project that point 0 in to the top view. 00:19:41.326 --> 00:19:45.366 A:middle It should intersect 2, 3 here at the same vertical projection. 00:19:45.916 --> 00:19:49.986 A:middle Therefore 1, 0, the top view which is here, 00:19:49.986 --> 00:19:50.846 A:middle should be true length. 00:19:51.006 --> 00:19:53.296 A:middle And the next thing we need to do is take the line 00:19:53.296 --> 00:19:57.106 A:middle of sight parallel to that true length and that's equivalent 00:19:57.416 --> 00:20:01.226 A:middle to taking a reference plane perpendicular 00:20:01.666 --> 00:20:02.276 A:middle to the true length. 00:20:02.366 --> 00:20:06.106 A:middle So as soon as you find 1, 0, its true length in the top view, 00:20:06.606 --> 00:20:10.656 A:middle take a reference plane here labeled horizontal on the left 00:20:10.656 --> 00:20:16.316 A:middle and 1 or primary auxiliary view on the other side. 00:20:16.786 --> 00:20:19.756 A:middle This will be our reference plane perpendicular 00:20:19.756 --> 00:20:20.746 A:middle to the true length. 00:20:20.746 --> 00:20:28.036 A:middle And then what you need to do is find locations of point 0 and 1, 00:20:28.036 --> 00:20:30.976 A:middle 0 and 1, in the primary auxiliary view. 00:20:31.586 --> 00:20:34.136 A:middle 1 and 0 must be anywhere along this projection. 00:20:34.576 --> 00:20:36.916 A:middle How do I find how far they are 00:20:37.046 --> 00:20:41.686 A:middle from the fold line here which is H? 00:20:42.406 --> 00:20:45.166 A:middle I borrow the depth of -- 00:20:45.166 --> 00:20:50.486 A:middle the height of 1 as measured here in the front view. 00:20:50.486 --> 00:20:52.946 A:middle This distance H here should be the same 00:20:52.946 --> 00:20:54.316 A:middle as this distance H here. 00:20:55.006 --> 00:20:57.766 A:middle And, as you can see, the distance -- 00:20:57.766 --> 00:21:00.026 A:middle The height of 0 is the same as height of 1. 00:21:01.006 --> 00:21:04.796 A:middle Okay. And 1 and 0 are in the same projection which means 00:21:04.896 --> 00:21:11.696 A:middle that 1 and 0 will end up being a point view because they are 00:21:12.426 --> 00:21:14.036 A:middle in the same projection here 00:21:14.456 --> 00:21:19.056 A:middle because I made this projection plane perpendicular to line -- 00:21:19.326 --> 00:21:21.106 A:middle the true length of line 1, 0. 00:21:21.106 --> 00:21:22.976 A:middle And they are the same height here. 00:21:23.826 --> 00:21:25.856 A:middle So the same distance away from this fold line. 00:21:26.456 --> 00:21:30.186 A:middle I'll do a similar thing for point 2 and point 3 here. 00:21:30.606 --> 00:21:33.676 A:middle So I construct the projection of point 2 here, 00:21:33.676 --> 00:21:36.056 A:middle the projection of point 3 here. 00:21:36.506 --> 00:21:38.296 A:middle Both projection lines perpendicular 00:21:38.296 --> 00:21:39.396 A:middle to the fold line here. 00:21:39.926 --> 00:21:42.756 A:middle To find point 2 here, I'll borrow the height 00:21:42.906 --> 00:21:45.996 A:middle or distance away from this H here as seen from the top 00:21:46.566 --> 00:21:50.566 A:middle as being the same as this distance of 2 away from this H 00:21:50.876 --> 00:21:54.666 A:middle as seen in the auxiliary view. 00:21:54.666 --> 00:21:56.316 A:middle Same thing with point 3 here. 00:21:56.316 --> 00:21:57.946 A:middle This height here is the same as this height. 00:21:58.746 --> 00:22:01.376 A:middle If I did everything correctly, the 3 points, 00:22:01.686 --> 00:22:04.636 A:middle 1 here, 0 and 1 are here. 00:22:04.956 --> 00:22:06.526 A:middle It's the same location as this. 00:22:07.246 --> 00:22:10.556 A:middle If I did everything correctly, 3, 1 and 2 should line 00:22:10.556 --> 00:22:12.496 A:middle up along a straight line 00:22:12.786 --> 00:22:15.116 A:middle and that is the edge view of plane 1, 2, 3.