 So, last Tuesday I had introduced to you two methods to find the intersection between two solids. The first one was the select line method or the generator method and the second one was the cutting plane method. So, here is a little revision. So, the first one is the generator method, important points or salient features. Step one that is something which is critical you have to figure out the view where the intersection points are obvious and the way to figure that thing out is to figure if the intersection points are going to be lying on one of the solids. Look at the front view, look at the top view and look at the profile view and you will figure that your intersection points will be lying on this circle in the profile view. So, once you have figured that thing out the next thing you would do is you would generate a bunch of lines that would represent the surface of the other solid. You would generate again a bunch of lines that would represent the surface of the other solid and that is the reason why the name the generator method or the select line method. Step two use generators to represent the other solid. For example, this line would be a line on the slant surface of the cone. So, this line is 3 11 line 3 11. In fact, these are two lines the first one is towards you and the second one is away from you that is behind the cone. So, first one is towards you the second one is behind the cone and the best way to visualize this in the front view is this. So, this is the first line line 11 and this is the second line line number 3. So, this line that is on the slant surface of the cone is in fact, representing two lines in the front view and in the top view of course, if you project this guy over here this would be line 11 and this one here would be line number 3. Step three and this is something which is important you have to exercise care in labeling and following them. If you do not label these lines properly and these intersection points properly it will be difficult for you to connect the dots later on. Now, Vasu was asking me this before how did I connect these intersection points here and here. So, if I look at the intersection points on the circle in the profile view I go A B C D E. So, I go in order and I follow the same order in the front view as well as in the top view. So, I go A B and then C and then D and then E and so on like was here A B C D and so on. Now, if you notice from this height and above this part will be visible in the top view. So, this part is going to be visible in the top view and from D onwards from this height down below these guys will be hidden they will be invisible that is the reason why these lines are shown using hidden lines or dotted lines. So, you have to be very careful in following the labels properly because if you do not then you will mess up the connections between dots between the intersection points. So, once you have all these lines ready you figure out the intersection. For example, this is one of the intersection points I call it I project it on to the front view. Now, these are in fact two intersection points again one is towards you and the other one is behind the cone. So, this intersection points get mapped at these two points in the front view and these intersection points lie on lines or generators number 3 and 11 respectively 3 and 11. Now, which of these 3 and 11 generators is towards you is it 3 or is it 11. If you stand here and if you look at this view which one of these will be towards you all right. So, once you have these intersection points map them on to the front view. So, this one will be lying on generator 11 and this one will be lying on generator number 3. Importantly do not miss any intersection point all right. So, this was the select line method or the generator method and works only when the intersection points are obvious in one of the views. The other method was the cutting plane method and look at these views for example, and the method works better when the intersection points are not obvious. And I will give you an example later today step number 2 you should know what the intersection between the cutting plane and solids yield. For example, if I you know cut through this assembly of the cone and the cylinder through these horizontal planes the intersection between the cone and the horizontal plane will be a circle of course, and the intersection between the cylinder this cylinder and the horizontal plane will be a rectangle. So, eventually determining intersection would boil down to determining the intersection between the circles and the rectangles in the top view. So, this step is critical you should know what the intersection between the cutting plane and solid yields. Let us take an example. So, let us take plane D slicing the assembly of cone and cylinder. So, corresponding to this plane the intersection between this plane and the cone gives me a circle and the intersection between this plane and the cylinder gives me a rectangle. So, in the top view how many intersection points would you see you would see 4 intersection points. So, you need to compute the intersection number 1, number 2, number 3 and number 4 these intersection points are going to be lying on which plane in the front view plane number D plane number 4 project points of intersection to the respective planes. So, you are going to be drawing a bunch of lines a bunch of circles a bunch of arcs and you know somewhere in between you will be thoroughly confused and you will be like. So, be very very careful and that is where labeling happens to be very important. So, all you need to do is project these points down on to the respective plane and in this case these are the two intersection points in the front view. So, summary of what we had discussed on Tuesday notice here that you will have two other intersection points on the back side of the cone. Which ones are going to be in the front this one and this one which ones are going to be the back these two guys. Once again do not and do not miss out on any intersection point otherwise your connectivity will not be proper. So, the connection or this contour or rather these contours will not be proper alright. Another example and I will give you a slightly different angle to solve this. So, this is the example of determining the intersection between two prisons two rectangular prisons one is vertical the other one is horizontal. Inspect the three views are the intersection points obvious are the intersection points obvious yes or no yes from where do you think you would be getting the intersection points profile here wonderful. So, you can use the generator method to solve this, but I will give you another method and this is something along the lines of the lines and planes the concepts that you had learnt in previous lectures hopefully first thing you would want to do is start labeling. So, p q r s is this prison that is penetrating the vertical prison inside. So, p is actually a line and so is q and so is r and so is s. So, p is this line q is a line which is closer to you away from you which is closer to r and then s is another horizontal line which is away from you. So, if you go on to your top view line p appears here q appears here r again at the center and s on to the top all right label the vertical prison in the front view. So, a b c d. So, this is a b backside or the back edge c and d likewise over here a comes from here b comes from here the center c comes from the top over here and d again at the center step one you have to do labeling properly. Now, you can think about these two prisons as being composed of four different planes a b c d is composed of four planes and p q r s is also composed of four planes all right. So, the problem walls down to figuring out the intersection between any two planes. So, it is like a plane plane intersection now notice again that in the profile view p q q r r s and p s they happen to be in the edge view. So, this plane this plane and this plane here they happen to be all in the edge view in the profile view. So, it is easier for us to figure out the intersection between planes and planes if we have a view where one of the planes is in the edge view now all right. So, extend this plane p q where do you expect the first point of intersection. So, if you are trying to figure the intersection between two planes a b and p q. So, this is plane a b and this is plane p q where do you expect your first point of intersection to be here it is quite obvious now it is quite obvious. Now, this point of intersections lying on which of the edges which of the edges in or on the vertical present a and do not miss any intersection point a and would be a b and d. So, one point is here the second point is here all right. Now, this point the intersection correspond to this point will not be obvious and that is the reason why I had extended this plane. Now, it is easier for me to locate this intersection point. So, this intersection point will be lying on edge a point number 2, but the intersection is actually happening here, but before that plane plane intersection would give me what a line intersection between two planes implies a straight line. So, one line would be this one joining 1 and 2 the second line will be this one joining 1 and 2 all right. Now, my actual intersection is happening here right. So, if I extend this edge q s these are the two points where my actual intersection is happening all right. This is like a pseudo intersection point this is not the real intersection point this is where the actual intersections are happening these two points having said that it is actually the edge q which is going inside which is kind of you know hampering this intersection from happening. So, you need to extend edge q from both sides because edge q is going to be visible in the front view. Second guy intersection between which of the two planes q r and well are you solving the intersection between two planes or three planes notice this is q r and q r is intersecting with plane a b and plane a d on the back side. So, be very careful and that is the reason why labels help all right. So, extend this one obvious intersection point is point number three this point on the vertical prism is going to be lying on edges b and d. So, this point three will lie here this point three will lie here and this pseudo intersection point that will lie on a if I call it point number four extended a lie here again intersection between two planes will give me a straight line in this case that straight line will be joining points three and four I will get one of these and the second one of these. Once again this is not my actual intersection point in section actual intersections happening over here all right. So, my intersection lines they get extended only to this point r s point number five project that point number four and five they will be essentially here same point p q extend that get the pseudo intersection point number six project that your point number six will be here relatively simpler example. Now, notice that the intersection between plane r s and the back side of the vertical prism is hidden and. So, is the intersection line between p s and the back side of the prism and they are hidden precisely behind these lines and that is the reason why they are not visible. So, that is something that you will have to keep in mind all right. So, the horizontal prism is penetrating inside the vertical prism. So, you will have a hidden line there a hidden line there and a hidden line there notice how these two points they correlate with the top view. So, if you take the projections up from these two points this is where they occur won't there be a hidden line that is what my next questions think and analyze. What you have to think about what you have to say about these hidden lines this one this one and this one are they going to be there or are they not going to be there are they going to be there or are they not going to be there I for the drawing think about that all right. So, let me do a little what we say in I T Bombay Kira you know what Kira is. So, it is one of the linga words in I T Bombay. So, this was a relatively simpler example. So, let me rotate this horizontal prism by some angle and see how interesting my problem becomes I rotate it clock wise by some angle similar story in section between two planes four planes forming a vertical prism four planes forming a horizontal prism the labels happen to be the same you know p q q r r s and p s they all happen to appear in the h view in the profile view. So, it is kind of obvious for me to extract the intersection point information from the profile view having said that of course when I rotate this horizontal prism p q r s the corresponding projections in the front view and the top view they will change. Now, first intersection point we need to figure out the intersection visibility of both the prisms extend this prism. So, you will have one intersection point zero intersection point. Now, do you say that the first intersection point which is kind of obvious is going to be this one yes where is that intersection point going to lie b d all right this is my second section point zero intersection point not real project this guy on to b and d. So, this is one and this is one. So, notice that I am labeling my intersection points as I identify them and as I project them on to different views because if I do not you know what is going to happen project point number two point number two is going to be lying on edge a not c, but a intersection of two planes they give me a line. So, the real intersection is going to be happening here at point number three likewise over there at point number three. I am not formalizing my drawings as yet I will go forward I will extend q r now in this case I do not have real intersection points here and here. So, I will need to extend this plane on both sides till I hit the two edges of the vertical prism zero intersection point number four and number five four lies on edge a project that this is where my four is five lies on b and d project these and this is where point number five or points number five they lie straight lines joining four and five. So, if you extend this plane four and five will be the actual straight line between the two planes that will be representing the intersection between two of them this is point number six which is the actual or real intersection and point number six is going to be lying on four five here and here point number six is there and six is there with me let us continue extend this guy once again we will get two pseudo intersection points number seven number eight seven lies on a eight lies on edge c of the vertical prism project seven seven lies on a eight lies on c the back side the back side. So, seven and eight will be a straight line point number nine this is where the actual intersections happening project that number nine would lie on b and d b and d. So, nine would lie here on b and it would lie here on d and the actual intersection line will be the line joining points eight and nine there and there, but this one is the actual intersection point. So, if I project that identified as ten if I project that point number ten is here and here messy already messy and the final thing extend p s identify that intersection point is eleven this one as twelve project eleven eleven is going to be lying on the back side or the vertical prism c and twelve will be lying on b and d join eleven and twelve or may be join ten and twelve. In fact, they would essentially happen to be on the same straight line same straight line. So, join eleven and twelve the actual intersection is happening here thirteen thirteen lies somewhere over here and here once you have identified these intersection points. Now, you need to connect them which ones of these will be real which ones of these will not be real all those points which lie on the horizontal prism for example, one three six nine ten and thirteen they will be real intersection points number one number three number six number nine ten and thirteen. Once you identify the loop the connectivity you can connect the dots in the front view. Now, one two three is that line going to be visible one two three line one three is that going to be visible well of course, you need to project these intersection points up that is something which is going to be a little tedious, but you should be able to do that this would be an exercise for you all right. So, line one three that is going to be visible how about the next one three six three six is that going to be visible now six to nine will be a hidden line because it is going to be on the back side of this prism if you are looking at the assembly from here this guy is going to be behind nine to ten nine to ten visible not visible ten to thirteen visible not visible hidden thirteen to one thirteen to one you need to be a little careful because this guy here although it might appear to you that this would be visible actually these two points are lying on the back side of the vertical prism. So, be very very careful be very careful this would be hidden. So, you know the previous example was quite straight forward you know this line and this line they merged this line this line they merged and you actually got a very nice v shaped thing now this one is quite tedious. So, very nice example of how labeling helps you connect the dots better yeah yeah you can if you want to easier less messy yeah you can you can all right you figure out the visibility of the prisms now figure out the visibility of the prisms. So, you know the concepts from lines and planes the visibility try to figure that thing out I would not say a word I just keep flashing different lines which are going to be green in color this is going to be visible that will be visible this will be visible this one here will be hidden why is that why is that s is on the back side yeah this guy here this line here is hidden and so is this line mirror image the same thing and you will get the result this line is going to be hidden peace inside of peace penetrating the vertical prism and so is our what about T 3 10 10 this guy yeah yeah yeah yeah if they are equal in size yeah no it was no in that model it was the cone that was cut yes quite obvious yeah what come again yeah which one has been cut in which one has not been cut if they are of the same size no no yeah so there should be so you are saying that there should not be any lines here because you have you do not have much clue about which one is penetrating which one this could be a single piece yeah yeah yeah this could be a single piece yeah true true so the question is this so given two prisms the horizontal one the vertical one how do you figure which one is penetrating the other one okay number one question number two a related question is what if the entire thing is a single piece that none of the prism is penetrating the other one so if the latter is the case then these horizontal lines may not be shown but if the former is the case then you need to show those horizontal lines and the q has to be derived from the three views that are given to you the clue has to be taken from the three views which are given to you yeah I haven't actually shown the intersection the top you or have I yeah yeah so no no no so you guys are in a habit of getting everything on the plate no don't don't don't you figure it out there has to be some thinking and analyzing that you need to do so if you are going to be using or if you are going to be drawing these lines of course you need to figure out the corresponding lines whether they are going to be there or not in top view you figure it out I have to go how does the three views communicate to you if one of the solids is penetrating the other solid what if I give that information that one of the solids is penetrating the other solid that's going to be clear otherwise not going to be clear so assume that I am going to be giving that information to you alright so this one is quite interesting intersection between two cones a vertical cone and horizontal cone the three views are shown to you and my first question is are the intersection points obvious from these three views are the intersection points obvious from these three views is it possible for you to figure out the intersection points just by inspecting each of these views yeah order 9 no no so it is not going to be easy for you to use the generator method or the select line method so instead you would want to use the cutting plane method so the intersection points to me at least they don't seem to be obvious use the cutting plane method okay so use use a bunch of horizontal planes slicing the assembly of these two cones okay now if you slice the assembly using a bunch of these horizontal planes intersection between one horizontal plane and this vertical cone will give you what they give you a circle intersection between the same horizontal plane and this horizontal flown horizontal cone is going to give me what a hyperbola so your intersection problem your problem of intersection boils down to figuring out the intersection between the circle and a hyperbola okay alright now what I have done what I have done is I have you know use the information that the sections going to be over here in sections going to be happening over here and here correspondingly I have chosen the respective horizontal planes to slice the assembly it makes my work a little easier but I do not guarantee that my solution is going to be accurate okay for more accurate solution you should be using more number of horizontal planes slicing the assembly of these cones okay so the first horizontal plane is this one second one is this one third one is the center or third one contains the axis of the horizontal cone both one and the fifth one okay now follow this very carefully because things are going to be a little messy alright so plane number one if I look at the intersection between horizontal plane number one and the vertical cone I am going to be expecting a expecting a circle in the top view okay and this plane when cutting this cone will give me a hyperbola the vertex of the hyperbola is going to be lying over here so essentially I will be expecting a single point of intersection there and there okay okay stay with me stay with me stay with me stay with me stay with me now this one this one is just a fluke I do not need to actually draw this because I know the intersection points just going to be a single section point wait for the rest of the construction alright so plane number two intersecting with the vertical cone will give me this circle and when intersecting with the horizontal cone will give me a hyperbola the vertex of that hyperbola is going to be here okay we need to construct it we need to construct that hyperbola so for that for that how is the base of the hyperbola going to be how is the base of the hyperbola going to be so the two points which will be lying on this edge of the horizontal cone they are going to be separated by twice this distance from here to here and from here to here one and two once I know that and once I know the fact that my apex or the vertex of the hyperbola is going to be lying over here I can draw a box once I draw a box I would divide this edge of the box into equal number of parts and this edge of the box into equal number of parts okay now from this point I am going to be joining all the points on this edge and from this point the apex of the cone I am going to be joining all these guys alright okay so let us say equal number of parts I am just using a bunch of points for demonstration not so very many join this point or the box with that point over there on that edge okay likewise for the second set this one where this one and this one with this one okay so you get a bunch of intersection points so intersection between this line and this line which is this point here will give you the first point on the hyperbola and intersection between this line and this one here will give you the second point okay two points you already know that this point is going to be lying on the hyperbola the vertex draw this hyperbola draw the mirror image of that okay so once you have the hyperbola in order find the intersection between the circle and the hyperbola okay this guy projected down on to the same plane this guy projected down on to the same plane so this would be one section point two intersection points over here and this one okay plane number three and the assembly the vertical cone will give me a circle the horizontal cone will give me kind of a triangle so intersection points getting that is not a problem to intersection points over here and two intersection points over there being projected downward on to plane number three we will get these two points plane number four again the vertical cone is going to be giving me a circle the horizontal cone is going to be giving me a hyperbola so you need to draw that hyperbola I will quickly go through that so measure this distance make that distance over there and over there okay draw this box the same construction procedure okay divide this edge into equal number paths that edge into equal number paths okay start joining the lines find the intersection points you will have four intersection points in all of four points lying on a hyperbola in all draw that hyperbola draw the mirror image of that find the intersection between the corresponding circle and the hyperbola okay so of course one point will be here and the other point will be here and of course there would be a corresponding point at the top okay and the fifth point is quite obvious I mean it has to be lying here so you do not need to worry about that construction top it is it is quite messy so I can I can really understand the expression in the face yeah okay so once you figure out the intersection points as I said the number of planes that I have chosen they are not adequate okay so they have to be adequate for you to get the actual contour of intersection that's the first one that's the second one okay and if you realize if you map these intersection points properly in the top view these are going to be shown in red join them get the mirror image of these this is how your cone cone assembly is going to be looking in the front view and in the top view now here I'm assuming that the horizontal cone is absent so what it has done is it has cut away a portion from the vertical cone and like just gone for right okay and that's one of the reasons why this part is visible otherwise this part would have been you know below the cone so that is something that you need to be a little careful about okay and once again if you want to compare this drawing with autocad you know walk it out cone-cone interaction place your cones properly this is how it's going to look now we got a vehicle so I was in tracking with him yesterday right and he was like you know I mean you showed that cone hexagonal prism assembly yesterday or Tuesday and how did you cut the cone I mean for your hexagonal prism to be fit precisely into it and to which I said well if you want to learn how to cut it first you have to figure out the intersection contour between the two solids and use the information that one of the solids is develop in fact in the example on Tuesday both the solids are developable okay use the fact that the cone is developable cut the cone spread it out you already have the intersection contour in there cut that part out you know flap or fold your cone back into shape and then you have the slot so the first part intersection is something that is covered that has been covered today and last Tuesday development will involve two more lectures next week and then you will learn how to cut one solid to be able to precisely fit the other one