 1 minute pass 2 let me get started. So, if you can keep your murmurs down and allow me to start. So, those who were there last Friday in the class we were discussing perspective views using I would say a 2 station point method right. So, that is a generic method that allows us to draw a single point perspective, 2 point perspective and 3 point perspective maybe I will wait all right. So, this word is. So, if you do not allow me to start now if you do not permit me to start now our class is going to get spilled over maybe till about 3 15 3 30 because I am going to be introducing a new topic to you and it looks like I may have to do a little bit of explaining. So, if you allow me to start now I will start I will have to talk a little about perspective views and then I will switch on to space geometry all right good wonderful. So, for those who were there last Friday I had introduced what I call a 2 station point method that allows you to draw perspective of any kind 1 point 2 point 3 point depending on how you place your object rather how you place your top view with respect to the picture plane. So, if you have your top view like. So, you will get a single point perspective if you rotate your top view about an axis that is perpendicular to the screen you will get a 2 point perspective and further if you rotate your top view in such a way that none of the edges of the top view or none of the edges in top view are parallel to the picture plane you will get a 3 point perspective. The beauty of this method was that you did not have to specify any vanishing point. So, I was working on this example and we had reached up to this point where I was trying to figure if all the edges of mine they were converging to 3 different points or none. And then you guys said well the first 2 points the first 2 vanishing points they are not lying on the horizon line and that made me doubt myself. And I immediately went on a defensive and I said well I may have made a mistake and help me try to figure my mistake. And we are searching for answers and it looks like I have found an answer and I say that well the only mistake that I was making was I was probably not drawing these lines accurately. So, these green lines now are much more accurate and they allow me to see that these green lines they tend to vanish at 3 different points. I will tell you that they are indeed accurate and the top 2 vanishing points they need not lie on the horizon line I will tell you about that. So, I am going to be switching to a new slide and what you will notice is that I will retain these 3 vanishing points. And I will try to justify that they are indeed the positions that we will get when we are drawing a 3 point perspective here we go you see those 3 dots you do yes or no come on good. Now, this is what I will do I will try to verify these vanishing points I am going to go a little fast. So, try to follow me. So, this is my x axis of the object this is my y axis of the object the vertical is my z axis of the object this is in the top view and the corresponding station point where I am standing the position of the viewer is here SPT. Now, what I do is I look along the x axis at an object at infinity that would hit the picture plane. Likewise, I look at an object at infinity along the y direction and the z direction these 3 blue lines they are going to be hitting the picture plane and they are going to be releasing the respective verticals right. So, on these verticals I expect my vanishing points to line fine with me with me good I will do the same thing with my profile view this is what what axis are these x and y and this would be my z axis I take SPP the station point in the profile view I draw a line parallel to the x y axis and hit the picture plane over here over here and I draw a line parallel to the z axis and hit my picture plane over here from here I am going to be releasing what the horizontal projections right once I do that I realized that my vanishing points are essentially the intersection points between the respective vertical projections and the horizontal projections convinced convinced. So, I did not make a mistake last time now I solved another example and you would see those slides on my web page. So, in this example I took my picture plane to be passing through vertex a not vertex e, but vertex a both in the top view and in the profile view here what I got was this perspective the dashed lines are behind they would not be visible, but they are there for me to verify the vanishing points I start drawing these lines they tend to converge to three different locations. And if I do the same exercise if I retain these positions of vanishing points draw my x axis y axis z axis in the top view using SPT draw three lines parallel to the three respective axes allow them to hit the picture plane here get the verticals. And then using SPP the station point in the profile view draw raise parallel to these three axes here x y and z. And I release the horizontal projections from there I tend to see that my vanishing points are you know the intersections between the vertical projections and the horizontal projections. So, you said that you did not have the proof, but now we have one. So, it works let me tell you I have a wonderful team I am very fortunate to have a wonderful team of tutors and TAs and I am sure you are benefiting a lot in their presence and with the knowledge. So, this is something that I did not think about I did not analyze about. So, for sure help me think about this and so one whole very elegant way of getting a perspective the only problem is that you have to figure how to draw the true top and profile views that is the only problem that involves two rotations of the object. And once you have that perspective is straight forward. So, this what the bottom line was where we all got stuck Friday the top two vanishing points need not be on the horizon line or the picture plane yeah Ayush. So, Ayush sent me a link on YouTube teaching me how to draw a circle in perspective. So, I will let him teach you guys oh we cannot switch the laptop I have the ok. So, we will probably do that at the end of the class maybe, but perhaps what you can do is you can use this animation and explain how to draw a circle in perspective yeah because that would involve switching time I have the slides yeah wait I have to give you the mic what do you have this for I do not have this it is not like it is not mine suppose we have the right pattern yeah get the animation first. What I will say 8 is that I will take the hominids yeah ok. Then I will join the two diagonals and the two lines which are actually the mean diagonal this diagonal and this diagonal this line was supposed to this and this line was supposed to. So, what I have is this is the hominids where these are the two diagonal and these are the diagonal diagonal. Then I will join this to this point this to this point this to this point and this to this point this is hominid diagonal and I can I will then say that the point of which is this one this point this point and suppose this is the and same of all the other squares. So, actually speaking actually speaking what you have are two methods right in front of you to draw a perspective and to draw a circle in perspective ok method one is what you see on screen right. So, you have the bounding circle enclosing bounding box including a enclosing a circle you have 8 points on the circle ok. So, corresponding to this bounding box you draw this box in perspective you locate these points corresponding to these great points here the first method you draw these two diagonals and then locate these red dots on those diagonals. Now, corresponding diagonals in perspective would be these two diagonals joining the opposite vertices now you would know that this would be in true length you take this distance from here to here you take this distance from here to here measure these two distances here project array starting from here up to the vanishing point and then another one starting from here up to the vanishing point. These two rays are going to be intersecting the two diagonals at four red points that is method number one method number two is what Ayush proposed instead of doing that instead of having these join these four points you will be making a diamond and then you would have this square this square this square this square you will have a diamond here here and here and then what would you do Ayush you would have a midpoint you would have a midpoint of this. So, you would have a midpoint of this midpoint of this midpoint of this midpoint of this you will get these four points and then draw another circle yeah so the second method of course is probably not exact, but it is relatively faster than this method this method happens to be a little more precise yeah ok. So, straighten your back if you have seat belts buckle yourself and get ready to sleep a new topic space geometry one request that I would want to make is I will try to explain this as best as I can, but just in case if I am not clear enough to raise your hand and if I cannot see your hand do get up if I still cannot see you use the chair stand up on the chair raise both hands and say sir stop I did not understand and I will be happy to explain that to you. So, I want your attention I want your eyes and I want your ears over here and I want you to prompt me if you are with me every time things will be a little confusing alright I will start with the orthographic projections this is the x z plane this is the x y plane almost all discussion will be with the third angle projection scheme in mind although it is not important. So, which view is this which view would this give you front view which view would this give you top view wonderful the front view the top view the front view will be on a vertical plane. So, I would call the bottom plane as the vertical plane the top view will be on the horizontal plane and I will call that as the horizontal plane. So, there will be a switch in notations in terminology that you need to understand. So, this is the front view or what plane top view and what plane good imagine that I have a line whose projection in the top view will be is a horizontal line. And the same line in three dimensions whose projection in the front view or vertical plane is an inclined line a little bit about coordinate geometry very simple let us say this is x 1 z 1 let us say this is x 2 z 2 the coordinates of this vertex x 1 z 1 coordinates of this vertex x 2 z 2 in the horizontal plane let us say the coordinates are x 1 y 1 and x 2 y 2 what would be the length rather the true length of this line whose projections are given there this delta x the whole square plus delta y the whole square plus delta z the whole square square of that. Now, since the projection of this line in the top view is horizontal y 1 is equal to y 2 and if I set this to 0 my true length will be delta x the whole square plus delta y the whole square right sorry delta z the whole square delta z the whole square with me alright. Now, do you see the projection of a line where you can find this length do you see the projection of the line where you can find this length on screen yes which one is it the front view this one here x 1 minus x to the whole square plus z 1 minus z to the whole square this is in true length alright. So, if this is not clear I have this pointer here it is oriented in some way in space what you see is the front view of this what you see is the front view of this in the top view I would see a horizontal projection of this line this is what these two projections are with me good. So, this is just for your understanding in your laboratory exercises you will not be needing coordinate geometry you will essentially be working with geometry. So, this is in true length and I label this length as t l true length another scenario x z plane the front plane or the vertical plane and x y plane the top view or horizontal plane this time in the horizontal plane I have a line or the projection of a line inclined differently it is not horizontal and in the front view that projection is horizontal coordinates x 1 z 1 x 2 z 2 x 1 y 1 and x 2 y 2 the x s remain the same the z's and the y's they get changed. Now, what do you expect the true length of line to be in this case in the frontal view on the vertical plane z 1 is z 2 look at that equation of yours from coordinate geometry z 1 minus z 2 the whole square is 0. So, your true length is delta x the whole square plus delta y the whole square which actually you can get from here. So, this is t l now these two examples should tell you something if I have the projection of a line which is horizontal either in the front plane in the vertical plane front view or it is horizontal in the horizontal plane or the top view I would be getting the true length of that line in the other view do I make sense do I make sense good. Now, imagine this line space and this line is oriented in such a way that you get the projection which is not horizontal in the front plane in the front view vertical plane and it is not horizontal also in the top view the horizontal plane what you do in that case how do you find the true length how do you find the true length you can use the experience from the two previous examples. You can make more of the projections horizontal if you do that the other projection will give you the true length of that line. So, what we have is the method of rotation for that let us say you rotate the projection in the horizontal plane of top view by some angle that rotation is about what axis the z axis this axis over here. So, when you are rotating that what is going to happen to the line what is going to happen to that black line it gets rotated and you are going to make this projection horizontal. So, that you can get the true length of a line over here right now what happens after that what happens to the corresponding projection of the line here any idea let me get back let us say you are rotating the line about this point you are rotating the line about this point about the z axis. So, this point remains fixed let us go up what happens to the x comma y comma z coordinates of this point with the x coordinate change with the y coordinate change with the z coordinate change the z coordinate will not change come back to this view the z coordinate of this point could not change. So, you can expect the z coordinate to be lying somewhere on the horizontal green line true true and you already have the position of this point in the top view use this projection get this point. So, if you have rotated the line. So, that this projection happens to be horizontal this would give you what the length of the red line will be the same the length. So, if you are rotating this line the length does not change. So, this green line here would represent what the true length of this line in three dimensions do you agree do you agree who does not yeah. So, remember that these two black lines these are projections number one. So, if you are rotating the line about the z axis its projection will also get rotated what is your question yeah in the top view it is not getting rotated about the y axis perhaps let me let me come back to the other example let me come to the other example that would help you understand this better this is ok right this is ok the same example. Now, let us say you have this line now you see the projection of this line in the front view now I am going to rotate this projection in such a way that it becomes horizontal I am going to rotate this in such a way that this becomes horizontal this rotation is about what axis the y axis now of course, this length would not change this length what you see would not change right what would happen to the y coordinate here no change I am rotating the line about this point. So, this point remains fixed in space the y the y coordinate of this line does not change I already have this point located over here I take its projection this word give me the true length of a line. Now, what do I essentially mean by true length let us say I have this line and what you see is this line in this plane yeah now if I rotate this line. So, that this line becomes parallel to the plane of sight it is only then that you are going to be getting the true length of this line not otherwise right once again if I make this line parallel to the plane to the vertical plane it is only then you are going to be getting the true length of that line yeah what is that I cannot hear you yeah we are using both planes for a view in that example the line becomes parallel to the horizontal plane that is the reason why you get the true length of that line the horizontal plane in the previous example the line becomes parallel to the front plane that is the reason why I get the line in true length in the front plane you know you guys are tired I guess from the galaxy thing. So, I would suggest that you take out your notebooks and pens and start scribbling because this is not very difficult. So, with that you will also follow what I say a lot better and then I will keep you active and then I will keep me active ok. So, is this method of rotation clear to you all is this clear should I go back and go through these two examples again yes sir no sir everybody with me I will go back alright. So, have your sketch pads on in front of you have your pencils or pens in front of you draw the projections of a three dimensional line in front and top view like what you see done done ok. Now, the trick is very simple you want to see one of the projections either in the front view or the top view to be horizontal if you see that the other projection will ensure that your three dimensional line is parallel to the plane and therefore, you will be getting the true length of the line fine alright in this example rotate the top projection. So, that it becomes horizontal when you see that you are rotating this line about the z axis. So, the z coordinate of this point does not change you are rotating the entire line about this point. So, this point remains fixed in space if the z axis here does not change sorry the z coordinate does not change then this point has to lie on the horizontal if it lies on the horizontal you already have this point located in the top drop a projection you get the new point. And therefore, you get your line parallel to this vertical plane and therefore, in true length fine alright. Let us do the same thing, but reverse the same example now what I would want to do is I want to see this projection to be horizontal. So, I would rotate this projection about this point and about the y axis the length of this projection does not change. Now, if I am rotating this line about the y axis the y coordinate of this point does not change. So, this point has to lie on this horizontal fine and then I already have this point located over here I take the projection up I locate that point this line now becomes parallel to the horizontal plane straight forward this is going to be very boring or could be interesting depending on how you take it. So, given these two projections of a line oriented arbitrarily in three dimensional space let me try to locate the actual line in space. So, I have this box stay with me stay with this is my what was this front view or vertical plane that is my top view or the horizontal plane. Let me get this blue line let me work with the horizontal plane first and mark it over here little blue line mark it over there this length is the same as this length. So, this blue line over there represents this guy with me with me alright let us do the same exercise with the vertical plane the green line mark it there second green line mark it there and this green line is actually this guy here you got a three dimensional box you got two projections is it possible for you to look at the actual line possibly like this like this do you all agree that this is the actual line alright. So, let us say my method of rotation is boring it is not good enough how do I get the true length of this line how do I get the true length of this line the only way I can get the true length of this line is to make sure that I am visualizing that line from a plane which is parallel to the line not otherwise the plane has to be parallel to this line fine. So, I have to figure out that plane first of course that black line will be in true length fine. Now, imagine that I have a plane that contains the blue line on top the blue projection imagine that I have this plane here that contains that blue projection now my question is very simple would the actual three dimensional line which is the black line within that box would that line be parallel to this plane or no rather let me rephrase would that black line actually lie on that plane do you all agree. So, if I view that line from that from that plane dotted plane will I be getting the true length of the line yeah good you all agree it does not really matter where I locate that plane space. So, long as that plane is parallel to itself. So, instead of using that dotted plane perhaps I can shift that plane forward or backward. So, long as I ensure that the plane is parallel fine all right now what I do is I drop verticals from the corresponding projections on to that plane. So, these red verticals from this projection which is over here on to the plane and I can shift that black line to actually lie on this plane right fine all right I want you guys to be attentive here I want you guys to follow this carefully this is a hinge line that separates what from what the horizontal plane from the vertical plane good what I would want to do is once I have located this plane in some way and this plane is inclined to both the vertical plane and the horizontal plane. What I would want to do is I would imagine that there is a plane here and there is a hinge line between say the horizontal plane and this plane. And just as what I do in case of orthographic views I flip this what I call the auxiliary plane up. So, this red plane is the auxiliary plane here. So, this is what I do and if I flip it up and if I have that red line contained on that plane this one I would be able to get the true length all right. Let me go back and explain this again this is for you and good. So, this example here I got the projections of a three dimensional line on the vertical plane on the horizontal plane I try to locate the actual three dimensional line within a three dimensional box. So, what I do is I first draw the projection or I first draw that projection on the horizontal plane I draw the corresponding projection on the vertical plane. And then I extend those projections and I get the actual three dimensional line. And then what my proposal is or what my proposition is that if I draw a plane which passes through that blue line on top I would be able to capture the actual three dimensional line which is the black line within that plane and that black line of course is true line therefore, it would be having the true length. So, I would be able to capture. So, if I let if I draw a plane that passes to this blue projection which is this one here I capture this black line the actual three dimensional object on that plane. And it does not really matter if the plane is there or if it shifted by some amount I can always shift my black line and ensure that it still lies on that plane. And all I need to do is process what I call this new plane as the auxiliary plane in such a way that I can see the true length of the line. So, the first thing I would do is I would identify two hinge lines you already know one of them the purple one is the new one. Now, this is quite tedious this is quite tedious I would want to work with this figure over here and I want your attention. So, this figure I have it on the left for reference follow the color codes the brown hinge line is this one do you agree. Now, notice that this purple hinge line is parallel to the projection of that three dimensional line in the top view notice that do you agree good. Now, of course the projections are going to be perpendicular to the hinge lines wherever they are right. So, I draw projections from this projection of the line perpendicular to this purple hinge line fine all with me so far. Now, look at this line here look at this distance here this distance is of this point from this brown hinge line. So, this distance here what I would do is I would take the same distance measure from this point of the hinge line and get this point look at that small distance I would stay on this projection line and take the same distance from here to here and I will join these guys. And then I will say that this is the true length of my three dimensional line do you agree or you are lost good back again. So, far so good here so far so good here. So, you see I have two hinge lines one is the brown hinge line and the other one is the purple hinge line the brown hinge line is relating the front view and the top view the purple hinge line is relating the horizontal plane here and the red plane the auxiliary plane. So, I draw this purple line here. So, go back to your understanding about orthographic views. So, what you do here. So, you have your front view here your top view here you just split this guy over do not you do not you yeah. Now, imagine that you have your top view here and you have your auxiliary plane at some inclination to this like this yeah we are doing essentially the same thing we are flipping this guy over in relation to the horizontal plane we are flipping this guy over we are doing exactly same thing yeah. Now, look at these green lines and look at these green lines these green lines they measure the distance of this point from this brown hinge line. And they measure the distance of this point from this brown hinge line. So, far so good so far so good. So, these are my corresponding green lines fine fine now go back to your three dimensional box here and try to identify and try to verify if this distance is the same as this distance from the purple hinge line is it is it likewise for this try to verify if this distance is the same as this distance from the purple hinge line it is it is or not which is what I do here. So, what I have done is I have drawn an auxiliary plane and I have related that auxiliary plane to the horizontal plane. So, I have this purple hinge line here and I have taken this distance and this distance and measured this and this to be the same of course, the projections have to be perpendicular to the hinge lines. So, these great projections are perpendicular to the purple hinge line and if I flip that auxiliary plane over that auxiliary plane is going to be containing the two three dimensional line and it is going to be therefore, in true length you agree. So, a little bit of practice and you will be you know I can only imagine I can only imagine how you are trained in close coaching classes. So, this is that version of it the instructor pardon me if I am wrong, but the instructor in one of your coaching classes will say well you know what forget about this it is too tedious you do not want to understand that you do not want to waste time understanding that this is a quicker version you have these two views all right you have this hinge line yes or no good. Step number one is that true no whatever I am just enacting draw a hinge line which is parallel to the projection in the top view step number one. So, you have this hinge line it could be here it could be here does not matter fine step number two make projections perpendicular to this to this hinge line from both these points step number three measure this distance and mark a point on this projection with the same distance as this number four measure this distance mark this point over here on this projection line with the same distance step number five yeah step number five yeah you can go no stay back no stay back man yeah. So, somebody here ask me what is the benefit of drawing that purple line if you can draw the projections directly from that black line and I will tell you what the benefit is I will tell you what the benefit is. So, if you do not draw this line and if you take projections directly from here this is a very simple example, but in more complex examples you are going to be needing this view not once not twice, but for a few times when you need this view you do not want to lose this projection if you lose that you are confused two more minutes and I will let you go sleep and get ready for galaxy in the evening. I can follow the same five steps I can follow the same five steps with reference to the vertical plane not from the horizontal plane, but with reference to the vertical plane this is how well I have already done that yeah fine maybe I will yeah it does not really matter where you position this purple line you can position it anywhere you will still be getting the same two length of a line. So, this is one quick example. So, the purple line is closer now to the projection in the horizontal plane the same exercise five steps you can get the two length and if you compare with the line in two length from the previous example this was this. So, in fact these lines are going to be parallel I can do the same thing in reference to the vertical plane of the front view nothing changes step one step one no no yes sir step one what is step one draw a hinge line parallel to the projection in the vertical plane step two draw projection lines stay with me stay with me stay with me step number three measure that distance and mark that distance over here step number four measure this distance and you are going to be measuring these distance distances from the hinge line mark that distance over here you got two points join these two points you get a line which is going to be parallel to the auxiliary plane and therefore, it is going to be two length. So, this is a three dimensional line which is parallel to the auxiliary plane and therefore, in two length.