 So, lines and planes interaction between them intersection between them the way they interact is bind section. So, this would be I believe lecture number 19 if I am not mistaken all right. So, given a line segment and a plane and given the projections on a horizontal plane and a vertical plane. How do you figure if these two entities insect would they just by looking at the figure just by looking at the figure would they insect would you know or would you not know yes or no you would not know all right. How do you figure if a plane will intersect with the line or vice versa you have the gadgets with you you have the edge view somebody said edge view no how do you figure if the plane and a line would insect no plane a line. Now, the best way to figure that is to see the entire scenario in the edge view of a plane take this plane look at the edge view of the plane and see if the plane is intersecting with the line the two possibilities number one this guy is going to be piercing the plane number two this guy is just going to be like that there would be a gap. If the guy is piercing we are lucky, but if there is a gap we need to ensure whether we can do anything about this. So, let us figure step one to look at the edge view of the plane draw horizontal that would be parallel to the horizontal hinge line mark d h on b h c h take its projection down mark d v and. So, a v d v will be in true length make a hinge line perpendicular to that line a v d v shoot the projections out and measure distances that distance gets transferred over there this distance gets transferred over here and the third one gets transferred over here. And this is how you get the edge view of a plane something which is quite clear to you now c a 1 b a 1 what would be here a a 1 and d a 1 all right. So, both these points they will be at the same point in the edge view why because these distances they are the same. So, this line is parallel to the hinge line now do the same thing with the line segment p q you have the corresponding projection points over here p and q shoot the projections out measure distances transfer it down there. So, that is your p a 1 measure that distance that is your p a that is your q a 1. So, this is what you see in the edge view auxiliary plane of that plane of course. Now, with the insect you will know you will know just by looking at the edge view and just by seeing that the line segment intersects with the edge view of the plane over here it does not necessarily mean that the plane is going to be intersecting with the line and vice versa. Let us see if it really does or if it really does not. So, the point of intersection there project that point backwards of course, this point is going to be lying where is going to be lying on the line as well as it should be lying on the plane make sense make sense alright. So, project that point back m a 1 it has to lie on the line, but is it lying on the plane no. So, project it upwards it has to lie on the line alright image is it lying on the plane would the plane on the line intersect in this case no. So, that is the check that you would want to make. So, this point has to be a part of the line and also it has to be a part of the plane number 1, number 2 these 2 points they should be lying on the vertical projection. Let us take another scenario. So, I keep the plane the same I will change the orientation of the line in both views witness you know the answer witness the plane remains the same the orientation of the line changes the edge remains the same because the plane remains the same. Now, show the projections from p v and q v measure the distance this 1 transfer that distance over here projection out from q v that is p a 1 of course, measure that distance and transfer it over there p a 1 q a 1 join p a 1 q a 1 and you have a point of intersection between the edge view of the plane as well as this corresponding projection of the line. So, that is your point of intersection m a 1 transfer it back it has to lie on the line that is m v and now you realize that the point also is within the plane take it up again it has to lie on the line m h again it is within the plane. So, if you see these signs you know that the line and the plane they are intersecting otherwise they are not everybody with me, but promise will have a t session someday everybody alright. What is the answer to this question yes come on better another case and this is the edge view method edge view method why because we are looking at the edge view of the plane to figure out if the line and the plane are intersecting or not. Horizontal line the exercise remains the same horizontal line take it down true length look at a plane which is perpendicular to this line of true length show the projections out measure distances out from all the 5 points the 2 points p and q and the 3 vertices of this plane a b c measure distances transfer them measure distances transfer them measure and transfer edge view of a plane mark the points measure that distance from here to here transfer that over there measure that distance transfer that and this is your corresponding projection of the line p 1 q 1 point on section m 1 transferred back that point has to lie on the line that is m v project it up again has to lie on the line this is m h and of course, the 2 projections they lie within the plane. So, the line and the plane they would not set in this case as well when they intersect what is that mean a part of the line is going to be below the plane and the other part of the lines going to be above the plane right now you got the point of intersections. So, let us erase all the other parts and let me ask you a relevant question which part of the line is going to be visible in both views and which part of the lines going to be hidden in both views. Once again which part of the line you know you are looking at this segment of of course, this part and this part is going to be visible here this part and this part is going to be visible here, but here which part would be visible and which part would not be visible likewise over here which would be visible and which would not be visible would this be visible or would this be visible or that guesses would this be visible or this be visible in the vertical plane would this be visible or this wanted to see the edge view will come to that which edge view. So, what I will do is I will tell you something about the projection method it is a little weird, but works. So, stay with me pay attention. So, the best way to figure what part of the line is visible or not is this from the top view or from the horizontal plane. This line is intersecting this edge of the plane and this line is intersecting this edge of the plane over here. Now, pay attention follow the steps very carefully what I would do is I would drop a vertical from both these intersection points like for example, this one I drop a vertical. Now, in the frontal plane on the vertical plane what do I observe where would this projection of which part of this projection hit first would be the line or would be the plane in particular would be the line p q or would be the edge a b here. Once again from this intersection point I am going to be dropping a vertical and my question is that vertical what is it going to be hitting first the line p q or the edge a b in the vertical plane the line p q. If it hits the line p q then it implies that this part of the line is visible do not believe me I do not expect you to believe me, but I would say that this part of the line is going to be visible and. So, I show that using a solid line now coming back to that intersection point if I drop a vertical from there. Now, that point represents the intersection between p q and b c the edge b c here the question I am going to be asking is which is it going to be a line p q or is going to be the edge b c that vertical is going to be hitting first b c. So, that would mean that this part of the plane is above the line and therefore, the line is behind this part of the plane and therefore, I would show this using a dashed line I will do the same exercise starting from this vertical plane points of intersection that I need to consider is this intersection between p v q v and a b b v and this intersection between p v q v and b v c b v c d. I raise a vertical from here go on to the top what does this vertical hit first a b it hits a b first that means what this part of the plane is in front of the line and therefore, the line is behind that part of the plane and therefore, the line is dashed and if I raise a vertical from this intersection point up what is that going to be hitting first which line p q. So, of course, p q is above the plane and therefore, p q is solid projection method is it making any sense is it making any sense yes or no who says yes the first am I the how is it making sense to you where of course, yeah but how do you verify the last vertical that I no no no. So, look at this point here this is the intersection point between p q and the h b c. So, if you raise this vertical from here over there the question that you are going to be asking is which is coming first the line or the h b c of the plane not the plane, but the h b c of the plane. So, I will give you a clarification of this and I will come to the edge view, but this kind of works and notice what is happening at the intersection point two things are happening number one at the intersection point a part of the line is solid and a part of the line is hidden here as well as here number one number two the part that is solid over here is hidden over here. And the part that is hidden over here is solid over here do you expect that do you expect that to happen yes or no is it going to be true in all cases are you sure would you like to t a about this think and analyze about this do that most of the examples that I have solved I have witnessed this phenomenon do this. So, this is something very interesting. So, this part is hidden the corresponding part over here in the other view is solid this part is solid the corresponding point the corresponding part in the other view is hidden. So, the point of intersection what it does it changes the state of that corresponding line segment solid to hidden hidden to solid make sense make sense. Now, I think if I need to justify this projection method works I have to look at the edge view this is the edge view that you had drawn before. Now, just focus on this part this is the edge view of a plane this is the line this is the intersection point. Now, focus on this region focus on this region what is in front is it the part of the plane or the line is it the part of the plane or the line which is in front you guys are tired is it warm. Did you have a heavy lunch yes come on stay with me which part. So, here is the part of the plane in front of the line or is the line which is in front of the plane in front of the plane. So, you are looking from here. So, this is your direction of view is it the plane that is in front rather let me ask you slightly different questions. So, is it the plane that is closer to the hinge line or is the line which is closer to the hinge line the plane is closer to the hinge line what does that mean from here to here the plane is going to be hiding the line behind it from here to here the plane is going to be hiding the line behind it. Do you see that do you see that they were supposed to be two more points over here when they or here I drew the horizontal from here took this down took the edge view of that yeah. So, somewhere over here do you agree. So, from here to here the plane is in front. So, this part of the line is going to be behind the plane from here to here the plane is going to be hiding the line and therefore, it is hidden from here to here the plane is behind the line here to here plane is behind the line and therefore, this part of the line is solid make sense make sense yes or no good can you do the same thing for the top view you have to draw the edge view over there yeah and if you believe that this reversal happens once you figure this thing out go over there this part is going to be solid this part is going to be hidden and you are done if you believe that another example plane and a line edge view method horizontal you know by the time we are done with the lines and planes you guys are going to be so adept with this method that while you are sleeping you will be like yeah horizontal line true length you know horizontal line true length hinge perpendicular to the line of true length shoot projections out measure distances from where bottom or top measure distances from bottom transfer distances is my screen shaking get the edge view of the plane mark the points on the plane get the line in there well that is the same example just that I am now focusing on the edge view of the plane from the horizontal plane. Now, here the question is of course, points of intersections m 1 m b m h here from here to here which part I mean what is closer to the hinge line is it the line or the plane line is closer. So, from here to here what is the corresponding part there m 1 to b 1 m 1 to b 1 from here to here right from here to here which part of the lines going to be hidden which part of the lines going to be solid. So, the lines closer to the hinge line there do you expect that to happen you expect that to happen because the lines closer. So, the plane is behind the line and therefore, this is dotted and if you want to come down this part is solid. So, this part has to be hidden and the other part has to be solid yeah cutting plane method imagine a plane in the horizontal plane in edge view that contains the line p q. So, you have a plane here that is slicing this plane a b c and that plane is containing the line p q. So, imagine a plane in the edge plane that contains the line p q and that slices a b c all right. How would that plane slice the plane a b c in the vertical plane or the frontal view look at this point drop a vertical down there. So, this point lies on a b the point has to lie on a b look at that point drop a vertical down there this point lies on p q and it point it also lies on b c. So, we are interested in that plane intersecting with the plane a b c and of course, two planes intersect to give you a what a line. So, this is where the impression that line is going to be that line is going to be intersecting the line p q of course, this would be the point section m b project it up to get a match once again once again I am half asleep already plane a b c in the top view double a come up in stage. Now, did you get this t shirt printed before the galaxy results were announced or after the galaxy results were announced. They got this t shirt printed before this last year. So, these scores they are not the true scores they are from the last year. So, these are better this time 717715706 was that is this arrow signify no, but the arrow is pointing downwards yeah it is not a good idea not. So, it shows variance are going down well good idea for some not a good idea for others where am I all right. So, plane a b c hold this line p q hold this and. So, this is the horizontal plane. So, imagine that a plane that is containing the line it is intersecting with the plane a b c plane containing the line is intersecting with the plane a b c hold the line. Now, this is what the scenario is it is flip it by 90 degrees yeah thanks. So, imagine a plane which is containing this line flip this entire thingy by 90 degrees. So, you have got 2 points of intersection 1 over here the other one over here project these points of intersections down. So, the plane is going to be intersecting with a b intersecting with a b and the plane is going to be intersecting with b c it is going to be intersecting with b c over here 2 planes when they intersect they give you a line of intersection. This would be the line of intersection between the plane which is containing the line p q here and this plane a b c all right and the intersection point between the plane and the line p q has to be common to both p q and this dashed line which is this here. And we project it upwards the intersection point has to lie on the line p q this is the image simple in the previous case if the line in the plane would not have intersected would you have expected intersection to be happening here it would be happening outside the plane yeah stay with me all right I have no idea what I am doing, but let us see horizontal line true end edge view of the plane a b c projectors from p and q out measure distances from the bottom view transfer distances on the auxiliary plane get the edge view of the plane mark the points on the plane get p q also all right. So, that is the point of the section and one projected back on to the line p q that is image projected down that is m v that is all right. Now, if you take a horizontal line here get the true length here get the edge view of the plane using this as the helping view the same exercise get the edge view of the plane there and of course, the corresponding projection of the line. Now, look at this point of intersection here look at that point of intersection over there should they be giving you the same result they should be giving you the same result otherwise the views will be lying which is not a good idea. So, m 2 correlates with m v and of course, m h correlates with m v now if you do not want to follow the projection method to figure out the visibility of p q you can simply use the auxiliary views. So, if you do not want to use the projection method to figure out the visibility of p q and if you just want to use the auxiliary views then do this figure whether the line is closer to this hinge line or the plane is closer to the hinge line here the plane is closer to the hinge line. So, that part of the line will be hidden the rest here corresponding to this part of the plane which is behind the line that part is going to be solid once again once again alright question what is closer to the hinge line the plane of the line here the plane. So, this part will be above the line and this part of the plane will be below the line. So, the corresponding part from here to here will be solid from here to here will be hidden go on to the top and ask the same question using the auxiliary view here which part is closer the plane of the line the plane is closer right from here to here the plane is closer. So, which part of the line will be hidden the bottom part of the top part the bottom part will be hidden the top part will be solid and the switch has to be at the intersection point. So, keep that in mind the switch between hidden and solid it has to be at the intersection point otherwise it does not make sense. Now, if I do the same thing using the cutting plane method I hope I am doing that no no well I am doing projections that intersection point and this intersection point I drop a vertical from there P Q and A C over here which comes first P Q or A C P Q yeah P Q comes first. So, what part of the line will be solid there what part of the line will be solid there this one or this one yes get a Q remember this yeah I guess you could do that right. So, projection method is not very difficult. So, get the vertical line figure that P Q is above A C. So, P Q will be solid from here to here and the rest will be hidden and do the same thing from the bottom raise a projection line from here. So, this is the intersection between P Q and A C A C comes first. So, this part of the line will be hidden and the other line will be the other part of the line will be solid. So, this is a nice I will go but the auxiliary view method of the edge view method gives you a reason as to why a part of the line will be solid the other part will be hidden or vice versa. . .Cutting plane method a plane which contains the line P Q here in the edge plane assume that it is intersecting with the plane A B C get this point down over here get this point down over here this would be the line of intersection between that imaginary plane and the plane A B C that line of intersection the blue dotted line is going to be intersecting the line P Q here. So, you would realize that M V and M H they are the same points whether you use the auxiliary view method the edge view method or the cutting plane method they would have to be the same. Now, ask yourself this and this is going to be a tricky question ask yourself this and this is going to be a tricky question. So, if I am having a plane here that is containing the line P Q and that plane is going to be intersecting with this plane A B C here which part of the plane A B C will be in front of this plane and which will be behind. Once again if I have a plane here containing P Q and this plane is intersecting with the plane A B C which part of A B C will be in front of this plane and which part of A B C will be behind this plane think about that. And in particular if you flip this scenario if you flip this scenario and come back to the vertical plane what happens then. So, that is that imaginary plane the light blue is behind the plane A B C the dark blue is in front of the plane A B C this guy is behind A B C this guy is in front of A B C I believe. So, A B C is in front of the line in the light blue area and behind the line in the dark blue area the same thing from the frontal side imagine a plane which is containing P Q here it intersects the plane A B C how do you find the points of intersection raise the vertical from here to here raise another vertical from here to here that is your line of intersection between that imaginary plane and the plane A B C and of course, this is your point of intersection. So, whether you start from the horizontal plane or the vertical plane does not really matter and again if you ask the same question which part of the plane will be in front of this imaginary plane which part of the plane will be behind the imaginary plane there is something that you need to think about. So, the light red part is behind the dark red part is in front. So, this part of the plane is in front of this plane. So, what will happen to the line what will happen to the line that think about this questions zombie.