 So, let us continue with the isometric views. So, last time we did some math. So, what we had was a unit cube and let me take a block in my hand what we had was a unit cube. So, this was the x axis of this, this was the y axis pointing towards me z axis and what we did was we rotated this unit cube about the z axis by an angle alpha and then we rotated this cube about this horizontal axis by an angle beta. So, when we did transformations we rotate the object of the unit cube about the z axis then about the x axis and then we perform the projection on the vertical plane that is the screen plane. When we did the transformations we pre-multiplied the rotation matrix about the x axis with r z and then we pre-multiplied the projection matrix and we got the new coordinates or the projected coordinates of the points p q and r. So, this point p q which is pointing inside the screen and r over here we computed the lengths o p o q and o r we equated them o p equals o q o equals o r we did some algebra and we eventually got our solution beta to be equal to plus minus 35.26 degrees and alpha to be equal to plus minus 45 degrees. So, if we plug in let us say the positive values alpha equals 45 and beta equals 35 degrees 35.26 degrees over here in our transformation matrix we get g equals 1 over root 2 0 1 over root 6 minus 1 over root 2 0 1 over root 6 and 0 root 2 over root 3 and last time we figured that the foreshortening the unit lengths o p o q pointing inside and o r they all got foreshortened by a factor root 2 over root 3. So, this is the x coordinate of point p on the projected x z plane your screen plane the z coordinate likewise the x coordinate of point q the z coordinate x coordinate of point r and z coordinate. So, we figured we figured that o p o p would lie along the x axis which is at 30 degrees to the horizontal likewise o q which is pointing inside would lie along o y which is at 30 degrees from the horizontal again on the other side and o r which is vertical will remain vertical here. Let us say this is the unit cube 45 degrees 35.26 degrees alright how will the bottom plane of this cube show up how will the bottom plane of this cube show up like. So, how would this plane show up in the figure like. So, how would this plane show up parallel to this plane now once you figure one once you identify the o x o y and o z axis of the object it is fairly straight forward for you to identify a point if I give you the coordinates. For example, if I give you the coordinates of any point as 30 40 and 50 30 for the x coordinate 40 for the y coordinate and 50 for the z coordinate how will you locate that point on the plane. What you would do is you would go 30 from the origin parallel to the x axis right let us say you have gone here and then you would go 40 parallel to the y axis over here may be right you are still on the x y plane and then to plot 50 you will stand here and go vertically upward alright that is how you would locate your x y z coordinates as 30 40 and 50 respectively with me with me not yet good yeah. So, how we call says why go 30 along this 40 along this and 50 along this why are we not using this foreshortening factor and he is absolutely right and at this point I would like to differentiate between something called the isometric drawing and isometric view in an isometric drawing you would be using that foreshortening factor in an isometric view it is ok for you to use to length the object just gets scaled by root 3 over root 2 that is all ok the other way around alright. So, last time we were doing this example let us redo this again at this time let me say that this is the length direction of my object. So, stay with me here let me say that this is the length direction of my object let me say that this is the height direction of my object and if I go to the top view this would be what the width once I identify my length height and width directions it is much easier for me to draw the isometric drawing or isometric view let us do that what did I say this was length about this about that width good I am drawing x y and z the other way around now this is my x direction which is still at 30 degrees to the horizontal my y direction which is still at 30 degrees to the horizontal and z direction is vertical now I am keeping my length direction along the x axis stay with me here I am keeping my length direction along the x axis that would mean that all features along this direction are to be drawn along this direction in the isometric drawing of you I am keeping my width direction along the y axis all features from here to here along that direction are to be drawn here and z I am keeping for the height. So, if you look at this object you would see that this is a fevicol joint. So, if you look at this object it is composed of two features this one is a block a slanted block and this one is like a tapered rib. So, let me draw the bounding box for the slanted box origin this is the origin what is this length 40 wonderful what is this great what is this wonderful what are the coordinates of this point 2040 30 great not very difficult that is precisely how you need to draw isometric drawing or isometric view the bounding box for these slanted or sliced box ready. Now, let me draw the bounding box for the tapered rib from here to here the length is 60. So, I will go from here to here 60 and then from here to here 15 15 15. So, I will go 60 from this point along this direction along the x direction and cell phones off please and I will go 15 from here till here and then I will draw a line parallel to the y. Now, let us focus on this part here let us focus on this. So, this face of the rib is coinciding with this plane what is this length 10 what is this distance 15 this distance is also 15 right I join these two points likewise I have another square here and I join these four vertices to get a block within which the tapered rib is going to be. Now, notice that I am only using the construction lines I have not yet begun to draw the actual solid right there is a reason for that and I will tell that reason to you in a little while what this edge be visible I do not need to draw this what this edge be visible no I do not need to draw this either how about this right my rib is ready you draw the isometric drawing all the isometric view the way you see it no hidden lines absolutely no hidden lines right how about the sliced block behind the back or at the back of the rib you see this edge did you see this one you do. So, you will see only you would see that edge only partially not completely yeah and since there is a slant I get this since there is a slant which is very similar to this since this plane is the same since this plane is the same this line have the same direction cosines as this line am I done as simple as that no rocket signs what you say about this edge would you actually see this edge no why do you why do you say that discontinuity because the planes are different this plane this plane is not the same as this plane. So, you would see this line so a lot of people came to me last time and asked me well sir why only this orientation why can I not have the object placed differently sure you can. So, instead of showing the object like this you can turn it around about the z axis by 90 degrees and have this rib show along the y instead of along the z perfectly fine or you can turn it around by 180 degrees and show the rib going the other way around along the z along the x. So, there are multiple possibilities in which you can show the isometric drawing of you what of caution no hidden lines are to be shown like we do in case of orthographic views no hidden lines are to be shown. We stay as realistic as possible when we show the isometric drawing of you. So, when a drawing isometric views you need to be very careful you draw a solid line only when you are absolutely sure that you will see that line otherwise you do not otherwise you will end up using your eraser and wasting time and spoiling the sheet isometric views are realistic we show what we see unlike in case of orthographic views where we show even those features that we do not see by means of hidden lines ignore that no that scale is different that scale is. So, maybe what I did was I scaled the 60 to 40 maybe that is what I did. So, at this time ignore that. So, you draw an isometric view with the help of orthographic views right, you do not have to strictly adhere to which is the left view which is the right view which is the front view which is top view those three views those three views are to give you an idea of how the object is going to look like. Once you get the idea once you get the idea you can choose the orientation of the object the way you want to show as many features as possible in solid lines. For example, had I shown this feature tapered rib at the back would it have been a good idea no because then a major part of that rib would have been occult major possibly all I do not know would it have been a good idea for me to rotate this by 180 degrees and maybe not see this slant on the plane no yeah. So, you need to orient the object in such a way that you get maximum you get see maximum features isometric drawing foreshortened isometric view or isometric sketch true values alright something very important. So, remember in the first class or the second class and definitely in the first lab you did a lot of exercises on ellipses we are going to be drawing a lot of elliptical arcs or ellipses in isometric views you see that unit cube on bottom left imagine that you have a circular feature imagine that you have a circular feature here you have rotated this object twice and projected on a vertical plane. Now, let me turn this object around you will have a circular feature here how about that circle look are you sure absolutely positively definitely I thought just in case you were insured I would back myself up with a little bit of math. So, this was the projection matrix alpha equals 45 beta equals 35 minus 35.26 if you have a circle on the x y plane which is the x y plane which is the bottom plane of the cube the coordinates of a point on the circle will be a plus r cosine theta b plus r sin theta and 0 a and b are the centers of the circle the center coordinates circle as the various plug in these values over here pre multiply this thing by this matrix get the new coordinates do some algebra eliminate theta and then compare it with your generic equation for a conic get what a is what h is what b is and the test is the litmus test is for an ellipse is less than interesting right choose any value of a any value of b any value of r h squared minus a b will always be minus 12 in case the circle is on the bottom plane. So, a circle gets converted into an ellipse right same thing the circle is on the y z plane the x direction the y direction the z direction the circle is here ellipse do the same exercise if it is on the y z plane your y coordinate will be non-zero your z coordinate will be non-zero do the same exercise and you will find that your litmus test holds in this case h squared minus a b is equal to minus 3 which is small 0 and the same thing if the circle is on the x z plane same thing the interesting part is irrespective is what b is what r is these values remain constant h squared minus a b it remains constant let us get to a tougher problem shown on the screen is the third angle projection of an object I will spare you the effort I will tell you that the object looks like this 10 seconds to clear your throats now if you realize you can actually break this object into three parts the part on the back which is a resting feature a part on the right. So, this is the first part this is the second part and this part would be the third part you can break this object up into three parts divided in rule that helps you can have a bounding box corresponding to the part which is resting at the back and you can have one for this one and you can have another bounding box for this guy here now for us to understand let me explicitly mention the x y and z axes now let the x direction have all the features along this duration let the height direction have all the features along this duration and let the length drawn along the y axis let us get started let us first draw the bounding box corresponding to this guy here what is this height 7 0 what is this 60 what is this 10 once you have identified the three dimensions of a box or of a block you can draw parallel lines along the x y and z and go ahead and draw the block. So, I am drawing a block that would encompass this feature here what is this length 5 0 right. So, 60 60 minus 10 what is this height 4 0 the same dimension got one face if I got one face all I need to do is draw lines parallel to the y and get this block how about the third one this length is the same as this length now what is what is this length here from the origin what is this length 7 0 I am looking at all this length from here till here 40 and what is the radius of this 25. So, 25 plus 40 is 65. So, this would be at 65 from here maybe is it going to be difficult for you to get this block is it going to be difficult for you to get this block no belly straight forward alright. Now, let us try to get this slant here this slant here and that would happen or that would appear on this face the corresponding slant will appear on a face parallel to this on the other side right. Now, focus your attention on this this line is this line here likewise this line is this line here these vertical lines are this and this and notice that there is a slot here. So, I have taken away a piece of block from this the width of the block is 40 the height of the block is 30 30 and 10 well I said the block well yeah. So, if you want to call it a trapezoid yeah, but it is equivalent to taking a block of 40 by 10 40 by 30 by whatever distances anyhow. So, this is the critical aspect of the drawing identify this length or this height as 30 go from here parallel to this identify this height get this point and get this point likewise go parallel to the y axis identify that vertical line get that face. So, this face would correspond to the face correspond to this face here at the back now start drawing now start drawing I am working with features on the slant surface now I am ready to start drawing solid lines I will see this will I see this will I see this I would see this face I would see this face here on the slant this line this line this line this line I worked out the details on the slant here I will see this vertical line I will see this vertical line this one I will see this face I will see this vertical line I will see this vertical line how about the line here I will see a partial line correspond to that a complete line. So, do appreciate how judiciously I am drawing my solid lines without spoiling my sheet and that can happen only if I am prepared if I am not prepared I will start making mistakes here comes the interesting part this circular arc will be appearing as an elliptical arc on which plane on this plane here on the x y plane or on a plane parallel to the x y plane do you remember the 4 center method you do yes for that I need to draw a rhombus that would bound the circle or bound the ellipse right the length of the rhombus is going to be what the length of each edge of the rhombus is going to be what 50. So, this length is 50 I have one length this is 50 this is 50 this is 50 I have got a rhombus on the top face of the block identify the longest diagonal from this vertex join the midpoint of this edge from this vertex join the midpoint of this edge you have two centers you have the third center here you have the fourth center here with this as center this as radius drawn arc with this as center this as radius draw another arc. So, since this is a circular arc I do not need to draw the entire ellipse I only need to draw half of the ellipse the rest I can join using straight lines with the main part here comes the trick what you say what we are going to be seeing here a parallel ellipse now ok, but you are a little ahead of me now do you think that drawing a rhombus to draw that ellipse would be a good idea no no absolutely. So, what you do and then you are going to be drawing only half the ellipse you have the center for this one which is here and if you have to draw an arc which is parallel to this arc what you need to do you need to go down by what distance by 10 you need to go vertically down by 10 identify the center here the same radius from here to here draw this arc if you start making a rhombus to draw this ellipse you will be wasting time that is the trick that you need to keep in your mind. Likewise from this center go down by 10 go down by 10 identify the center with this as radius draw an arc now this arc is not going to be complete this arc is going to be partial this arc is going to be partial because there is this vertical line here alright and then join this guy the rhombus that we used to make the ellipse what you mean to get the ellipse here this length is 50 this length is 50 this length is 50 this length is 50 this angle is 60 this angle is 60 this angle is 120 degrees this angle is 120 degrees you have all the conditions necessary for you to have an ellipse enclosed within a rhombus what lens you got the block now you have this block here and of course what is your question again if the diameter of a circle is 50 what would be the length of the bounding square of the square that is bounding the circle 50 length and width would be the same same in case of rhombus absolutely same in case of rhombus because what is happening is if you are rotating object about the x about the z and then projection of projecting your square is getting shaped into an into a rhombus so far so good so far so good it has to be tangent to the arc. So, if you draw this arc using soft pencil using construction lines your arc will probably be here it will probably go some somewhere like this ok it goes somewhere like this and the point where the vertical line intersects with the arc that is why you need to stop I would need 10 more minutes be patient so this vertical line is tangent to both these arcs hold on hold on you can do precisely the same thing for this block here what would be the length of the rhombus for this 60. So, this I know is 60 I can I can identify by 60 is from here I can identify by 60 is from here I can draw this line which is parallel to this line I get another rhombus I join or I identify the longest diagonal the same method for center I get for centers here first two centers the first center the first center again I need to draw only half an ellipse with this point as center this has radius I draw this arc with this point as center this is radius I draw another arc and to draw the same thing at the back or on the back face I need to essentially control C control V control C is for copying control V is for pasting ok, but I cannot do that here. So, what I would do is this was my center for this arc I would go 10 parallel to the x identify this as center with the same radius I draw an arc at the back from here I will go 10 again parallel to x here I will draw an incomplete arc and then try to figure out a line which is tangent to both the arcs and then this vertical line is going all the way down see this feature here please repeat are you with me do you think now it is going to be easy for you to draw the ellipse or rather ellipses correspond to this wide this circle all you need to do is locate the bounding rhombus same method for center method get the ellipse am I done there would be one more elliptical arc and that would give the impression of the perception of the depth of the void for that the same trick I go 10 down with the two centers what you mean for the second arc it is a void it is a cylindrical void along the z direction I do not need a line a few last things actually I never needed this this was only for our understanding and appreciation that I had marked the x y z coordinates or the axes explicitly we do not need this we do not need this in your actual drawing or in reality on your sheets this is how the isometric what did I draw drawing sketch of view view isometric sketch or isometric view because I used the true scale I did not use the asymmetric scale that is even harder and that is one of the reasons why we at times somebody asked me right. So, that is one of the reasons why we prefer to use true scales at times thank you.