 This is example number 10. So, here we try to figure out the intersection between a cone and a prism whose cross section is a diamond. So, this is the profile view, this is the third angle setup, profile view, the front view and the top view. So, I have deliberately marked these surfaces as dark lines, ideally I should not have done that, but nevertheless I have done that. So, that I can distinguish between the projector lines and the surfaces clearly as I draw, as I try to or attempt to figure out the intersection between a cone and this prism. Let us look at this front view here, is it possible for us to figure out directly where the intersection points are going to lie, by inspection possibly not. In the top view also by inspection possibly it is not easy for us to figure where the intersection points are going to lie, but if you look at the profile view, the intersection points they have to lie on these 2 surfaces of the prism rather these 3 surfaces of the prism. So, once we realize that maybe we can take this profile view as a base and then whatever section points we are going to get from here, we are going to be transferring those points on to the front view and on to the top view. Let us get started at this time it is important for us to you know make some basic construction and label, but before that let me summarize that there are 2 methods to try to find the intersection points between 2 solids or inter penetration points or inter penetration curves between 2 solids. One is the select line method and the other one is the cutting plane method. Now, over here it is obvious that the intersection points are going to be lying on these 3 edges. So, perhaps it is easier for us to represent the cone using a set of lines and then try to find the intersection between those lines and these edges of the prism. So, let us get started let me start working with the top view here and try to divide this circumference of this base of the cone into equal number of parts. So, perhaps I will be dividing this into 12 parts each or rather 12 sectors with an intermediate angle as 30 degrees. Let me ensure that my drafter is aligned properly and then my first 30 degree line is going to be of course, passing through the center I am going to be taking or I am going to be using a 2 H pencil for that. And my second line is going to be again passing through the center at 60 degrees. So, I am using a 30 60 90 set square and try to appreciate how I am positioning my set square in such a way that I get my 30 degree angle first and then now 60 degree angle. So, I got 1, 2, 3, 4 divisions likewise on this portion of the circle and if I want to get a 60 degree here perhaps it is a nice idea for me to flip this over or may be my drafter is getting locked over there. So, I will try to figure a better way to do that yeah of course, better way do that as you know just take horizontal projection from this edge get this point there take horizontal projection from this point get this point here and then join these using a straight line of course, has to pass through the center. Now, at this time it is a nice idea for us to label these 12 points on the circumference of the base of the cone. So, let me label them as A, C, F, G, H, I, J, K let me darken these construction lines a little. So, that they are you know clearly visible to you looks like they are now. Now, let me transfer these points 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 onto this base of the cone here and this base of the cone here. Now, for that perhaps it is a nice idea for me to work with the longer side of the ruler align that with the center line just make sure that is kind of vertical and start drawing the verticals from here. Let me draw all the verticals and then label the corresponding points on this base let me realign my drafter. So, point A is going to be here point B and L points B and L both are going to be here C and K, G and J, E and I, F and H and finally, point G here this labeling is important because we do not do that then there are lot of chances that we will get lost midway while we are constructing while we are trying to locate the points in section. So, it is a nice idea for me now to project these points horizontally on to this 45 degree line. So, maybe I will start doing that from here from C even excess B you have got A there and then L possibly I over shot by a little amount and then K here and then let us get these points on to this base of the cone 10 lines you know perhaps nice idea for me to realign my drafter. So, that the longer ruler becomes a vertical now make sure that I have got the vertical properly then from there we go down here up to the base of the cone from here again up to the base of the cone and finally, from here we will get me go to this center line or this horizontal line then from there extend this vertical double line there, but not a problem and let me realign my drafter. So, that my longer ruler becomes the horizontal now while labeling these points will have to be little careful. So, D is coming here C and E go down here F and B here A and G along the axis and then L and H I and K down here finally, J here once done let me call this apex of the cone as point O. So, once done the next step for us is to represent the surface of this cone by using what we call as the select lines or generators for that let me not worry about aligning my drafter again let me rather use my set square and very carefully join these points on the base of the cone with the apex or the vertex of the cone. So, these are called generators or select lines got to be very careful and in the front view as well there I am. So, if you now look at the profile view the intersection between this prism with the diamond as cross section and the cone will primarily be those between these generators and the three edges. Now, if you notice this vertex and this vertex these are going to be special vertices created specially. Now, if you notice we do not have a generator passing through this vertex and likewise passing through this vertex. So, perhaps to get the intersection points corresponding to this vertex and this vertex we would need additional generators and let me draw them as well in the profile view right there. And let me name them as say M likewise let me draw another generator from here and let me name these two points as P Q. Now, if you would ask me why two points M and N here and P and Q here well if you project this point upwards and further upwards and then make sure that horizontal is properly aligned make horizontal projection just about there may be a little perhaps here. And you know using a set square aligning this edge aligning with this edge in a point here. So, M and N would correspond to one of these points each. So, let me name this point as M and this as N all have to do is project these points down up to this and for the down to the base and let me call this M likewise I project this down up to some horizontal line over there and then for the down let me call this point as N. Now, also notice that you know there is a point of intersection between this vertical projector and the circumference of the base of the cone let me call this M prime. So, that we have M prime here likewise we will have N prime here and N prime here. How about P Q well the same treatment project this on to the circumference of the base of the cone. So, I got P here and Q here project these points downward make sure I have my vertical line properly and likewise from here Q. Let me have P here and Q here. So, too many points that we have to deal with now starting from the profile view we start figuring out the points of intersection between the cone as well as this prism. Of course, this is my first point of section let me name it as 1. This is the second one, two, third one, three, fourth one, four, fifth one right there, five, sixth one between the vertex of the pyramid or pyramidal well vertex of the prism and this special leader here six and then seven again between the vertex of this prism and the special leader then eight and this one as ten. So, working with prisms it is relatively easy, but if you are actually trying to figure out the intersection between a cone and a cone then things become little difficult. I will try to take up this example later, but for now let us focus on the intersection between this prism and the cone. So, once we have the intersection points let us try to transfer them on to the front view and then on to the top view. The transfer has to be very careful you will have to keep configuring your drafter if it is you know disturbed frequently. So, I have this horizontal there and then from point two got to be quite careful point three point four these lines are going to be parallel of course, but they are going to be quite close to each other and figuring the intersection points over here will be a little tedious that is the reason why labeling becomes quite important point five there well let us start working with point seven onwards. So, point seven is going to be here point eight is my drafter aligned properly little adjustment just to make sure that my horizontal is alright yeah and then point eight perhaps there and nine finally, but not yet ten still have to figure out the projection from the intersection point six well let me draw well the intersection point six has to be lying on this line anyways. So, I already have a part of it. So, there we go now if you notice point one is going to be on leader or generator j. So, this horizontal intersecting with j j would be here. So, this point one so let me name this as point one here intersection point two will be on two generators i and k. So, k is this one i is this one so will be here and here let me name them as two point three will be on h and l. So, going back this is generator l and generator h. So, once again going to point three and this would be my third intersection point at l and at h. So, let me name them as three it is little outside the cone intersection point four this is on special generator p and q. So, I need to draw those generators in the front view as well point four on q it so happens that point four and point seven they both happen to lie on q and p well point four may not be that important, but anyways since we have it let me mark it anyways on p and q. So, point four would be here and here on p and q. So, let me name this number five this will be on b and f. So, b is the third one got to be quite careful and f is again third from the left. So, this is intersection point five point six this will be on special leader m and n it is going to be lying on this line wherever in six m m would be. So, I need to draw another set of select lines or generators in the front view to get point six. So, once here and the other one is. So, back to point six this would be on m and n. So, let me name them as number six number seven this would be on p and q again. So, seven so this would be on the base line of the prism and p and q. So, p is here q is there. So, we have point number seven number eight this would be on h and l l is here and h is here number eight got that labeling is of utmost importance otherwise we lose the track of the sequence of these intersection points. Number nine this will be on i and k. So, k is there i is there. So, this is intersection point number nine and finally, ten will be on j. So, this is j. So, great. So, we have a sequence of intersection points and looks like it is going to be. So, there they are kind of arranged symmetrically from the vertical and it is quite obvious because you know the intersection is going to happen on this side of the cone and the other side of the cone. So, that is one of the reasons why we have symmetric placement of these intersection points. Now, always a nice idea to use you know different mechanical gadgets. So, that we can get a smooth curve that would join these points. Let me give it a shot. So, finish curves four at a time. Let me try to first work on the symmetric right part perhaps use my edge pencil to finalize this curve and maybe flip it over, work on the left symmetric part. Well, the slope here is going to be horizontal. So, it is continuous curve and then try to join these guys, flip it. So, and perhaps try to join this is a difficult part curve fitting. I got this part right seems if I flip it should be getting well let me try the inner side. It is probably not going to be looking symmetric, but perhaps I should have used that part as well over here. Well, let me erase this and see if I can do any better you know if you are not careful things will probably not work out to your satisfaction getting these leaders or generators back well they are not leaders they are generators more. So, I already had this point let me mark this is 5 and 6 back. So, I had these 2 points already and perhaps let me think of joining 4, 5 and 6 through well maybe this would be a better should turn out to be symmetric. Let me try free hand how about that. So, I will use a construction pencil and then from 6 to 7 and then maybe let me join for 7, 8, 9 and 10 perhaps using the French curve would that that would be easier seems it looks like that would be easier and a gentle curve frustrating or maybe I have it I just about have it may be a little shift down just about there any half from 6 to 7 you think we should probably have you know another point in between them. So, that getting a curve becomes easier. So, how about this point perhaps. So, maybe let me call it 11 point number 11 and this would be the intersection between this vertical generator a g and this point here. So, let me call this 11 take horizontal projection from here there we go and 11 would be lying on a and g. So, well close to here start on the surface. So, it becomes a little easier for me to complete this part. So, this is 11 and this is again 11 back to the use of French curves I should be using it rather I should be relying on my skills to draw free hand, but it looks like how these 3 points lined up a curved up pretty nicely using French curves. So, notice that no part of the curve it should be lying outside the surface of the cone perhaps likewise I could do the same thing here just about there. So, it is going to look some like that perhaps I can dock on this intersection curve offline and you will come to know about that. So, that things become lot more you know clearer, but before I do that perhaps it is a nice idea for me to take the projections of all these points up over there or I could as well get the projections from the profile view on to the top view and get all these points over there. Let me go offline for a while and try to you know dock on this intersection curve. So, I took a little break went offline and then using free hand I kind of drew this thick curve and then I realized one thing you know if this prism was to be there then this part would have been hidden and correspondingly these curves they should have been shown using dashed lines. While this part was going to be visible in the front view and therefore this curve should have remained solid this remains solid alright, but I had also kind of represented these three curves using thick lines. Now what would that mean that would mean that this curve is a result of this prism having cut the cone and the prism now is taken away. So, this is essentially solid cone minus solid prism that you would be seeing this curve all in solid in the front view. So, I have to keep that in mind and you know to remind myself that this is actually the cut solid solid cone I mean the prism cut from the cone to remind myself of that what I will do is I will kind of thicken the cone surface as well. So, look pretty neat I am going to do that later I am going to thicken the lines later over here I am not doing that here because I need to transfer these intersection points back on to the top view, but let me finish this line first there I am looks pretty much in contrast. So, I will come back and darken the relevant features essentially this feature here assuming that the prism is no longer there, but before I need to transfer the intersection points back on to the top view and for that let me realign my drafter. So, that I have the longer ruler as my vertical having done that let me quickly transfer these intersection points on to the 45 degree line haste makes waste one very good friend of mine Roger Sauer of Arvid Hachen University came up with the Deutsch phrase a la mitwele I tried to remember that after. So, I am going to slow down here and very patiently transfer all the intersection points. So, 5 of them already done 6th 7th is already there sorry 11th is already there and so 7th perhaps 8th now and then 9th and then finally my 10th intersection point back to the horizontal my first intersection point is on J. So, it goes up from there I mark my intersection point right there on J I also label that let me darken this projector line let me darken this horizontal projector better. So, this is point number 1 well right there number 2 goes up till there 2 lies on I and K generator I is there K is there. So, intersection points 2 indent that little bit 3 goes from here 3 lies on H and L just want to make sure of that. So, intersection point number 3 always good idea to label number 4 lies on P Q. So, I need to have the corresponding generators joining the center of the cone and Q and P and the center of the cone looks like my horizontal is right this is going to be a little interesting. So, these are points on Q and P intersection point number 4 number 4 number 5 is lying on B and F and F number 5 here number 6 is lying on special leader M and N for that I need 2 generators joining the center and M and center and N. Of course, point number 6 is going to lie here going up there well looks like this is not that parallel is it anyhow. So, let me take care of point 6 first and then get back to intersection point number 5. So, 6 is on M the same as on N need to be very careful my drafter is getting logged over there and that is the reason why I am not getting a line which is parallel which is what I expect from here looks like I should have gotten points much lower than what I got over there. So, perhaps what I could do is I could erase this part erase this part make those lines again and try to relocate my intersection points number 5. So, looks like they are going to be there well I would just keep that mind that these 5's they get shifted I would not bother to erase the number and now to intersection point number 7 should be lying there somewhere and on to this line and point 7 lies on P and Q. So, this is point 7 would lie and likewise over here we name these points as 7 3 more to go number 8 horizontal and 8 lies on H and L this can be intersection point number 9 goes from here and 9 lies on I and K. So, point number 9 here and here and finally, point number 10 would lie on J. So, if you realize all these intersection points they happen to be symmetrically placed about this vertical axis as I said before if you are looking at. So, if you are looking at the assembly of the prism and the cone from the top you know this part is going to be visible while this part is not going to be visible. So, this part is going to be hidden. So, if I draw a curve from 1 up to 6 is what I am going to see and from 7 onwards from 7 onwards this part is going to be below the prism. So, they are not going to be visible, but if I consider the cone or the prism having been subtracted from the cone then this entire curve will be solid which is what the case was here. So, having transferred all the intersection points perhaps it is a nice idea for me to you know darken these lines representing the prism subtracted from the cone will probably give you a pretty good look. First let me work on the intersection edges and the rest of the cone this comes out really nicely does not it there you go. And finally, working on the top view is my intersection point 4 placed correctly is what the question is should be placed over here because if I project the intersection point 4 from here should be positioned over here somewhere. And if I look at point 4 it should be lying on p and q. So, it looks like if I take it up go back yeah I need to be a little careful with intersection point 4 lies quite close to 3 3 seems to be all right 4 does not seem to be ok. So, was it my drafter let me double check. So, if I take it up over there yeah. So, looks like it was my drafter that made a mess of it. So, point 4 should be lying very close to 3 right there right there. So, this is point number 3 this is point number 4 will not be there. And if I extend that to p it is going to be here. So, 4 is going to be here just at the base of 3 point number 3 is there. And if I verify that pretty much get it from here number 4 pretty much get it from here also number 4 just about there just very close to 3 ok. So, 4 is not there 4 is there 5 is not here 5 is here. So, having said that maybe let me sketch this curve free hand I will have to get up from my stool make way for myself perhaps try to sketch this free hand I am using multiple strokes just to get an idea. And then from 6 perhaps I should have located in section point 11 as well. So, 11 goes 11 lies on A and G and lies on this vertical. So, 11 would lie on A and G as in this vertical. So, perhaps if I can project 11 from here for that I need to realign my drafter never a good idea to disturb the drafter once it is set, but for demonstration seems should be point number 11 was here somewhere just on the cone. So, if I take a projection point number 11 will be there and from here well it will be here let me name that in section point I should not be using my drafter anymore, but let me name this in section point is 11 both sides and then 6 11 7 and then 7 8 9 and 10 I have to see if 8 is positioned all right. Let me realign my drafter to be horizontal 8 was here it looks like 8 is positioned all right. So, perhaps yeah. So, just to get an idea as to how the curve of intersections going to be looking in the top view as I said had this prism been there then this part would have been visible. So, this part would have been visible from 6 up to 10 all these intersection points would have been shown using hidden lines because they were going to be below the prism in the profile view. So, having said that, but so since we are considering a solid that represents this prism subtracted from the cone this intersection curve will be looking like a solid perhaps I can take a risk or may be not first try to draw this using thick solid lines slope here is going to be 0 slope here is going to be 0. And finally, I will be using my sketch pen let us see how it goes try single strokes let me set my drafter side a little bit and finally, this is how the intersection curve in my top view is going to look like. I would have also loved to draw this entire base of the cone using my sketch pen. So, that if I have given you a very nice picture maybe I will try that I do not have a compass well I have a compass, but I am not sure if I will be able to fit this sketch pen there maybe I will try that let us see how it looks. Should I or should I not touch the boundary of the cone well this is how the entire things going to look like prism cut away from the cone intersection profile in the front view intersection profile in the top view and entire thing in the profile view.