 Louder, once again I read that as loads of practice, loads of practice space geometry it is all about practice if you do not practice then things will be a little difficult for you to comprehend. Read this, once again, read this, loads of practice once again. So, 1 to 1 map lines and planes loads of practice once again what is this louder still not with me lines and planes to me means loads of practice what does it mean to you lines and planes you do not need practice fine planes and planes to me would mean practice and practice practice and practice unless you practice things will be a little difficult for you to understand. So, Tuesday I was here I was sleepy you were here you were sleepy I did not know what I said I did not know what I explained and then I went back and I was like did I explain things right the answer that question to myself was perhaps not. So, maybe I will give another shot and then I thought what would I do if I were you leaving the stage sitting down over there and trying to understand thoroughly space geometry. Then I said well if I were you I would remind my life some 15, 20 years back back to 1991 which is how many years from now 20 what no kidding really I was just thinking 15 anyhow. So, I would remind my life back and assume that I was you first year and I will try to understand the intersection between a line and a plane and a plane and a plane that I am going to be talking about today. So, these are some very nice handy models that would help you understand the concept better if you have gadgets if you have cameras on your cell phones feel free not now, but after the class is over feel free to take a peek at these two models you know if I were you I would look at the frontal plane take a picture look at the top plane take a picture look at the edge view of this take a picture go back home or to my hostel room or to my lab and try to walk out the example with these pictures. Likewise frontal plane take a picture top plane take a picture edge view take a picture and do the same thing they will help you understand this better. So, feel free after the class alright. So, lines and planes. So, this is what I did. So, on Tuesday I was there in the lab was a Tuesday was Wednesday Wednesday I was there in the lab and I asked Ashwini ji to make a little model for myself. So, he made a little cardboard model which is little different. So, in fact, he painted red and green. So, very nice of him. So, he made a cardboard model like. So, a plane and a line passing through that plane. So, this is the front view top view and then I also got the two edge views in my camera and then I thought maybe I can explain this little better with these pictures. Now, realize one thing on Tuesday I said if this part of the pencil would be visible or this part of the line would be visible most likely this part of the line will be invisible. Did I say that on Tuesday yes or no this is an example that negates my statement. So, it is not necessarily true that if this part of the line is invisible this part would be visible. So, this is a counter example and a pretty nice one. So, it is as well all right. So, I had these pictures let me start with the front and top view. So, a triangle a triangle same basic thing projectors just to make sure that the pictures are. So, note that note two things these pictures are not accurate because they are actually in perspective they are not purely orthographic. So, these pointers they actually spread out when I take a picture when I take a photograph. So, note that. So, there would be certain errors, but if you discount them if you just ignore them and just follow the example here it is horizontal line to length hinge perpendicular to length projectors out distance is measured and transferred. This is the edge view of that yellow plane that you see and if I do the same thing for the pencil this is what my pencil looks like and if I look at the edge view of this. So, this is what my pencil looks like of course, this is a point of intersection if I project this thing back onto the top view and the front view. So, this matches pretty well with the pictures that I have over here and then if I look at the edge view and I superpose almost like accurate almost accurate. So, I said there are errors because this plane is also ward so almost like accurate and then if you figure out or if you recall the logic behind the visibility that I gave you if you see this part of the pencil this part of the pencil is lying in front of this plane closer to this hinge line. And if you look at that part and if you compare if you know that this part is going to be visible you realize that this is indeed the case. And part of the pencil which is behind this plane or away from this hinge line that part is actually concealed behind this plane right. Is the edge view going to be unique from where I look yes or no perhaps perhaps not well if I take the edge view using the vertical plane or the front view as a reference do the same mechanical stuff edge view the pencil. And if I compare this with the picture that I have again quite accurate not very much quite accurate this is the pencil you have this is the plane and this is the edge view of the plane little angular difference. And if you compare if you if you think about the visibility this part is closer to the hinge line is in front of the plane and if you go here that corresponds to this part and you realize that this part is actually visible. And the pencil the part of this line which is behind the plane if you look at the corresponding line here this part is not visible. So, are you with me on this example everybody who is not good. So, on Tuesday I introduced the concept of the cutting plane method probably I did not explain I did not get time to explain that pretty well, but hopefully I will do that better today. So, imagine a plane imagine an imaginary plane in the top view in the edge view which is in the edge view here. So, just imagine that this plane is containing this projection of the line and it is kind of vertical to this top view plane it is going straight in just imagine that question 1 would this plane contain the actual line would it would it not it would. So, this plane is going to be intersecting with this triangle a b c at this point and at this point. So, this point is on a c this point is on a b. So, if I project these points down get to the corresponding point on a c here if I project this point down get to a b here what would this line correspond to. This line would correspond to the line of intersection between a b c and the imaginary plane in the edge view here would it would not it yes and how would you find the point of intersection. So, this line has to be intersecting with the corresponding projection of this line p q and of course, this would be the point of intersection. Now, the second thing that you would want to imagine and this could be a little difficult. So, imagine that this is the plane and if you flip this entire figure if you flip this entire figure which part of a b c would you see towards you which part of a b c would you see towards you or more clearly which part of a b c would you see in front of this imaginary plane would it be this part or would be this part what it say. So, it is this part of the triangle which will be in front of this imaginary plane in edge view here and this part of the triangle a b c will be behind that imaginary plane. So, once you understand this and realizing that this imaginary plane is containing the actual line the projection of which is here in the front view if you look at the visibility of course, this part of the line will be visible how about this part would this be visible because a b c is in front of that imaginary plane which is containing the line. So, this part is visible this part is not visible, but this part will it be visible. So, if you are having a hard time understanding my second shot back to the cube example. So, the top view on the horizontal plane the front view on the vertical plane this is the actual three dimensional plane and if you look at the imaginary plane that is in edge view in the top view a horizontal plane that is the actual line p q that imaginary plane is going to be looking like that the yellow plane. Of course, it is going to be containing the line it is going to be containing this line. So, this is the point that I was trying to make using the cube example and if you would further want to work out this example you would see that this imaginary plane is going to be intersecting here and this would be the actual point of intersection if you project this point of intersection over here in three dimensions this would be the point of intersection. So, nothing much to say, but going back to the board example. So, imagine that the plane is in edge view that is containing this projection of the line in top view and if I flip this and look at the front view this is how my front view is going to look like right are you there sleepy sleepy stay with me have some water. So, if I take this projection down from here to here if I take this projection down from here to here this is where my imaginary plane is going to be intersecting with the real plane a b c this is where my actual line on the pencil is and clearly the intersection between the red dotted and green solid is the actual point of intersection between the line and the plane fine. What I did with the top view you could do the same thing with the front view. So, imagine a line or rather imagine a plane that is piercing the front view along this line and the plane is in the edge view. So, it is just kind of piercing this thing over here and if you flip this over and look at the top view same thing if you project these points up that imaginary plane is going to be intersecting a b c at this line and this line is going to be intersecting with the projection of p q at this point. So, this is the component section that you have and of course, if this is a plane and if you flip it over to look at the top view which part of the triangle will be in front of the plane in top view. So, this part will be in front of that imaginary plane and this part sorry this part will be behind this part of the triangle will be behind that imaginary plane and realize that the actual line is contained within that imaginary plane. So, which part of this line is going to be visible here which part of the line is going to be visible would it be this part which is visible or would be this part which is visible the actual line is on the plane on the imaginary plane. So, with that part that is going to be visible because this part of the triangle a b c is going to be in front of the imaginary plane. It is a little difficult for you to appreciate that at this time, but with practice I guess you will be better. So, planes and planes change the angle between the planes, if I change the angle when you say if I change the angle what do you mean. So, now focus on this the plane is in v number 1 and in edge view what does it mean. So, I am actually restricting my plane to be in edge view here. So, I do not have the freedom to change the angle of the plane here. So, this is probably I kind of got it wrong yesterday. So, my plane is going straight in vertical to this that is the reason that is how it is going to be in the edge view otherwise it would not be in the edge view. Can I repeat how my triangular plane a b c the imaginary plane that is containing the line. Now, what you see in the front view is this may be a projection of this do you do not you this plane this imaginary plane is in the edge view this is my triangle that is what you see in the front view. Now, if I want to look at the top view of this what would I do. So, should I go this way or should I go this way. Now, if I go this way are you sure should I go this way or should I go this way. So, what you see in the front view and what do you see in the top view a part of a b c is going to be in front of this imaginary plane the other part of a b c is going to be behind the imaginary plane. And this well direction flip. So, it it comes with a little bit of practice and visualization. So, I I cannot maybe I will go back. So, let us take a look at the this case. Now, this plane is going in this plane is going in if you want to take a look at the front view of this how would you flip your thing how do you flip your assembly you would want to flip it this way. Now, if you want to go back from front to top how would you want to flip you want to flip this way simple. How am I figuring that out that that is what visualization is in this visualization is. How am I figuring that out. So, anybody anybody have an answer to this what is the name Vinay. So, Vinay says well fine answer you have your imaginary plane going in intersecting with the plane. And then you flip when you flip how do you figure which part of the triangle is in front of the plane and which part of the triangle is behind the plane. That is why the question is right you want to come you can imagine it how many of you can imagine this raise your hand not very many not very many that is what visualization is. So, makes things difficult this is my model I will take it out. So, your green plane is a b c and let us say that this is your imaginary plane. Let us say this is in the top view in the top view you should be looking at the edge view of this you see that this is the line you see that this is the plane. If you want to look at the front view of this what you do or maybe perhaps this would be a better example edge view triangle a b c. If you want to look at the front view flip it up which part of the triangle is in front. So, it is always a good idea to make these models physical models and later on with practice mental models. Otherwise it becomes a difficult for anybody to explain this anyhow I was never a big fan of the cutting plane method. So, if you ask me I would say well if I were given a chance I would probably not want to use the cutting plane method instead I would want to go with the edge view method which I am a lot more comfortable with. It is not so easy to imagine this, but anyhow. So, comes with practice all right. So, let me go forward planes and planes everybody ready for this same stuff same stuff. If you want to figure out the intersection between planes and planes. So, given two cases well not two cases same case two projections the frontal projection like the front view and the top view two planes a b c and d e f. You want to find the point section or rather line of intersection between these two planes two planes intersect to give you a what line how would you do that how would you do that planes and planes how do you find the intersection the line of intersection between these two. Look at the edge view of one of the planes. So, if you look at the edge view of one of the planes I will take it out. If you look at the edge view of one of the planes then your line of intersection will be on this back project and you should be able to get the two points. Let us see how horizontal line this is a very mechanical exercise from now on I guess two length shoot projections. What are we looking at we are looking at the edge view of which plane a b c great. So, you understand relativity a little bit all right. So, this is the edge view of a b c and then you would want to do the same thing to the plane d e f would that d e f be a triangle in this auxiliary plane it will same thing pretty straight forward. Can you look at the points of intersection now one would be this the other one would be this right pretty straight forward those points of intersections are going to be lying where look at this guy here e f one of them is going to be lying on e f the other one is going to be lying on back project these intersection points one lies on b well one lies on e f and the other one lies on d. So, this is one m b n m h and the other one lies on d f right there. So, this is the line intersection between two planes in the front view line section between two planes in the top view do you agree that this is actually the line intersection between two planes yes or no yes why do you say that does this line belong to both planes or rather projections of both planes in both views. So, if these projections of the intersection line are contained within the respective views of the two planes in both views they seem to intersect right you can do the same thing. So, this was with the front view as a reference to get the edge view of a b c you can do the same thing with the top view as reference to get the edge view real quick no rocket science points of intersections this is intersecting triangle a b c at this point and at this point. So, this point is actually on a c and this point is on b c project these intersection points back. So, irrespective of whether you are looking at the edge view of a b c plane a b c or d e f you should be getting the identical result same thing now comes the interesting part visibility I want you guys to pay attention to this you think that all of these lines are going to be solid for both planes both views some will be hidden some will be solid some will be behind the other plane some will be in front of the other plane how do you figure that out how do you figure that out if you stay with me there is something which is really interesting. So, the same concept here d e f is in my edge view right now there would be a little part of a b c that will be in front of d e f in edge view that would be closer to this hinge line there would be the other part of a b c which would be behind the edge view of d f which would be away from the hinge line. So, from here and here from here you can figure out the visibility of a b c for this from here the same thing there would be a part of d e f which would be in front of the edge view of a b c and then there would be a part of d f which would be behind a b c. So, look at this edge view and whichever part is in front go back over here mark that part solid whichever part is behind mark that hidden now let us take it loop wise find a loop of the plane d e f above or rather in front of the plane a b c in edge view when I say in front I would rather mean that the part of a b c which is closer to the hinge line flipping now which loop is in front of d f which loop of a b c is in front of d f a 1 and 1 stay with me stay with me and 1 b 1 in front I should have been looking at that anyhow. So, which part of d e f is in front of a b c that is e 1 d 1 m 1 n n 1 e d m and n that is in front. So, that will be visible therefore, that would be solid the other part of d f will be behind a b c d f the rest of this guy will be behind a b c, but not all of it will be behind a b c only the part which is covered by this plane a b c will be behind it. So, would this part be solid would this part be solid hidden the rest of the plane d f will be below the plane a b c in edge view all right. Now, just solve a counter example if this part of d e f is behind a b c of course, that corresponding part of a b c has to be in front. So, this part would it be solid or hidden of a b c this part of the loop d m n e is solid and that is kind of hiding a part of a b c what you say of this edge of a b c how about this edge and the rest of a b c will be solid. Can you do the same thing from the other side help me out here. So, I will first figure out the visibility of a b c with respect to d f because d e f is in the edge view and then I will figure out the visibility of d f with respect that right. So, which part of which loop of a b c is visible a m n b a m n b because this is relatively closer to the hinge line and the other part of a b c is behind it has to be behind, but only a part of that would be visible the other part is going to be hidden. So, which part of a b c is going to be hidden behind d f this guy here right. So, this would be hidden this part this little part will be solid now go backwards if this part of a b c is hidden the corresponding part of d f has to be hiding it. So, this part of d f will be and if this part of a b c is in front of d f what you have to say about this about this and the other straight forward. This is one of the reasons why I like edge view a lot better everybody with me clear enough if you remember I had discussed the projection method sometime. So, they got two lines and planes you can use that you can use that to verify the visibility and the trick of the trick to using that is consider one plane at a time and see the other plane as a set of three lines. So, you will actually break the problem into a lines and planes problem right once again one plane at a time. So, take a b c as a plane take a b c as a plane for example, and d e f as a set of three lines d e f and the third one and then apply projection method. Well, what I said about the projection method holds true, but this is a little. So, instead of treating or instead of seeing a planes as loops you can actually see planes as lines and figure out the visibility of a line with respect to the edge view of the plane. So, this was just an example of that. So, but that is that is something which should be straight forward alright. So, this is another example two planes a b c and d e f little animation same stuff, but just for practice who is not with me so far yeah you are not with me. So, come close to me anybody else not with me anybody else not with me should I take it as a sign that you understand what I discuss today or you know your feedback is important. So, if I see smiles and sparkling eyes I am like everybody understands if I see heavy eyes rupee eyes I am like yeah you guys are still thinking about the quiz at three o clock. What is it what is the feedback what is the feedback quiz alright. So, another example you see the two edge views this corresponds to a b c in the edge view and this corresponds to again a b c in the edge view a point to note I wanted to actually discuss quite a bit, but maybe I will just leave you here wait, but before I leave you here you see only a part of a b c is intersecting with d e f. But here a complete part of a b c is intersecting with d e f how do you find the line of intersection how do you find the line of intersection. So, maybe I just take m n n over here I will project them back this is what my line of intersection would be I will project these guys down this is what my line of intersection will be m n n, but is it the true line of intersection why not why not because the entire line m n is not common to both the planes right only a part of that line segment is common to both the planes which part is it which part is it it is going to be this part and that part. So, the entire line is not common to both the planes. So, keep that mind. So, maybe I will just leave you here all the best for your quiz.