 Last time I asked you to come up with a single funniest ever statement that would relate four words late, elate, played and slate anybody, anybody sir I am late because no how about Thursday can I get some input from you guys Thursday a single statement a single statement that works as an excuse for you you are late because and you are going to be using all these three words elate, played and slate let us see try to take on my funny bone all right. . So, let us get started bored with orthographic views already are you yes third angle first angle three views FTP front top profile one learn something new yes no may be all right. So, talk about axonometric views and particular we will talk about isometric views you have done this exercise before what did we do we took this solid and we drew the orthographic views in the third angle projection right what if I do not give you this object what if Mr. Ashish can you come up yeah and Vasu is there. So, what if Ashish gives Vasu these three drawings and asks Vasu to interpret these drawings would be difficult or would be easy not that easy. So, would you need this drawing or would you not need this drawing you do not know. So, the question is given the three orthographic views and nothing else is it possible is it feasible for you to interpret the object who would say yes would say no anybody who says yes would be difficult yes or no it would be difficult, but if I show this object to you you will be like fine you have an idea of what an object would look like in 3D and you also have an idea as to what the object would look like when looked at from three different directions three different orthogonal directions. So, would you say that this picture is complete the three orthographic views and a pictorial view or a 3D view would you say that yes or no another example what if I do not give you this even more difficult yeah, but if I give you this then you have a fairly good picture of what an object would look like for example, if I do not give you this things might be a little difficult for you to interpret, but if I give you this you are fine you know what these are orthographic views what is this what is that this is what we are going to be learning today isometric views to draw a single source peak three dimensional picture of the object and I plan to bow you guys today with a lot of math well if not a lot then a little bit. So, let me read what is written there through multi view orthographic projections we observed how the object looked when parallel projections were taken on three mutually perpendicular planes and down below an exonometric projection the object is oriented read as rotated twice a priory and appropriately. So, that all its three dimensional features are clearly visible on the vertical plane of projection an example this is a chalk box what do you see what do you see a rectangle. So, what I am saying back there is if I rotate this object once and rotate this object the second time. So, when I rotate this object once how many planes are visible to you two and then when I rotate this object the second time three yeah. So, you see plane one plane two and plane three and you also see that there is a pothole in this chalk box and a few chalks. So, the entire trick in trying to get the three dimensional picture of this object is to rotate this once rotate this twice. So, that all the three dimensions of this object are visible to you. So, we will try to learn the math behind this let us say we have a unit cube how many edges cube 12 lengths all them one what you see is the vertical plane of this cube and let us call the plane as the x z plane one of the vertices of the cube is at the origin of the axis this the x this the z where is y outside or inside inside right hand rule. Let us practice let us orient this cube in such a way that you are able to see the three planes right now you are seeing just one plane let us first rotate this cube about the z axis one angle alpha when I do that I see two planes right fine I am calling this x projected ok, but it is essentially containing the projected length of this edge and also the projected length of this edge right, but the actual reference frame happens to stay stationary it is fixed and then. So, you would actually see two of these vertices p p and q p at first you are seeing only this vertex right now you are seeing two planes and then by the way let me ask you this let me go back. So, this length is one right when I rotate what happens to the length does it get decreased or increased by what amount decreased by cos alpha decreased to decreases to cos alpha 2 cos 2 cos alpha. Now, we rotate the object about the x axis when we do that rotation angle beta we get to see the third phase of the cube forget I am do not worry about that and then you also see three vertices very distinctly p p q p and r p would you agree with this our three dimensional features of the cube visible to you yes starting with a single plane to two planes to now three planes yeah yes or no what is happening to these three lengths of the unit cube when you perform one single rotation you said the length reduced to cos alpha in this case any idea no idea we need mathematics for this now, but would you say that these lengths are equal to one or less than one or greater than one all three of them all three of them less than one fine yes or no who says no yeah why do you say no z component is not changed is this true length is this true length yes it is totally vertical now if I rotate it towards myself what happens to the length decreases by how much by cos at this angle cos at this angle. So, all length will change and it so happens that all lengths will get reduced now depending on what angle alpha is and what angle beta is we will have to figure out by how much do these lengths actually get reduced for that we need mathematics you would agree with this now all lengths are different for arbitrary angles alpha and beta now it so happens that for a certain set of angles alpha and beta you get these three lengths to be totally equal and when you are looking at that projection it is called the isometric iso means same metric means scale isometric projection when all the three lengths are equal when two of these lengths are equal it is called die metric projection die two metric size and when none of these lengths are equal it is called a tri metric projection you guys are getting bored already I see so many yarns. So, this is something that you would need to keep in mind, but in particular we are going to be looking at isometric projections as a part of this course. So, you have a plane x y you have a particle or a point which is at fixed length with respect to the origin. So, this length is fixed so you can consider this to be an edge and you are going to be rotating this edge about the axis that comes to you by some angle. So, the coordinates of that point are x j and y j the original coordinates the new coordinates are x j star and y j star the initial angle made by this H o p j is alpha o p j is of length l this edge is being rotated about this axis the axis that comes to you by an angle theta right have you come across this before yes or no yes straight forward. So, this projection this horizontal projection is x j the x coordinate of p j this vertical projection is y j the y coordinate this horizontal projection is x j star this is y j star given the angles theta and alpha can you relate these projections you can x j star is l times cosine of theta plus alpha which is if you expand that is l cosine of alpha cosine of theta minus l sin of alpha sin of theta l cosine of alpha is x j l sin of alpha is y j. So, x j star is x j cosine of theta minus y j sin of theta likewise you can find y j star as l times sin of theta plus alpha at the end that is x j sin of theta plus y j cosine of theta and in matrix form you can write x j star y j star equals cosine of theta sin of theta minus sin of theta cosine of theta times x j y j that is something that I think you may have come across before you have. So, far so good. So, this matrix this matrix is called the rotation matrix cosine of theta minus sin of theta sin of theta cosine of theta what happens to the z coordinate of the point if this is the x axis this is the y axis what happens to the z coordinate of the point remains same or different same coming back to the cube one of the vertices is called O let us say this vertex is p with coordinates 1 0 0 q is inside the screen with coordinates 0 1 0. So, y axis is pointing in words and r is 0 0 1 we are going to be applying the result the rotational result on this rotation about the z axis if you look at this block this 2 by 2 block that would rotate your x and y components and your z component remains the same. So, you have 0 0 1 here and 0 0 1 here. So, the old value of x coordinate the old value of y coordinate the old value of z coordinate pre multiplied by this matrix will give you the new value of the x y and z. Everybody with me yes no good what if I want to now find rotation about the x axis what is the entry here 1 cos is it too trivial no anyhow got right. So, remember let us not lose the physics come back a single plane rotation about z by some angle that rotation matrix is given by r of z. And then this is my x axis rotation about the x axis the second matrix give you gives you the rotation. And the third step is whatever you see here you have to project that on the vertical plane what happened then your vertical plane is what x z plane your vertical plane your screen plane is the x z plane right. So, your x coordinate will remain the same your z coordinate will remain the same what happened to a y coordinate 0. So, this is your projection matrix yes or no no step 1 step 2 whatever the x y z coordinates are this is my x z plane this is my x z plane where y coordinate is 0 if I project if I project this guy over here x and z coordinates from these two rotations and the y coordinate will be 0 fine. Now, this is the tricky part when you perform transformations you have to follow a certain order first you are performing rotation about the z second you are performing rotation about the x and third you are projecting when you follow this order when you follow this order rotation about z then about x and then projection the y z plane you will have to pre multiply matrices the first matrix on the right is rotation about z then rotation about x and then projection on the y z plane. And when you multiply all these three matrices you will get the result on the right hand side top row cosine of alpha minus sine of alpha 0 middle row 0 0 0 which explains that your actually you have actually gotten the image of the object on the x z plane. And therefore, all the y coordinates are 0 and the third row sine of alpha sine of beta cosine of alpha sine of beta and cosine of beta little bit of math would be boring what you have to say about this these are the coordinates of what these are the coordinates of p these are the coordinates of q and these guys are the coordinates of r right. If I want or if I ask you what the new length of op is what would you have to say new length of op x squared plus y squared plus z squared cosine alpha squared plus sine alpha squared times sine beta squared all right new length of op q square this up square this up and new length of op r enough of math a little more axonometric projections you have three categories isometric diametric and trimetric what happens in isometric all three lengths are equal what happens in diametric two lengths are equal trimetric all lengths are different. If I want to work with the isometric case I have to equate or have to ensure that op is the same as oq and does the same as or how many equations two three two how many unknowns. So, if I say that op equals oq and or equals op I should be able to solve for alpha and beta how many solutions do you expect would you expect a unique solution would you expect more than one solutions if so how many more than one if you do the algebra which you can in your hostel rooms you will see that beta comes out to be plus minus 35.26 degrees and alpha comes out to be plus minus 45 degrees. So, in fact you have four solutions in fact you have four solutions. So, you have positive rotation and correspondingly you have negative rotation in both directions for alpha and beta enough of math. So, this is the crux, but in T a 101 we do not worry about the intermediate steps we only worry about the final result and this is what the final result is that if you incorporate those angles alpha and beta what it essentially means in terms of drawing is you get to know the orientation of the x axis the y axis and the z axis of the object of the cube the x axis is oriented at 30 degrees from the horizontal the y axis is also oriented at 30 degrees from the horizontal and the z axis remains vertical lunch followed by mathematics is not a good combination. So, in your labs or when drawing isometric views do not worry about the math you can just start from here pick a point on your sheet draw x axis at 30 degrees from the horizontal draw y axis at 30 degrees from the horizontal draw the z axis which remains vertical. Once you have identified the three axis you should be able to draw the three dimensional picture of the object I will see one example louder plus minus plus minus 35 30 in what range there should be four values corresponding to those values corresponding to those values you will probably figure that you will get one of the axis is this no matter what you will get the other axis is this no matter what and your z axis will remain vertical no matter what you can try it out. So, it could be as well 35.26 there would be something else after 6. So, in that sense maybe it is an approximation, but cosine of beta is under root of 2 by 3 that is the precise no no this is exact how did we get that you see this result better let us focus on let us focus on the peak ordinates peak ordinates are given by cosine of alpha 0 cosine of alpha 0 and sin of alpha thank you this guy right let us let us just focus on p. So, if you plug in the value of alpha what you get let us just take the positive values for now what is cosine of alpha 1 by root 2 y is 0 what is sin of alpha what is what is sin of alpha times sin of beta. So, this would be 1 by root 6 now fine fine alright. So, my point p will be here somewhere remember that this is the x z plane the vertical plane is the x z plane my point p will be here somewhere what is the x coordinate of that 1 by what is the z coordinate of that. So, what is the cosine of the angle what is the cosine of the angle which is what 30 degrees that is how you get it and likewise if you consider q you will get 30 degrees on the other side alright. So, the lens get shortened or foreshortened by how you got the what last part of what step 1 rotation about x rotation about z then rotation about x and then projection on the x z plane. So, the screen plane is the x z plane. So, you are actually seeing all three dimensional pictures of an object on a vertical plane which is the x z plane fine fine. Do you agree with the math if you plug in the values of alpha and beta let us take just the positive values you will get the new coordinates of p as 1 over root 2 0 and 1 over root 6 new coordinates of q as minus 1 over root 2 0 and 1 over root 6 new coordinates of r as 0 0 and root 2 over root 3. So, your p will be lying here your q will be lying here right and your r will be lying here, but this is what x coordinate of p z coordinate of p. So, the x coordinate is 1 over root 2 the z coordinate is 1 over root 6 this is 1 over root 2 this is 1 over root 6. So, this over this is what tan of this angle which is what you can you can work out the same for q. So, the x coordinate is minus 1 over root 2 the z coordinate is 1 over root 6 tan of minus 1 over root 3 which is 30 degrees on the other side all right what happens to lens they get shortened they get foreshortened by what factor or to what factor how. So, o r folks can I have all ears here. So, the initial length of o r was 1 the final length of o r is root 2 over root 3. So, o r has shortened by root 2 over root 3 and so has o p and so has o q right. So, the lens in all three dimensions they have shortened by equal amount which is root 2 over root 3. So, this is something for you to understand. So, through this lecture you can understand the math, but the take home message is this part when you start drawing your isometric views pick a point on your sheet draw the x axis at 30 degrees to the horizontal draw the y axis at 30 degrees to the horizontal draw the x axis draw the z axis vertically upwards message 1 message 2 that your true lens they need to be factored they need to be multiplied by root 2 over root 3 to get the isometric lens and we are going to be working on how to draw the isometric scale horizontal you have a line you have another line this line is at 45 degrees this line is at 30 degrees mark the true lens on your 45 degree line true lens on the 45 degree line. If you take the horizontal projection of let us say length l which is the true length on the 45 degree line 45 degree line then the horizontal projection will be l times cosine of 45 what is this length this length over this length is cosine of 30. So, you can find what that length is l cosine 45 over sine of sorry over cosine of 30 which is l times root 2 over 3. So, in effect what is happening is that you are scaling the lengths on the 45 degree line by root 2 over root 3 if you are taking vertical projections of those points on the 45 degree line on this red line mark these points draw verticals on the red line from the green line. So, this is the true scale this is the true scale and this is the isometric scale. So, this is the second take home message from today's lecture. As for TA 101 is concerned you do not need to worry about the math that we had done what you need to worry about is how to draw the x y and z axis on your sheet and how to come up with the isometric scaling. So, in your lab assignments and homework assignments when I say isometric drawing when I say isometric drawing I want you guys to draw with isometric scale with this scale when I say isometric view or isometric sketch I would want you guys to draw isometric view, but with true scale. So, keep this in mind. So, let us draw the isometric view from these three orthographic views given third angle first angle third angle how do you say that third quadrant fine how do you think or what do you think the object is going to look like how do you think the object is going to look like in three dimensions. Let us get back to our exercise first thing get one of the axis the x axis now realize how I am drawing the axis I am drawing the axis opening towards you not opening away from you x axis y axis and the z axis first thing both x and y third degrees from the horizontal z axis vertical. I know the dimensions from the orthographic views I can draw the bounding box I can draw the bounding box the bounding box three dimensional bounding box which plane is this plane x axis z axis this is y axis. So, this plane here is the y z plane how about this plane x z x z or x z. So, the break in drawing isometric views is to identify three different planes distinctly and work on these planes from now on I will not utter a word I will just have you see what is going on.