 I am Professor Keshav Valesi from Valtran Institute of Technology, Solapur. In today's session, we will discuss about 2D homogenous coordinate transformations, particularly some examples we will try to solve. Coming to the learning outcomes, at the end of this session, viewers are expected to solve the problems on geometric transformations, particularly using the concept of 2D homogenous coordinate transformations and again in that regarding commonly used transformations of translations scaling and rotation. Let us revise few basics from earlier video regarding 2D homogenous coordinate transformations. In translation as you all know, we represent the mathematical relationship as a p dash is equal to p into t for point to be transformed, wherein p dash happens to be new translated point, p is original point and t is the translation matrix. This is the relationship for a point wherein x dash y dash indicates coordinates of new transformed coordinate, wherein this one is the augmented part of the algorithm for homogenous coordinate transformations. So this is p dash is equal to this is p into this 3 by 3 matrix represents translation matrix in homogenous coordinate transformations, wherein 3 by 3 matrix is required for actually 2D working, wherein this third row and third column they indicate the augmented part for the requirement towards algorithm of representation in homogenous coordinate system. Coming to scaling, we know the relation p dash and new point is equal to p original point into s, s will be the scaling matrix. So scaling matrix in this case happens to be diagonally s x and s y, wherein s x and s y are scaling factors along s and y and this third row and third column is the augmented part, wherein this is p dash, this is p. In a rotation p dash is equal to p into r, wherein again r the rotation matrix is given in 3 by 3 for 2D transformations in this particular form, wherein there is a normal 2 by 2 part required in rotation, whereas this third row and third column are the augmented part towards homogenous coordinate system. At this point I expect the viewers to imagine some objects and try to apply these concepts of 2D homogenous coordinate transformations. Let us now solve some simple problem. The text of the problem reads, wherein we are given 2 points passing through coordinates 1, 1 and 2, 4 and we are required to translate a line passing through these 2 points, we are required to translate through 2 units along x direction and 3 units along y direction and this is using homogenous coordinate transformation. This is a given problem and we will see how we can go about this in homogenous coordinate transformation system. We know earlier we have discussed further point, now let us talk about the line. We are given a line here, so line is a 2 by 2 matrix as you all know normally we represent. So, L dash will be the translated a new line, L is the original line and T is the translation matrix. So, with this relationship this could be the way otherwise we could have normally represented the matrices for a given line 2 by 2 passing through these 2 points, whereas this is the translation matrix with the 2 and the 3 as the translational distances along x and y as given and we have to find out L dash new line. We will see here how we can do it with homogenous coordinate system. L dash is equal to L into T we know and in homogenous coordinate system we can represent the same equation in this style wherein L dash will be replaced here with x 1, y 1 and 1 as well as x 2, y 2 and 1. So, these x 1, y 1 and x 2, y 2 are actually the 2 by 2 part of 2 points through which the line is passing and this 1 the third column 1, 1 these 2 elements are the part of augmentation required towards homogenous coordinate system representation for a line. So, this is for the new coordinates is equal to L dash, this is L dash is equal to L, L will be represented in this particular form wherein 1, 1, 2, 4 these are the coordinates of original point this is 1, 1 this is 2, 4 this is the original line. So, this original line here is represented as 1, 1, 2, 4 and third column is the augmented part taking values of 1 and 1 in third column. So, this is L into T the translation matrix here will be diagonal elements 1, 1, 1 and here we will have T x is equal to 2, T y is equal to 3. So, this T x is equal to 2 is the translation distance along x axis whereas, T y is equal to 3 is translation distance along y direction. So, this is the actual representation in homogenous coordinate transformation system in that pattern with new algorithm and the result of this is put up here 3, 4, 1, 4, 7, 1 wherein dropping this augmented part third column we get 3, 4 and 4, 7 as the x 1, y 1 is 3, 4 x 2, y 2 is 4, 7. So, it is indicated here new positions of these two points are given here 3, 4 and 4. So, this is the translated line with homogenous coordinate transformation. Coming to scaling here again we have the same problem, a line is given which passes through 2 points 1, 1, 2, 4 and the scaling factors given here are 2 along x as well as along y and we have to do this using homogenous coordinate transformations point to mention here is this scaling factor may be different, but here what we are given is it is a same along both the axis that is 2. So, the relation we have is L dash the new scaled line is equal to L into S as usual L dash is the line after scaling L is the original line and S is the scaling matrix normally otherwise we would have indicated L the line in 2 by 2 style as this 1, 1, 2, 4 2 by 2 matrix and the scaling matrix could have been like this style 2, 0, 0, 2 S x and S y is equal to 2 scaling factors, but here we have to do it with homogenous coordinate system. So, L dash new line is equal to L into S if we put up in homogenous coordinate form we get this as we have discussed earlier this is your L dash this part is L indicating 1, 1 and 2, 4 as 2 given points through which the line is passing and this is augmented part as usual and this is 3 by 3 our scaling matrix where in diagonally this 2 is S x representing scaling factor along x axis this is T y or S y here scaling factor along y axis and this is S y. This is augmented part representing 0, 0 and 1 values in third column and third rows the resultant of this comes out to be 2, 2, 1 and 4, 8, 1 where in 2, 2, 4, 8 this is the part of 2 coordinates that is 2, 2 here and 4, 8 here. So, this is the scaled line with this as the augmentation coming to rotation again we have the same example line passing through 2 points 1, 1 and 2, 4 and angle given is theta is equal to plus 30. We have L dash is equal to L into R the normal notation form we indicate as usual here and this R is equal to cos theta and this value this particular matrix is rotation matrix in case of 2 D transformation otherwise without homogeneous coordinate transformation. But here we are using homogeneous coordinate transformation system with that L dash comes out in this form as usual L again as usual we have put up as in earlier cases and this newly the R rotation matrix in augmented style for homogeneous coordinate transformation comes out with this basic 2 by 2 transformation matrix and augmented with this third one and third column and the result of this for theta is equal to 30 if we put and solve we get x 1, y 1 and x 2, y 2 as these values with this as the augmented part. So, these 2 points are indicated here. So, this is first point x 1, y 1 and this one is x 2, y 2 thus we get the rotated line here. So, these are the transformations using 2 D homogeneous coordinate systems. These are the 2 books as usual we have written rather we have referred Grover and Ibrahim Zaid we have taken all these things from these 2 books. Thank you.