 So, this is the final example on development. So, shown here are two figures, this is the front view of a cone and this curve over here is the intersection between the cone and a rectangular prism. So, I have drawn this curve, I have kind of transferred this curve from an example. This curve is slightly similar to the previous example, but in the previous example we had taken a diamond prism as opposed to a rectangular prism. This curve corresponds to that from rectangular prism and of course, I have also taken the top, we have not transferred these intersection points over there for a reason, but this example is on development. So, you should realize very well that a conical surface is developable and it so turns out that if you cut this conical surface and kind of flap it open, it would lie nicely on a piece of paper, lie nicely on a plane, which is what we are trying to do in this example. And we would also try to capture this intersection curve, this curve in section on that developed conical surface. Now, think about sheet metal work, this is precisely what they do. So, what they do is they would actually develop the conical surface, they would transfer this intersection loop on that, then they would kind of mold that surface into a cone and when they do that, they would get the intersection curve just like that. And you know you can actually place the rectangular prism over that or inside that or within that you know little void that is slipped. So, we will try to do that primarily you know prepare for sheet metal work. So, enough of talking let us get down to business. So, to develop a conical surface of course, this would be you know if you kind of cut this cone and if you flip it open, flap it open on the plane, you know this is conical surface is going to open out as a sector. And the radius of that sector is going to be this much you know the slant length of a cone. So, what I will do is I will first try to capture that, I have that pretty much I will probably take any convenient center and perhaps somewhere over here and then draw a little sector, maybe I will start from here and you know just draw a little sector here pretty much like that. Now, what I also have is I say that all right let this apex of the cone be you know cut by a horizontal plane. So, that we see a little void here. So, let me also draw that corresponding arc is probably going to be of this radius let me draw this right here just about there and let me make a line to start with. So, I am not going to be labeling as of now, but perhaps later I will do that. So, this my start line for example. Now, one of the problems that I need to address is I need to you know capture this void as a single loop within that developed surface. Now, if you look at for example, i j segment i j here and segment i j there and if you go back to your theory you would observe that this segment is parallel to a hinge line. So, I do not have a hinge line here. So, maybe I will just show that this is going to be parallel to this. Well, perhaps I should be using my rafter switch to my 2 h pencil just kind of you know draw a little horizontal line here perhaps little darker and then use my set square friend. So, this is my hinge line, but the point I was trying to make was that this segment is parallel to the hinge line. And therefore, this straight line is going to be true length. So, if I assume this arc to be of you know approximately the same length as this line segment j k or i j for that matter or any other line segment for that matter, because for all these line segments the corresponding projections they are all parallel to the hinge line. So, perhaps what I would do is I would take this as or this to approximately represent the length of this arc and with this length I will start cutting down this larger arc. So, if I start from j for example, then 1, 2, 3, 4, 5, 6, 7, 8, 9, well 10, 11, 12. So, perhaps 12 of these arcs or perhaps 1 more let me just go ahead and keep cutting maybe I will count later. So, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 perhaps 1 more 14. So, of course, when you open up this conical surface it would not be reaching to a 180 degree thing, but it will be little smaller than that. And assuming that this is my starting surface now if I go back to my example and perhaps give you a little peak on the top view of this here. You will see that sectors C O D and D O E they do not contain any part of the intersection. So, C O D D O E. So, there is no intersection or there is no part of intersection curve in there. So, perhaps what I will do is I will start you know nomenclating or naming these points from say C because that would allow me to have my intersection curve some over here and then I will go cyclically. So, let me start with C and then D F I J K L over here back to A C D well I do not think I would need D. So, perhaps I will stop down to here C to C I would erase the rest of this bigger arc as well as the small arc. I will try to make sure that I do not leave any trace of the bigger arc as well as the smaller arc having done that and of course, this is my apex point O or apex point how so ever it is pronounced. And from this point I will start joining you know the points on the arc I should be using my 2 H pencil for that because these are construction lines. I will limit well slowly but steadily I am not sure if you are realizing that my stable my table is unstable slightly unstable rhymes pretty well my stable my table is unstable or my table is not so stable. It is a bit shaky, but I do not think I should be complaining I have to take care of some special leaders or generators O Q O P O M and O M. So, maybe let me draw O M because my intersection points also lie on these generators or these lines. Let me try to locate them so P is in between G and H and of course, G P I can assume it to be in true length because G P over here is horizontal. And I will assume that P and Q they are lying on the same horizontal they should be lying on the same horizontal. So, I measure this arc and from G towards H it is going to be very close to H from G towards H I make a cut and call this P and likewise from A towards L I make another cut with same radius and call this P and Q with the center and let me take care of M and N. So, from I I can measure this arc length from K I can measure this arc length they should once again be the same M and N they should be lying on the same horizontal just verify this quickly looks like they do and then I measure the arc length from Q this arc length. So, it is going from well maybe I should be measuring it from L it is even better. So, from L towards K from L which is here towards K and this is my point N let me name it and on the left side from H towards M let me make sure that I have this length right just about from H towards I from H towards I my point M should be here let me name this point and then join O 2 M and O 2 N forced N and then M are having all the generators on the developed conical surface, but I have a scar on my sheet I am not sure if you notice it probably you do not nevertheless. So, having these generators on my developed surface it is time for me to transfer these intersection points on the respective lines. Now, notice that all these intersection points are lying on the respective generators in the front view. Now, if I want to get the true length if you notice this would be the true length this would be the projection of that length projection of that length projection of this length projection of this length and so on and so forth. So, to get the true lengths of these intersection points from O I would need to project all of these intersection points on to this line. So, let me call this line as a true length line may be shift align my drafter I do not think I have been doing my drafter very often in this example. So, perhaps I can you know mention this over here that is my true length line and then I align my drafter with the horizontal and let it rest I have to get up you know just type true length should have come out all right and I will raise the rest of this make sure I wipe the residue from the sheet. Well I was wrong when I said that I will not be using the drafter this is a critical part I just want to make sure that I have the longer ruler aligned with the horizontal the critical part is to transfer all these intersection points on to the true length. So, point number 1 point number 2 point number 3 4 already lies on the slant length of the cone and number 5 6 7 8 also lies in the cone now very carefully I am going to be measuring these lengths from O and then transferring them back on to the respective generators or selected lines I am going to be using my compass for that ok. First is O 1 right there O 1 lies on A. So, this is where my intersection point 1 is we label it just double checking if I have everything outside because I do not want any part of my intersection loop to be lying away over here even if it does it does not really matter I can always you know extend this part of the cone then erase the other part seems all right I mean well nevertheless you know what is C O B going to be having a part of the intersection curve back to my top view C O B looks like it will be having quite a bit of this loop perhaps this is nice idea for me to extend this I will take care of that later ok. So, point 1 lies on O A point 2 it lies on C and K. So, C there there and K would be here K. So, this is intersection point number 2 all right. So, point number 3 extend this further point 3 lies on it looks like point 3 lies in Q very close to be here. So, this where point 3 would lie L and Q are quite close to each other and B and L B L Q now go here. So, this where my point 3 would be would be on P as well where if I take look at this 3 should be lying on L and H if I am looking at the top view. So, L and H. So, L is here H would be here I am not sure if I am doing this right, but well we will come back to that later. So, let me not label 3 for now 4, 4 definitely lies on A and G enable this that is what I am sure of point number 5 lies on B and F yeah number 5 lies on B and F is here F is here intersection point 5 6 lies on special leaders N and M first let me get the length here that lies on M which is there and N which is here number 6 number 7 where does that lie point number 7 is here looks like it lies on B and F again. So, let me measure 7 true length of that here and it lies on B and F. So, F is here and B is there let me call this 7 number 8 again lies on A and G remember I still have to come back for the intersection point number 3 because I was not sure where it would lie it should not be very difficult for me to figure it out nevertheless number 9 if you look at the previous example or even if you do not want to that is fine, but 9 lies on Q special leader and P. So, I have my Q here I have my P here. So, 9 would lie here somewhere ok. So, that is on Q and P. So, this is number 9 let me indent this point a little bit. So, that I do not confuse between these 2 leaders these 2 leaders number 10 that lies on B and F again little shorter in length this is 10. There is a possibility that I may have to extend my developed surface on this side there is a possibility, but I am not really sure at this time 11 it lies on C and K now C is just about there just about there 11 true length is right there lies on C and K C's and K name this as 11 11 and number 12 of course, is on D and J. So, J is here D is here that would be smaller in size of course, this is 12 this is 12. Now, if I have my intersection points here it is not going to be possible for me to get the entire loop within this region. So, I will probably have to do something either I need to grow my sectors along this duration or a simpler thing that I can do would be to erase this and renumber this. So, that I get this entire loop within here somewhere perhaps that would be better I have already transferred these distances. So, that should be a problem. So, I will just go ahead and erase this and then rename this just want to make sure that I erase these nicely it is quite possible that I need to locate my special leaders again that should not be a problem. Now, before I rename these vertices let me take a look at the top view of this figure and notice that it is this sector C O D and D O E that does not contain any part of the intersection curve. So, is it safe for me to start with D or perhaps is it safe for me to start with E if I start with D and go does not matter. So, if I start with D let us say from here and then go anti clockwise I probably would be missing I probably would be fine over here and then I will probably expect my loop to be on the left hand side as I traverse anti clockwise. So, maybe it is better if I start from D here and then of course, over here I will be back to D. So, if I start with D from here then this would be E this is F this is G this is H J this point over here would be K L there and then A this is C no problem there let me adjust this radius draw this sector nicely it was this part that got erased and you know I will do a little thing which I think it is smart, but later on it will be confusing. So, what I will do is I will make you know all these marks over here. So, this is for the intersection point 1 point 2 well I have not yet marked 3, but I will start with 4 5 this is not very difficult at this time, but when I am going to be marking the final intersection points there are chances that I might make mistakes number 6 number 7 remember that I do not need to do this I am just doing this to make my life a little more complicated. Number 8 10 will be slightly smaller 11 still smaller and finally, 12 too many acts nevertheless first things first let me try to locate M and N again. So, if I measure from I so from I towards P or from I towards H from I towards H this is where I am going to be getting M let me name this point and then from K towards L same radius K towards L yeah this is N let me join M and N to O let me talk on this a little bit even though I know that I do not need this and then how about P and Q. So, from A to Q that is the longer arc length easier for me to measure A to Q towards L A to Q towards L. So, that is my Q and from G to H from G to H K join my new Q to 0 or O should be using a different pencil and my new P to O looks like I am ready to mark my intersection points point number 1 for that let me use the top view easier for me to work with that point number 1 lies on J where is my J. So, this is the arc for 1 this is the point that I guess I will be getting number 2 lies on I and K. So, I is here K is here perhaps. So, that is number 2 that I get number 3 is on L and H perhaps on L L is there well I do not have the radius for that or maybe I do not sure that is number 3 that is the radius that I have alright. So, 3 would lie that is my L. So, it is probably going to be lying here let me name this arc as 3 over here and H it is going to be lying here and from here on I guess I will be ok. So, 4 lies on A and G 4 is this arc lies on A and lies on G 5 lies on perhaps B and L I will probably have to pull it up back 5 lies on B and F on B F does not lie on L ok. So, 5 lies over here and over here number 6 lies on M and N that is interesting M should be between H and I from H towards I from H towards I N should be from let us say K towards L from K towards L. So, looks like M and N are ok, but should the intersection points be here well let us not worry about 6 for the moment 7 lies on B and F again 7 lies on B that is for 7 8 on A and G physical 48 that lies on A looks like I am getting a loop, but I am still to decide on 6 9 lies on special leaders P and Q. So, this is the arc for 9 it is to be lying here on Q and on P 10 lies on B and F once again pull this up lies on well lies on L and H would have been nice idea for me to you know have this intersection over here as well, but let me take this as reference. So, B lies on H and L 10 section point number 10. So, this is L and this would be H. So, far so good 11 lies on I and K this is the arc for 11 that is K that is I and 12 has to lie on J. So, the arc for 12 lies on J once again tracing from J going all the way coming back all the way coming back. So, 8 point number 8 is somewhere over here 8 is on yeah A and G. So, should be yeah what I will do is I will take a little break for 5 minutes and then come back and try to locate the point that I have not been able to look at so far. So, point number 6 looks like I may have missed something see you know why. So, while I was taking a break I was trying to figure where did I make a mistake because I was not able to get the 6th intersection point and if I look at the top view that I have and if I look at the top view that is there I immediately realize what my mistake was what my folly was and the folly was that if you look at my specially the point M here should be up there not here likewise N should be up there and not here. So, if you follow the previous example you would be able to realize it better that was where my mistake was and of course, these generators are therefore, not correct. So, that is what I would do I would make that correction right now. So, my N is here in between B and C and my M is here in between E and F having said that well let me call this N prime I do not want to erase anything further let me call this M prime. So, let me join M prime with O with the 2 H and N prime with O with the 2 H. So, M prime is now line between E and F if I start from E it is more towards F let me capture this arc of course, if I look at this corresponding projection is horizontal and it is parallel to the hinge line likewise this is horizontal and parallel to the hinge line therefore, this would be true length you know about that from the theory. So, center as E and towards F I am going to be drawing this arc let me call this M prime and to locate N prime center from center as B rest my point B and towards C yeah. So, center as B and towards C this is my N prime ok. So, having located these two new leaders I am going to join them using a 2 H I am going to join N prime to O which is instead and now N prime to O becomes easier for me to locate the intersection point 6 which I was unable to locate last time. So, the arc for the intersection point 6 is here this is where it is going to be lying on N prime and on N prime it is going to be lying here now that is the arc for 7. So, this would be the arc for 6. So, I have to be very very careful especially when I am not leveling these points. So, this is intersection point number 6 and here this would be the intersection point 6 right there now that I have made so many mistakes in this lab perfectly all right one should rather invite mistakes that is how you learn. What I would do is I would kind of circle these intersection points just to give you an ideas to how they look. So, number 1 is here maybe I will just use a smaller circle number 2 is here number 3 is perhaps here number 3 is on L and B, B, L and possibly F is it or anyhow. So, number 3 is yeah H this far less H. So, that is number 3 I have to be extra careful because if I get it wrong then I will have to erase the entire thing perhaps I need I would then need to do this lab all over again. Number 4 is on A and G that is on A here here somewhere 5 is on special leaders P and Q special I do not know why I keep saying them leaders why keep calling them leaders they are generators 5 is on P and Q is that or it is on B and L looks like it is on B and L getting my top few back F rather B and F. So, this is my fifth intersection point my sixth intersection point is on N prime and N prime this is my sixth hope I am getting it right and here all right my seventh is on B and F is it seventh on B and F looks like this is my seventh there eighth is on A and G. So, my eighth intersection point is on A and G there and there let me mark them my ninth is on special P and Q but my ninth here Q this is P ninth on P tenth is on B and F possibly or perhaps it is on H and L well let me get back to the top view tenth would be on L and H tenth on L two more 11th is on C k and E I looks like I have got k and I here now which one is my 11th and which one is my 12th. So, looks like this is my 11th all right. So, these two points are ok but maybe I did not mark number 12 correctly. So, these points are ok 11th is on k and I my 12th is going to be on J and it has to be here to verify let me take the true length for 12th it is my 12th my true length going back over there here is pretty much going back over there and this is my arc for the 12th. So, this should be my intersection point here on J. So, I just mark it there. So, having done that now I am going to be very careful and sketching that loop out I will have to get up go to the other side of the sheet and using free hand sketch it nicely. So, I will start with J this is point number 1 well let me see how it goes similar that perhaps and the slope over here would be 0 and then this is point number 5 I believe all this point number 6. So, 1 2 3 4 5 6 and then 7 2 8 2 9 and then this would be my 10th 11th and 12th. So, this is how the intersection loop is going to be okay. Now, let me darken this using the H pencil be quite careful doing this free hand ideally I should have used French curves once again here the slope is going to be 0. So, this is how this loop would probably look like bottom line there are chances of a lot of mistakes that one can make one needs to be a little careful even if one is careful even then there are chances that you might still make a lot of mistakes do not worry about that keep making mistakes keep verifying them keep learning from them. But try to ensure that once you learn from them you do not make more mistakes from them.