 a circle break it down to two parts rotate this guy about the z axis rotate this guy about the y axis or x axis combine them together. You see this is with regard to as much as drawing can do, but there is a very there is a very interesting philosophy behind this. There are a bunch of sea words there are a bunch of sea words that if you understand and if you think and analyze about those sea words you will be great throughout your life from now till the age you retired and beyond. This was what my last lecture was going to be about it is still on if you guys are interested I will have to figure out a time it is got nothing to do with the drawing that we are learning, but since this course is about thinking and analyzing I thought it may be a nice idea for me to introduce to you those sea words that you can cogitate upon sea word cogitate upon, but that is for later not now. I will start by saying thank you this is my penultimate lecture 25th officially my last lecture is on Thursday this week. So, I will start by saying thank you to all who have helped all of us in making us you including me think and analyze together was the tutors was the tutors. So, I will start with civil engineering frankly I am not really sure how effective the slides have been on my web page and making you guys learn. So, a little feedback whether effective whether not effective whether good whether useful whether not useful there is a story back in 2003 2004 when you were still in great school Prasad Vinod Thare he was standing here and I was sitting over there as a tutor. And the idea that I should be using animations in trying to explain things to you actually came from him. So, he used animations in his presentations I was very very impressed with and I thought maybe I will continue with that idea he was a five star instructor he is still a five star instructor. So, I am not really sure if I will still be able to you know compare myself with him, but the idea of using animations in slides making you guys follow and teach actually came from him. So, he deserves another round of applause. Professor Pune Dubos, Professor Tarangukta, Professor Rajesh Sinha, Professor Javed Malik, Ria George from Aeronautical we have Bhadwaj and we have Shantanu from Mechanical we have Professor N. N. Kishore sitting right there he is going to be teaching this course in the coming semester. And Professor Kishore if I can see you thank you very much your feedback was wonderful and your idea about the three point perspective was absolutely fantastic. So, I enjoyed it I learnt it thank you. Professor Ashish Dutta, Professor Vasanthal Sharma and Shakti Singhukta I do not have the names of all the teaching assistants at this time, but they were there in the foreground not here, but in the labs and they were there for great help to you. So, thank you to all the teaching assistants a person you have not seen or you may not have seen. So, he is the person who has been doing all this work in the background preparing models preparing autocad stuff and you know helping me with a lot of things Ashwini Kumar. So, we will start with the new topic developmental surfaces you know I am exhibiting a few solids here some pyramid well 1 2 3 4 5 6 hexagonal pyramid which is truncated by a plane. Another something very similar a skewed pyramid here a skewed tetrahedral here this is this is a hexagonal pyramid again regular hexagonal pyramid this is a very nice structure. So, which is again I think a hexagonal pyramid, but it has a cylindrical void and it has been truncated by a plane another something very similar. So, what I will do is I will pass these things on to you you can take a careful look and return them back to me do not take it to your host rooms, but what is very interesting about all these solids is they have been made by paper and importantly a single piece of paper a single piece of paper. Now, let me revert this question and ask you this how would you want to prepare this. So, that you get the solid how would you want to prepare that corresponding piece of paper. So, that you get this solid and this is what this lecture is about yeah pass it on do not do not keep the solid with you can you come from somewhere there anybody anybody you know where Northwest is you know where Northwest University is you want to anybody all right. So, here we go penultimate lecture you know I was asking God as to whether it has whether he has or he has many many examples on developments of solids and I came across this nice setup nice PDF from I guess I T Guwahati I well Arringham day is the name who taught this and I liked a few pictures in this and these are the pictures. So, what you see is a box or a block a cylinder a tetrahedron a cone a sphere and well this is a tetrahedron this is pretty much like a pyramid yeah and what you also see is that they may have been made by paper and how would you want to kind of unfold the paper. So, that you can lay it down flat on a plane. So, perhaps you can cut one of the edges you can make one cut over here you can make one cut over here another cut over here open up and lay down on a plane and once you fold them back you would get this keyboard for example, same thing with this lender you know. So, cut this face cut this face apart you know and make a cut along this direction lay down. So, essentially the idea is that all of these solids except for perhaps sphere they have been made by a single piece of paper and the question again is how would you want to prepare that piece of paper. So, that you can get the solids back all right. So, this is quite straight forward right the unfolding is quite straight forward the unfolding over here is quite straight forward. So, the method is the parallel line development you know lay it open cut and lay it open parallel line development this is pretty much like a radial line development pretty much like you know for example, when you cut the pyramid you open up. So, speak a sector not a rectangular piece of paper, but a sector same goes with the cone and with the tetrahedron you open up this piece of paper into triangles. So, just an example. So, example 1 2 3 4 and 5 they happen to be examples where you prepare your piece of paper in such a way that you get the exact solid back. This one is like an approximate example we will talk about that. So, certain developments are exact certain developments are approximate. So, except for this example all the other developments are exact this one is approximate and this one is this point seems to be important. So, for example, if you have a little wide here on the surface or may be perhaps a wide here or wide here or any feature when you are opening up that corresponding piece of paper what do you expect should that feature be in should the feature be in true dimensions or it should be in some sort of you know untrue dimensions. So, speak or projected dimensions what do you expect true dimensions everybody this is what the intent is. So, if you look at these surfaces all features on these surfaces they have to be in the true dimension and that is the point that you have to keep in mind when you are developing a solid or developing a surface true lens or true shapes. So, this is an octagonal pyramid let us try to lay it open on a plane we have given you have been given 3 views the front view the top view the right view well the question is to form an octagonal pyramid from a single piece of paper. So, we have to figure out how to make cuts on that single piece of paper now look at this length this horizontal length in green here what is the corresponding projection this guy here yeah. Now, what you have to say about this length is this length in true dimensions why is that because the corresponding projection is parallel to the hinge in one of the views right. So, this projection is parallel to the hinge in one of the views and therefore, this one is in true length. So, I will call it T L 1 look at this projection over here in red and look at the corresponding projection in the top view is this projection in true length again the same principle. So, this projection here is parallel to the hinge line in this view and therefore, this one should be in true length. So, I will call it T L 2 the entire thing that you have to keep in mind is to figure out or keep figuring out the true lengths is this information good enough for you to work with a paper on a plane is it information good enough for you to transfer the entire information that this surface has this solid has on a single piece of paper. Yes all right let us see. So, you have this true length T L 1 all right taking one of the ends of T L 1 as the center and the radius as T L 1 draw an arc and taking T L 2 as dimensions with this vertex as center draw in fact, cut the previous arc with the new arc and keep cutting these arcs how many times are going to be cutting 8 or 9 keep cutting and all these lengths they are going to be what all these lengths they are going to be what T L 2. So, there has to be they have to be how many cuts. So, 1 2 3 4 5 6 7 8 corresponding to the number of vertices that you see in the top view there yeah and of course, join these points with the center. So, what you have is something very simple. So, if you make this sector out of a paper and fold it at these source peak lines you will get the solid back right pretty simple. Let us make our life our lives a little more complicated what if the pyramid is truncated what if I have a plane that is cutting the pyramid then what the base will be the same. So, we will have to be working with this entire picture over here just that we have to be a little worried about the intersection points between the optical and the pyramid and this plane yeah. So, let us try to figure the intersection points first you need to know where the intersection points are where are they going to be lying where are the intersection points going to be lying well I mean you have these edges right. So, the intersection points they have to be lying on a plane this plane and they also have to be lying on these corresponding edges all you need to do is like project those intersection points in the top view and the right side view. So, keep projecting. So, this one lies on edge number 6 this one lies on edge number 5 5 and 7 I know 5 perhaps. So, keep projecting and you essentially will be figuring out these intersection points in the top view all except for two of them those intersection points which will be lying on the vertical edge over here number 4 and the edge behind it number 8. So, 4 well 4 is behind. So, number 8 which is facing you and number 4 which is behind this how do you figure those points in section you can make this projection perhaps. So, if you can make this projection this is number 8 this is number 4 you will be getting the corresponding intersection point over here and over here project them backwards and on to the top view and get those intersection points or otherwise if you want to be a little more tedious maybe make any measurement make the hinge line make the same measurement over there make the hinge line you know and something that you have learnt in your lines and planes thing. So, measure that distance transfer that distance over here and measure this distance transfer this distance over there. So, these are your two intersection points and the second of course, was quite right. So, you can directly project those intersection points from the profile view and transfer them on to the front view. So, these are your intersection points I am not labeling them. So, labeling is something that you will have to do later and join those intersection points to get this contour. Now, when I say that the pyramid is truncated I essentially mean that this part of this pyramid is taken off and it is replaced by a plane covering the top portion of the pyramid. So, this is the top lid of this pyramid what is the bottom lid what is the bottom lid the octagonal face 1 2 3 4 5 6 7 at the octagonal face. So, you can as well project these intersection points on to the profile view get them on the respective edges of the octagonal pyramid and complete the contour. Is this contour in true shape is this contour in true shape what did I say. So, if you have to transfer everything in such a way that you have to lay down those features on a plane you have to get the corresponding things in the true shape now well. So, this is the actual truncated pyramid that you see the actual truncated pyramid that you would see the top portion of the pyramid is gone this is the top lid of the pyramid which is covered. So, the pyramid is not open and of course, the bottom face of the pyramid is covered by that octagonal face alright always label the points. So, I made a mistake of not labeling the points, but you should be always labeling these intersection points every point that you are working with in your intersection and development exercise always label them. So, now I am labeling those points a b c d e correspondingly a over here the two b's a b c d e would go you know from here to here a b c d e would come from here to here alright. So, always label your intersection points alright. So, I am done pretty much now I need to get this contour in true shape yeah. So, something that I already told you it always has to be the true length and true shapes that need to be transferred on to the piece of paper and we need to find them as a part of the exercise and development. Now, let us worry about this little later let us worry about this little later let us worry about the rest of the edges look at this edge this edge would lie on which edge of the octagonal pyramid 6 or 2 what is the corresponding projection what is the corresponding projection in the front view this one. So, this is h number 2 is this in true length. So, if this is in true length it should be possible for you to transfer that length over here there is there any two that you are seeing here no that is fine how about this entire thing this is on which edge number 6 the corresponding projection the corresponding projection here is this in true length you need to transfer that on to h number 6 there. So, the intersection point over here would be a and the intersection point over here would be e always label let us look at this edge in section point c all right. So, where is it can you identify the corresponding projection in the profile view c would be on 8 and it would also be on 4. So, 4 and 8 4 and 8 yeah is this in true length all right. So, it should be on 8 is this also in true length great. So, it should be on 4 wonderful how about the lens on edges number 1 7 5 and 3 lens on edges number 1 7 5 and 3. So, 5 7 are here 1 3 are here. So, if you are looking at the corresponding edge over here and projection there. So, this is slant and so is this. So, is it possible for you to extract the true dimensions directly from these 2 or from the information given it is possible no algebra no fractions just geometry no root 2 root 3 root 4 root 5 no pi just geometry I will come I will come there I will come there all right. So, these are intersection points c all right what about the rest you are trying to address that. So, look at look at this projection and try to figure the corresponding projection in the top view b 7 b 7 is this one b 5 is this one yeah I will do a little trick. So, what I will do is I will displace the line in both views I will displace the line in both views I will displace the line in such a way that 1 of the vertices of this line the true line lies on vertex number 6 you follow that you follow that just rigid body displacement. I have kept the lens of these corresponding projections the same just rigid body displacement. And just in case if you remember the method of rotation to figure out the true length precisely if you do that if you rotate this if you rotate this then what will happen to this projection you take a projection from this point here down this height would remain the same and eventually this guy would come here just using the method of rotation yeah. So, what you have done so let me let me go back. So, you have identified these projections made rigid body displacement you know shifted the line such that 1 of the vertices of the line lies on number 6 h number 6 follow the method of rotation when you do that this guy gets rotated to lie on this land surface yeah. Now, you have the true length you have the true length for what this guy here b 7 and b 5 as well yeah take that true length and perhaps transfer there likewise I can do the same thing for this identify this projection the corresponding projection is 1 d and on top it is 3 d not 3 dimensions but 3 d shift follow the method of rotation this guy gets rotated to lie on this .landage yeah this is just for no this is just for you to understand one method that I am going to be talking about now it is going to be much simpler. So, what you realize is whatever lens you have here whatever lens you have here if you just project them in such a way that they all happen to be on the slant edge of this pyramid essentially you are going to be getting the true lens over here this is where I am coming to realize that to get the true lens you need to project the intersection points on to the extreme right or left edge of this solid yeah. So, this point is lying on this edge this point projected over here from here to here you will get the true length this point again projected on to the extreme left get the true length likewise for d and point is already there right. So, if you project all these intersection points so that they lie either on this edge or this edge the corresponding lens they will all be in true length that is a much simpler method. So, these are the corresponding projections either on the left or on the right. So, once you have these projections transfer them on to that piece of paper draw the radii or draw the arcs if you want to you do not have to, but if you want to draw complete arcs, but what is of interest to us are these edges edge number 1, number 7, number 5 and number 3 this is where we want the true lens to be yeah alright. So, let us work with edge number 1 this length is what this length is this point getting projected over here. So, it is this length right it is on edge number 1. So, one here and the other one here and so is the case with edge number 3 also yeah. So, this is 1 and this is 3. So, 3 of these lens they get transferred how about number 5 and number 7 be projected over here. So, it is this point here now alright. So, of course, label and this is the true length on edge number 5 and this is true length on edge number 7 same are you with me all of you by the way my T cup how many of you think that this is actually a torus a torus torus torus this is actually a torus is that true or not have you ever played ring that ring is a torus do you remember playing ring when you were kids or even now that is a torus. So, this cup is actually that ring do you believe me do you believe me you do not believe me louder do you believe me alright, but that is true this is actually a torus anyhow you know this something called topology topology science of connectivity or discipline of connectivity. So, you learn things about connectivity theory about connectivity in there. So, many of the guys in electrical computer science they will be learning about networking network that is connectivity in a way. So, you know in your third year in your fourth year when you learn about topology when you learn about homeomorphism that is where you will actually figure that this is a torus, but that is too late for you now this time at this time it is engineering drawing. So, once you have transferred once you have transferred all those true lengths on the respective edges and labeled the intersection points properly alright. So, I will make a little change. So, what I will do is I will change the color I will change the color of these lines and also will I change the representation of these lines instead of denoting these lines as solid lines and denoting them using hinge lines why why because that is. So, those are the locations where I am creating folds those are the locations where I am creating folds to get the solid back to represent that the paper is bent or folded at these edges and then of course, close the contours. So, if you take a piece of paper cut it in this fashion fold it at the respective locations that are denoted over there you will get that solid back with the difference with the difference the bottom plane and the top lead they will not be there. So, it is going to be an open truncated pyramid let us walk on that again. So, you are going to be working with graphite pencil. So, your drawing is going to be gray not colorful to get the top lead once again if I may remind you that you have to get two features and transfer them on to the paper plane right. So, you need to determine the true shape of plane a b c d e d c b a. So, this plane right here you know how to get the true shape of a plane you see the edge view of this plane you see the edge view of this plane which view is it in which view is it in one view draw a hinge line shoot projectors perpendicular to that hinge line transfer distances transfer distances you know that. So, I am not going to be going through it once again transfer distances and then get the true shape of that top lead. So, where is this lead going to be of course, you have to label these points also. So, where is this lead going to be somewhere over here or somewhere over there where is this lead going to be well it is going to be somewhere over here yeah let us worry about that in a while alright. So, by convention by convention this is what we do we start developing we start developing the surface in such a way that we start with the shortest edge on the surface. So, if you look at these edges if you look at these edges and if you figure out the lengths of these edges you will find that this would be the shortest in length and you start with that. So, the edge you start with you have to end with the same edge because you have to glue the corresponding edges. So, this is what I have done. So, instead of starting with edge number 1 I figured that edge number 2 was the smallest 2 e was the smallest. So, I started and ended with edge number 2 e conventionally start development with the smallest edge and end with the same alright. So, rest of the information is taken away what is important is the lateral surface of the solid the top lead of the solid and the bottom face of the solid. This one constitutes the lateral bounding surface of the solid constitutes the top bounding surface of the top lead the bottom face that is of course, in true shape what you need to do is rotate these guys and align them appropriately on to the main developed thing. So, notice where I am aligning this face over here. So, notice a b and looks like I am aligning the 2 a bs together right likewise rotate the top face and align the corresponding edges appropriately. So, I align 1 a together. So, I align this guy with this guy right now that is a single cut that you need to make to get the entire solid back. A single cut you need to make on a piece of paper to get the entire solid back this is what development is. If you cut this paper folded and glue appropriately you will get the bounded truncated pyramid back is the location of these 2 planes important are the locations of these 2 leads important these 2 faces important. Can you locate them anywhere theoretically can you locate them anywhere with the location of the top and bottom bounding surface make any difference. So, for example, if I choose to locate the bottom face there and the top face here should be should be do not forget to introduce the scenes those are little extra things that you would want to provide over here. So, that when you fold the solid back or when you fold the solid you should be able to provide enough paper to be able to glue the solid back not like. So, these are the scenes right. So, if you fold this thing over here if you fold this thing back you know. So, you are going to be gluing at these little pieces of paper over here and over here now figure what the problem is figure what the problem is. So, if you allow for a little extension at this location this is interfering with this lateral develop laterally developed surface. This location here. So, you got to leave sufficient gap or space for the scenes. So, something that you will probably want to keep in mind yeah or you want to say that this is this to be represented by technically all right. So, if I may repeat what is the name. So, what is asking is whether this line is to be represented you using the hinge line convention with the same logic that well we are also folding we are also making a fold here all right. Any other question where you with me throughout where you with me throughout where you with me throughout out of the blue yeah all right. Can I have my solids back can I have my solids back please should not there be a true E here there should be true truncated top view yeah. So, do you agree wait do you agree that the important thing for us is to show the true shape of the open lid of the top lid it is not really matter how you get it. So, long as you get it any other question