 development of surfaces which are non-developable or oblique. So, these are the surfaces that you cannot cut and lay down flat on a piece of well lay down flat on a plane. So, we need to pursue or follow some tricks in this. So, that is on development keep that in mind. So, let us look at this sheared hexagonal pyramid the top view shown and so is the front view it is kind of sheared. So, what you do is you actually have your solid like this you just kind of shear the top surface with respect to the bottom surface that is how this solid is and we would like to develop it. Step one in development always look for true lengths true shapes because unless you do that it will be difficult for you to develop any solid. So, you got how many edges 1 2 3 4 4 that you see and 2 are behind. So, 1 2 3 and 4 5. So, these are the edges that you see and then 2 edges are behind these 2 edges. Can you identify edges in true length if. So, which ones are those which ones are those come again 6 5. So, this is true length because this projection is horizontal what else 6 7 6 7 well that is in true length also because horizontal horizontal once again any other 1 8. So, you see this projection and the corresponding projection for this is horizontal over there. So, you have a bunch of edges which are in true length and if you want to develop this how would you want to go about this how would you want to go about developing the surface well start with the hinge line. You know that edge 1 8 is in true length and if you kind of cut your solid let us say at this edge if you cut this solid at edge 1 8 and open it up open it up perpendicular to this. This is how your vertices 1 2 3 4 and 5 6 7 8 will appear they will be appearing on these corresponding projections. So, once again scissor this edge open this surface out these vertices are going to be appearing on the respective projections. So, once you have figured these projections out start with edge 1 8 and follow the connectivity 1 8 is connected to what. So, 1 is connected to 2 8 is connected to 7 2 and 7 they are connected. So, follow this connectivity in from both the views from 8 what do we have 7 and this length is the same as this length right and this length from 1 what do we have 2 and this length is in true length of course. So, you got 2 here got 7 here 2 and 7 are connected and keep going forward 2 is connected to what 3 2 3 and 7 is connected to what 6 alright. So, you can figure the 2 lengths of 7 6 and 2 3 keep going forward because 6 7 is going to be in true length both the projections are horizontal and. So, on this is a relatively trivial example 1 8 to 1 8 fine relatively trivial example nothing very rocket science about this alright or weird looking cone what about the what about the what well yeah. So, if you if you want to consider this to be an open solid just the surface then you do not have to worry about the ledge, but if you want to consider that to be yeah. So, weird looking cone the apex is shifted not marginally, but really in a weird fashion and we would like to develop this. Now, is it possible for you to develop this weird looking cone the same way as you would develop a regular cone is it possible once again is it possible for you to develop this cone the same way as you develop a regular cone. Yes, assuming that is not possible for you to develop this surface you know in a regular fashion what you do you approximate the surface and the way to approximate the surface is via triangulation it is possible to develop the surface accurately may be not. So, you need to go for approximate development and the way to that is to triangulate the surface and this is how divide the base into equal number of paths label these vertices 1 to 12 and draw generators from each of these vertices to join the apex O. So, project those vertices down on to the front view the apex over here of the cone is O join each of these vertices to that apex. So, the surface of this weird looking cone is now approximated by a bunch of triangles on the surface of this solid source P. To develop this cone now implies how you treat these triangles now I am the same procedure try to figure the two lengths or two dimensions from both of these views. So, we will be needing a true length diagram to figure out the two lengths of these edges 0 1 0 0 2 0 3 up to 0 12 and of course, 1 2 3 4 5 6 7 8 and so on and so forth. Now, look at this segment here 8 9 not the arc, but the segment line segment would that be in true length how about 6 7 would this be in true length. So, in fact all these line segments joining these vertices they will be in true length because the corresponding projections in the front view are all horizontal right step 1 fetish clear. Step 2 how do we figure the two lengths of 0 1 0 2 0 3 up to 0 12 is what the question is and for that we need the true length diagram. Now, look at 0 1 and look at the corresponding projection in the front view is that in true length look at 0 7 and look at the corresponding projection of 0 7 in the front view is that in true length 0 1 is in true length and 0 7 is in true length well start with a right angle triangle that I have drawn over there to find the true length of the rest of these segments what you do method of rotation right. So, rotate 0 2 to lie on the horizontal take its projection down or measure this length take its projection down over here or what you could do is you could transfer this projection over here directly and this is 0 2 in true length likewise 0 3 measure this length transfer it down here and join vertex 3 to 0 and that will give you the true length of 0 3 and. So, what the same procedure for 0 4 0 5 0 6 simple method of rotation measure these guys transfer them on to the true length diagram that is in true length that is in true length that is in true length and that is in true length all right. So, once you have the true lengths of 0 i's where i goes from 1 to 12 and all these other guys you can start developing this surface and this is how start with 0 7 draw a knock with radius the same as 7 8 or 7 6 depending on which side you are developing. So, if it is 7 6. So, this distance is the same as this distance and then to get this length you need to choose what distance from here up till here yeah 6 fixed number 5 with this distance cut an arc and with 0 5 cut another arc with this pointer center 5 located and continue yeah. So, this is not a true development, but an approximate development and the way we did that the way we perform this development is through triangulation we approximated the surface of a solid by means of a bunch of triangles by means of a set of triangles as simple as that all right and everybody with me everybody with me first example of the method of triangulation. So, if you remember our discussion last time on Tuesday you go for parallel line development in case of cylinders prisms etcetera those kinds of surfaces solids you go for radial line development in case of cones pyramids regular solids cones pyramids and you go for triangulation for other complex or non developable surfaces like in the previous example. As I said last time they all some way all the other involve determination of true lengths and true shapes without that you cannot do anything this is another pre interesting solid the front view given the top view given yeah this cross section is circular this cross section is rectangular or square. So, this surface is actually a blend between the circular cross section and a rectangular cross cross section can you imagine this all right a linear blend between two different shaped cross sections. How do we develop this how do we develop this triangulate no that is manufacturing that is not developing you are far ahead of me first means you develop this and then in your ta 201 if you get a chance to make this then make that yeah. So, we triangulate the surface it is a non developable surface we triangulate that and the way we do that as we divide the circular cross section to bunch of paths and we actually work with the four vertices of this square and we join these vertices in such a way that we get triangles approximating the surfaces over here and the surfaces over here look at the way I am joining the triangles now I am joining these vertices to get the triangles you follow the red lines bunch of triangles easier to work with the top view just transfer the triangles on to the front view all right. So, yeah of course, I am approximating these this curve by a bunch of linear segments also ok. So, triangles 11 3 4 11 4 5 5 6 12 6 7 12 7 12 9 7 8 9 1 8 9 1 9 10 so on so forth they all approximate the surface transfer the vertices on to the front view and transfer the connectivity as well on to the front view ok. The trick is or the notion is to work with as less number of triangles as possible ok, but necessary enough number of triangles that would allow you to develop this surface properly. So, you do not want to compromise on the shape of the surface what do we need next the two lengths for that we need the true length diagram notice that all these green edges they happen to be true lengths yeah how about the rest what you have to say about the lengths of 8 9 2 10 11 4 6 12 are they the same are they the same. So, just what same, but yeah of course, well the true lengths of these edges will be the same won't they the true lengths of these edges 2 10 4 11 6 12 and 8 9 they would be the same and likewise the true lengths of 1 10 1 9 9 7 7 12 12 5 5 11 11 3 3 10 they will all be the same. So, we need to find the true lengths of only a set of lines and we will be happy with that now all right all greens in true lengths that is ok need to determine the true lengths of edges like 1 9 2 10 and 1 8. So, I am going to be preparing my true length diagram over here length 9 10 is in true length length 9 10 is in true length yeah to get the true length of 1 9 I would rotate this projection of 1 9 in the top view. So, that this projection becomes horizontal and then I would project that down onto the front view and this length 1 9 will then be my true length method of rotation as simple as that to get the true length of 2 10 I would rotate 2 10. So, that 2 10 becomes horizontal. So, this guy here I have rotated it about vertex 9 this guy here I would rotate it about vertex 10. So, 2 10 becomes horizontal I will take the projection down cell phones off please I take that projection down and 10 sorry 2 is here all right. So, this guy here that would be my true length 2 10 what else am I left with 1 8 1 8 is already in true length I do not need to worry about that. But, if I still want to figure out the true length of that I rotate 1 8 about 1 take its projection down and this is my true length 1 8. So, once I have the true length of all the edges all these triangles that approximate the surface I am ready to develop the surface pick a compass pick a pencil and get started start with 9 10 and of course, when you are developing make sure that you are following the connectivity here and here following the topology right 9 is connected to what 9 is connected to 8 9 is connected to 1. So, which triangle am I going for first I start with 9 10 I start with 9 10 I will go with 9 1 and I will go with 10 1. So, this triangle is something that I have now on my 2 dimensional plane yeah 9 10 9 1 1 10 which triangle now perhaps 1 2 10. So, with 1 as the center 1 2 as the length draw an arc with 10 as the center 10 2 as the length draw another arc you will get triangle or you could go with 1 8 9 1 8 9 is here same thing. So, you have what x 9 here with this length 8 9 with 9 as center draw an arc with this length 1 8 with 1 as center draw another arc you will get 8 and keep on following this procedure we get 1 2 10 make sure that you are following the connectivity right 1 2 10 and then next triangle would be 2 3 10 and the next 1 will be 3 10 11 next 1 3 4 11 and then 11 4 5 keep on continuing this is straight forward. So, the challenging part for you is this two length diagram and understanding the file that you have to approximate a non developing surface using a set of triangles yeah you cannot, but cutting out a circle is not a problem, but your question has a context what is the context yeah not the sheared well sheared cone yes yes yes yeah well the accuracy of development will depend on the number of triangles you use to represent the surface the more the number of triangles you use the better the accuracy of this development will be again. So, merely cutting out a circle out of a piece of paper is not a problem right. So, if you just want to cut this circle out from a piece of paper perfectly fine, but if you are wanting to cut this circle with reference to the entire surface around it that is the context. Now, look at all these vertices lying on the circle 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8. And then you have to of course, have one more triangle over here I think yeah right. So, these guys where would they lie. So, if you if you close if you glue this 9 8 if you glue this 9 8 with that 9 8 and if you kind of close the surface all those guys they will be approximating a circle from this example yeah. So, my point is if you want. So, if I understand your question correctly if you want this part of the surface to be a precise circle is this what you want is this what you want. If you want this part of the surface to be a precise circle you will have to have more number of points approximating that circle and then well I mean you have to account for those points by means of triangles. Those points on this circular curve over here they have to participate in triangles they have to be a part of the triangles thinking and analyzing ends well you have lab number 13. And then you have your exam it better not end and even after this course your thinking and analyzing should not end C for M for did you say M for midsims M for marks M for money M for what math C for this is Brahmagyan Brahmagyan nothing to with T a but everything do with thinking and analyzing that I will share with you. If you want to take it take it if you do not want to take it fine C for contentment satisfaction one word that each and every one of you is looking for throughout your life yeah does anybody disagree C for contentment C for currency money many few guys think the more money you have in your pocket the more content you are is that true louder not true but still trust me once you are in your fourth year you are going to be going for companies were going to be paying you higher which you should not you should be going for companies that of you jobs of your choice C for choice your choice that is where things change the more choices you make the more confidence you get C for confidence the more choices you make the more confidence you get in life the more compromises you make compromises would be like without you wanting to do something you are forced to do something some way or the other the more compromises you make the less confidence you get two factors which are going to be important throughout your life after your 26 or 27 may be later carrier C your spouse your companion C your life will revolve around these two factors you choose the right carrier your content you choose the right companion your content you choose the wrong carrier you choose your wrong spouse or companion C words children direct source of contentment look at a one year old you like to hold him or hold her in your hands yeah see for college you can figure out a bunch of C words that if you cogitate about if you think and analyze and this is why perhaps subconsciously I name this course think and analyze 101. So, if you think and analyze about these bunch of C words you will see how these words might change your life later on keep intruding finally, you always say you know the party is not good they are doing no it is a democratic country I can say whatever I want well within limits. So, you guys tend to censure people you guys tend to blame people you know for things which are not right around you guys want to change C word you guys want to change for things to be better around you are the change you take a right step forward and the world will follow you do not wait for the world to take the right step forward and then you follow the world the other way around you take the right step forward and the world follows from again impressed imbibed taken in all right. So, with that said lab number 13 is on your exams are on on the 26th 9 to 1 a single minute late doors are closed keep that mind for both badges no sharing of equipment information anything during the exam or in between when the badges are getting or when the badges are in transition anything else was a pleasure teaching you and for that thank you in advance.