 So, we have 4 plus 1 more lectures to go and after that what 4 plus 1 the 5th one I will figure if I need it if I need it then I will give that lecture great day today. So, hot outside. So, hot outside you would have had your lunch now you are here in L 7 and it so happens that what should I say. So, you had your lunch it is a hot day outside you are here in L 7 it is all chilled out here air conditioned out here in front of you are a bunch of T A 101 slides which I am going to be you know going through ideal ideal situation for you to doze off is not it is not it is that the reason why you are here all right. You know I would not blame you if I were you I would probably do the same. So, I will try my level best to keep you awake all right. So, let me get started this is about 10 minutes past 2 almost new topic interpretation of solids as I said we are going to be covering 4 plus 1 lectures more the 5th lecture is going to be very interesting all right already. The 5th lecture if I get a chance to take it is going to be very interesting it is got nothing to do with engineering drawing, but a lot to do with what you are going to be doing later on in your lives if I get a chance to take it that is anyhow. So, last time I ended up by posing this question to you. So, given 3 points is it possible for us to draw a hyperbola is it yes or no yes who says yes given 3 points is it possible for us to draw a hyperbola that passes through these points. So, if those who say yes want to think a little further generic equation of a conic 6 constants 6 constants how many conditions you have 3 you think it is possible still I would know let us see how do you get a hyperbola how do you get a hyperbola you have a cone well. So, I am not really sure if I would be needing additional conditions to have a hyperbola pass through these points, but let us investigate further. So, if you think about how we get a hyperbola this is how we do that. So, we have a cone and if we have a 1 vertical plane need not be vertical if you have a plane that intersects with the cone cuts it out then the intersection between the cone and this plane the red plane is going to be a hyperbola. So, this plane can be vertical it can be parallel to the axis of the cone or it could be slant what could it not be it should not be passing through the if I have this situation what do I get we get a parallel otherwise we get a hyperbola. So, this situation so we have a cone we have a plane that cuts the cone and imagine that these 3 points they happen to be a result of this process. So, the first point over here is obviously the intersection between the cone and the plane to get these 2 points we draw the base of the cone. We project this intersection point down over here we measure this distance and if we rotate this cone by 90 degrees these 2 points are going to be lying away from the axis of the cone by the same distance on both sides. Now, if you have this so to speak additional information that you have a cone you have a plane and you have 3 points which are a result of the intersection between the cone and the plane in red it is possible for you to draw or construct a hyperbola. And this is how it could be done so construct first a rectangle the length of the rectangle is the distance between these 2 points the width of the rectangle is essentially the height that this point lies above the base of the cone. The right vertical edge divide that edge into equal number of parts the bottom right edge of the rectangle divide that also into the same number of parts of equal length segments. Now, this is what is important from this point first you number these points from the top of the rectangle to the bottom of the rectangle and from the centre or the axis of the cone up till the end of the rectangle. So, the sequence is important top to bottom on the vertical side and left to right or centre to right in the in the or on the horizontal. And from this point start drawing line segments that join this point to all these guys 1, 2, 3, 4, 5, 6, 7 what you do about these points you start from the apex of the cone. So, from the apex of the cone join this point now the point of intersection between this line and this line would give you a point on the hyperbola. Likewise the point of intersection between this line and this one here would give you a second point in section between this and this would give you a third point and so on. So, once you have the intersection points points that line on hyperbola join these points and you get a hyperbola. Now, this is only possible if these 3 points they happen to be such that you can extract the information about the cone and the plane that is cutting cone. Let me ask you an inverse problem of course, this hyperbola symmetric. So, you would see the other side as well. So, do you think the additional information is required in terms of the cone and the plane that is cutting the cone in addition to these 3 points do you think that. So, if I give you 3 points and if I give you nothing else is it still possible for you to draw a hyperbola or if I give you these 3 points in such a way that you know the information about the cone and the plane that is cutting the cone is available to you would be able to draw this hyperbola. Yeah this is what my second so speak inverse questions now given these 3 points is it possible for you to find a unique set of a cone and the cutting plane. In other words is it possible for you to figure if the information about the cone and this cutting plane is implicit within these 3 points that is the inverse problem and think about that. Think about that because you know last time a bunch of you guys came to the board you know took the mic and started describing different techniques to construct a parabola given these 3 points. So, I thought maybe it was possible yeah think about that. So, for today's lecture I am going to be using a special pointer. So, this is my T cups and this is my pointer and the point that I am trying to drive through is interpenetration of solids this is the first example that we are going to be doing. So, imagine that this is a cone imagine that this is a cylinder and imagine that you have put through the cylinder I mean put the cylinder through a cone. So, of course you would have made some cuts on this side of the cone and this side of the cone the problem is very simple figure out these cuts what the shape is an extension of this is what you are going to be possibly doing in T a 2 o 1 when you are working with sheet metal. So, a cone is a developed surface is it yes or no. So, a cylinder. So, you can actually cut the cone out and spread it out on a plane how is that going to look like it is going to look like a big sector. What it is going to look like a big sector likewise a cylinder for example, you can cut it out and laid out on a plane and it is going to look like a rectangle. So, first things first given the positions of the cone and the cylinder depicted by these 3 views and it does not really matter first angle or third angle you know cone cylinder here, cone cylinder here, cone cylinder here figure out the intersection curves or in a single word interpenetration between the 2 solids. A question that you might want to ask yourself where or which view would you possibly use to figure out where the intersection points would lie. In other words let me rephrase this question. So, which view will tell you the best or which view will give you the best information about the intersection points between the cone and the cylinder would be the top view would be the front view or would be the side view. Side view wonderful where would the intersection points lie where would the intersection points lie. So, you say side view all right, but where would the intersection points lie would I be correct if I say that the intersection points are going to be lying on this circle. Now, the second question is how do I extract these intersection points how do I extract these intersection points I will have to do something to represent my cone right I will have to do something to represent specifically the slant surface of the cone to do that this is what I would do I would divide the base of the cone in the top view into equal number of parts and this is where things become very important this is where labeling becomes very important. So, you need to be very careful when you are labeling because if you are not then you will mess things up I number these points anti clockwise 1 2 12. Once again I divide the base of the cone into let us say 12 equal parts could be 15 could be 9 could be 10 whatever they have to be equal right they need not be equal as well. Now, once I do that I start taking projections of these points on the base of the cone on to the front view be very attentive be very careful. And then I start labeling the points on the base of the cone now which is this point great how about this point wonderful 2 6 great what 1 7 next 8 12 next 9 11 next 10. Now, I will do the same exercise I will represent these points over here on to this base of the cone the profile view and for that I am going to be using this 45 degree line. So, essentially I am going to be taking these projections you know and transferring them on to the profile view the reason why I said the numbering is going to be very important because here you would know that the numbering will change of course. So, when I do that I am going to be joining the apex of this cone to all these points on the base of the cone same thing now what would this point be great 8 and 6 next 5 and 9 next next next last great. Now, what you have noticed is 2 things 1 that is going to be very easy for you to extract the points of intersection from the profile view the points of intersection are going to be lying on the circle and once you have these what I call generators or select lines they essentially represent what the slant surface of the cone right your intersection points are going to be the intersection between the slant surface of the cone and the circle he is a little late but that is ok. So, your intersection points are going to be the intersection between these lines which represent the slant surface of the cone and of course, the circle your first point of intersection will be here what is this point intersection between which line on the cone on the surface of the cone 4 and 10 and the circle project that on to call it point a project that on to the front view now in the front view how would this show up how would this show up in the front view it would be on what slant lines 4 and 10 where is 4 here where is 10 there. So, point a is going to be on these 2 lines in the front view right right. So, of course, in fact this timing was just perfect. So, imagine that this is a cylinder imagine that this is a cone and this is how you are actually seeing the interaction between the cone and the cylinder like. So, there would be a point of intersection here and there would be a point of intersection here right there. So, what I would do is I would keep this on the chair in front of you. So, that while I am working with this slide you can imagine how the intersection is going to shape up maybe I could use stable better. Now, the other 2 intersection points are going to be here and here. So, point b would lie on the slant line 5 9. So, go on to the front view 5 is here 9 is here and it is at this height. So, you will have point b here and correspondingly point b there likewise there would be another intersection point on the other side. So, let me call this point l l will be at the same height from the base of the cone and l is also going to be lying on 3 11. So, 3 11 is here and here. So, be very careful where the intersection points are going to be lying because tracking that is very important. Once you understand this third intersection point on the circle lies on leader 6 8 project that call it c project that on to the front view identify where leader 6 is or where generator 6 is here mark that where generator 8 is mark that there would be an intersection point on the other side of the cone call it k and k would essentially be lying on the same generators here and here in the top view. Now, this is so in the third angle projection this is what the profile view. So, if you turn it by 90 degrees this way this part is something that is going to be facing you here. So, this part is something which is going to be facing you this part is going to be behind you. So, understand that. So, this these two intersection points they are facing you and correspondingly the other two intersection points are behind the cone. So, understand that rest is a little mechanical. So, d j they lie on 6 8 project that identify 6 and 8 e and i they would lie on 5 and 9 3 and 11 5 and 9 would be here 3 and 11 would be here and eventually the last point g that will be lying on slant line 4 and slant line 10. So, your intersection profile pretty much as might went off no your intersection profile pretty much as done in the front view join these points and this is how your intersection profile which show up. So, once you have drawn these curves transfer these points on to the top view once again be very careful these points they have to lie on specific slant lines on the cone or generated lines on the cone. For example, this one where would this lie 4 great this one would lie again on 10 and these are points a how about this one 3 and 5 careful how about this one one would be on 5 the other one would be on 3 likewise this one would be on 9 and 11. So, 9 here and if you project them upwards. So, labeling is important b would be on 5 and 9 respectively and l will be on 3 and 11 respectively project this point where would that be. So, think about this point they actually represent 2 points c and k. So, this point would lie on 2 and 6 let us see where it is 6 and if you go on to the profile view point c lies on 6 and k would lie on 12. Now, k would lie on 2 likewise from the right these points would lie on leaders or generators 8 and 12 and so on so forth. So, keep projecting be very careful those intersection points they have to lie on specific slant lines I will go a little slow. So, that we can follow and labeling is very important. So, if you mess up the labeling then you are going to be messing up the entire procedure. So, labeling is very important all right. So, looks like you have all the intersection points in top view now let us look at the profile view now. Now, if you are looking at the top of this assembly of the cone and the slender which points are visible to you which of these points are visible to you the points above the. So, these points all the points above this axis this horizontal axis of this slender they are going to be visible all the points below this axis they will be invisible. So, this part will be visible a b c d on both sides and this part will be invisible or hidden all right. Now, if you want to figure the rest of the visibility of the cone slender this how it is going to be. So, this part of the slender is visible of course, this part is visible this part is inside the cone. So, it is hidden of course, these vertical edges or these vertical circles are visible visible this part is invisible again because this inside the cone and this part of the cone is visible. Now, notice that I have not shown these edges using hidden lines why is that why is that they have to be removed. So, that the slender is accommodated within the cone likewise the same thing top view this part is visible this part is visible of the slender this is inside the cone. So, therefore, it is hidden visible visible visible this part is inside the cone again. So, it is hidden this part of the base of the cone is visible this part is below the cylinder. So, hidden visible and hidden now notice what is happening here likewise what is happening there the state of the line is changing right. So, the slender is visible up to this point and then it is going inside the cone and then it is coming out of the cone and it is visible again. And this is happening where at the junction of course, where the cone and the slender are interacting right at this intersection point just a double check. So, this is called the select line method or generator method. So, once again to summarize start by looking at the three views carefully start by observing the three views carefully number 1, number 2 figure out which view will give you the best information about the intersection points most direct information about the intersection points in this case it was the profile view. So, once you have figured that thing out in this case represent the surface of the cone using these generator lines. So, they are essentially the slant surface of the cone they would represent the slant surface of the cone and of course, the points of intersection would be between this circle in this example and these slant lines. Once you get the points of intersection label these points very carefully they have to lie on you know certain lines. Once you label them carefully transfer them on to the front view and the top view straight forward. The same example using a slightly different method and that is the method that I am not quite sure if you guys are still comfortable with the cutting plane method you guys probably did not do very well with the quiz with the cutting plane method. So, here it goes I will go a little quick with this. So, the idea is very simple. So, you have the assembly. So, you have the assembly of the slender and the cone just slice it using a bunch of parallel planes. Now, if you slice the cone with the plane parallel to the base of it what you get a circle and if you slice a cylinder with the same plane what you get a rectangle. So, your intersection points will be what your intersection points will be essentially the intersection between the corresponding rectangles and the circles this is what the basic ideas. So, you have this view for example, slice it with a bunch of horizontal planes the green planes are those cutting planes number them 1 2 3 4 5 6 7 8 9 from top to bottom. Now, these planes are going to be intersecting the cone in certain at certain heights project these guys up in the top view the intersection between the plane and the cone will appear as a as a what you guys are dozing off as a what great. So, each plane will intersect the cone at a circle sketch those circles in the top view and of course, do not forget to name or number those circles. So, from inside to outside they are numbered as 1 2 3 4 5 6 7 8 9. Now, from the profile view from the profile view you know that these planes are also cutting the cylinder. So, you would have this intersection point these two intersection points you know and so on and so forth. So, this portion over here will essentially be a rectangle project this rectangle on to the top view for each and every plane each and every cutting plane. So, this point here will essentially be a line rectangle with no area this would be rectangle b b rectangle c c rectangle d d and finally, rectangle e e and remember that the same rectangles will be below the axis as well. So, once you have identified or named those rectangles name them in the top view. So, a would be a line b b would be this one c c would be this one d d would be this one and e e would be essentially the rectangle corresponding to this plane here. Now, your first point of intersection is going to be between some circle and some rectangle can you give me that circle 1 and rectangle a first point. Second point is between 2 and b b how many points you expect 2 one at the bottom the other one at the top. Third one is between circle c and rectangle c c or circle 3 and rectangle c c 1 at the bottom the other one at the top. Fourth one is in between 4 and d d bottom and top and fifth one is of course, in between circle number 5 which is here passing through the axis of the cylinder and this rectangle here and do not stop here keep going down. So, this one is in between let us say d d or maybe c c rectangle c c and circle what circle what 6. Next one b b and 7 is it b b and 7 no it was c and 7 this guy here and this guy here c and 7. So, b b and 8 and of course, eventually a single point here a and plane number 9 this guy symmetric once again it is going to be these intersection points above the axis they are going to be visible the intersection points below they will be hidden they will be invisible. So, this is solid and this part is hidden. So, the intersection is symmetric mirror image and drop the corresponding intersection points down on to the front view where would you want to drop this to which plane which plane. So, here you have plane information here. So, which plane plane number what plane number 9 these 2 guys here which plane where the line where these points light on which plane to the light what plane number 8 great these guys plane number 7 plane number 6 keep working backwards eventually up to the plane number 1. And of course, you have this intersection point well this intersection is again symmetric. So, get the mirror image get this intersection point or intersection curve right. So, do you expect the 2 methods to give you the same result do you expect the 2 methods to give you the same result compare. Now, you have done autocad if you do 3 D modeling autocad take the cone of the same size take the slender of the same size performance section get the slender into the cone performance section. And if you try to compare that with what we have your identical all right. So, why is this exercise important why is this exercise important take a look at this example. Now, what I had done is I had made little modification to this cone I have cut a certain feature of this cone I have cut a certain feature away. And this feature is such that it allows this hexagon of the prism to go very nicely into this cone with barely any effort I am making some effort. So, if you do not know how to do this or if you cannot do this this is not possible for you all right. So, this method permits you to figure out the intersection between the hexagon of the prism and the cone. And eventually figure out the part that needs to be cut away from the cone. So, that when you fold it up it is a lot easier for you to let this hexagon of the prism pass through this cone something that you will be possibly doing in T A 2 O 1 of course, this is the portion which is cut away from the cone all right. So, this time we have a rectangular prism rectangular prism passing through the cone once again question number 1 which view will give you the right information about the intersection points profile view where would the intersection points lie where would the intersection points lie they will lie on this edges or these edges yeah. So, essentially the problem is to figure out the intersection between the slant edges of the cone slant edges that lie on the surface of the cone and these edges right. Same thing as in case of example number 1 previous example all right. So, this is your first intersection point it lies on this horizontal surface I am going to go a little fast all right. So, just follow this all right. So, there are two intersection points over here the second intersection point comes from here the interaction between the generator 5 9 and this edge correspondingly there will be two intersection points on the front view B C the third one comes from here lies on 6 and 8 fourth one would come from here possibly yeah that would lie on number 7 and 1 7 and 1 would be at the centre yeah all right. So, keep following this procedure. So, H and I would lie on 6 8 and 2 12 J and K would lie on 5 9 3 11 and eventually the bottom intersection point would lie on 4 and 10 all right. So, if you think about this face of the prison if you think about this face of the prison well of course, there would be another intersection point that comes from here and here visibility all right. So, a part of it is outside a part of the pyramid is outside the cone some part is inside. So, some part is hidden all right this part is visible this part of the cone is visible and of course, this edge is visible now if you think about this plane what would this plane give you. So, this plane on the prison what would this plane give you would give you ellipse yeah what would this plane give you it will give you parabola. So, the top part is the elliptical part and the bottom part is the parabolic part yeah this plane should be parallel to the what. So, what you are saying is this plane should be actually here somewhere oh this will give you an ellipse. So, both are ellipses are you sure it is a half ellipse all right. So, you guys are not sleeping after all good. So, this is so all right. So, project these guys up on to the respective slant lines on the cone be very careful with regard to where these intersection points are going to be you can figure this thing out this is not very difficult you can figure this thing out where the intersection points are going to be. Now, which part will be visible in the top view which part of the curve will be hidden in the top view which part of the intersection curve will be visible which part will be hidden what would this part be visible, with this part be visible no. So, everything from here up will be visible everything from here down will be hidden. So, this part of the prism is visible that is hidden this is visible visible visible that part of the face of the cone is hidden part of the base of the cone is hidden, visible, visible. So, the center part is going to be visible, because yeah and the bottom part or the other part will be hidden. So, the part correspond this region over here that will be hidden. Once again if you go back and if you walk it out on or a cad this is what you would see. Coming back you know I mess up my coordinate geometry. So, a plane that is parallel to the base of the cone if it cuts the cone it give you circle. A slant plane will give you an ellipse right. How do you get a parabola? One at time if it is alright. So, if it is parallel to the slant line anywhere you will get a parabola alright. Here you will get a hyperbola this one will give you a hyperbola. So, it need not be vertical. So, a plane a plane need not be parallel to the axis of the cone to give you hyperbola. So, it could also be like so till it is again parallel to the slant line that is when you will get a parabola. And of course, this will give you a what this will give you an ellipse alright. So, fix this.