 I am Dr. Kesho Valase from Walchand Institute of Technology, Swalapur. In this session, we will be discussing about two-dimensional homogeneous coordinate transformations. Talking about the learning outcomes, at the end of this session, viewers are expected to understand explanation on different types of two-dimensional geometric transformations more particularly about translation scaling and rotation and that to in homogeneous coordinate systems. Secondly, viewers are expected to solve the problems on same again two-dimensional geometric transformations with homogeneous coordinates. Let us revise few basic things on 2D geometric transformations which are covered in earlier videos. Geometric transformations are nothing but modifying the current position of the given object with respect to its shape, size, position or orientation. And how do we represent points and lines? You see basically in geometric transformations, we need graphical features to be represented in matrix form. I repeat again, graphics features we represent in matrix form. Thus, if we talk of a point, a point we perceive as 1 by 2 matrix. In geometric transformations, talking about again a line, line is perceived as 2 by 2 matrix. A line passes through two points, so x 1, y 1 and x 2, y 2. The combination of these two points in matrix form is what we perceive a line to be in geometric transformations. This particular fundamental aspect gives us the liberty to handle the data very comfortably using geometric transformations. One more important conceptual thing here is geometric transformations are fundamentally carried out on points and these points are those points which represent the object. So, in geometric transformations what we are doing is we are transforming the points and at those points again new points, new positions of the points to say it specifically we again redraw the object using its algorithm. Again this is a revision of three basic types of geometric transformations, very briefly if we talk translation, we have a P dash and new point will be equal to its original point plus procession matrix. Here in the point to be noted is we are adding two matrices in translation operation. In scaling we multiply to given point position P its scaling matrix and we get the new scaled position of the point, point to note here is we are multiplying it here. In scaling we multiply, thirdly rotation we rotate the object about origin by an angle theta. So here P original point is multiplied with rotation matrix here again we are multiplying to get the new point its coordinates. So to compile in a translation we are adding the matrix, in scaling and rotation we are multiplying the matrices, concatenation is by and large the need we have in geometric transformations wherein with any given single transformation many times we may not get the desired orientation of the object. Hence we need multiple transformations to reorient the given graphics object. So carrying out such multiple transformations at a time is normally referred as a concatenation it is nothing but more than one transformation to be carried out so as to have the orientation of the object the way designer wants. At this point I would expect the viewers to please recollect the basics of two dimensional geometric transformations and even concatenation that is the requirement of more than one transformation and try to relate these concepts to any application area any object you can think of. Now to summarize towards developing an algorithm translation matrix we are adding rotational scaling we are multiplying with this if we try to have some common way of representing these transformations as an algorithm form we may use this P dash is equal to m 1 into P plus m 2 wherein this P dash is the transformed state of a matrix this P is the original matrix and m 1 is the matrix for rotation and scaling which is used for multiplication and m 2 is the matrix used for translation which is added. So this kind of an algorithm we can use to achieve these three types of transformations eventually we are thinking but these types of operations are not efficient and they are little tedious taking more time in execution and we need many times these types of operations to be carried out wherein the processing time probably would be little more hence to have efficient and effective execution we need to have some modifications. Now these modifications are provided by way of what is called as the concept of homogeneous coordinate transformations. Now here this particular homogeneous coordinate transformation if we take in case of two dimensional things we augment that 2 by 2 matrix 2 3 by 3 for any kind of translation rotation and scaling kinds of geometric transformations required. We will see how one by one firstly let us talk of translation. In translation now with the new homogeneous coordinate transformation type of algorithm if we are using as I said for two dimensional things we augment a matrix by adding one more row or column as per the requirements. So here P dash is represented as x dash y dash and 1. So this 1 is an augmentation to suit to the algorithm of homogeneous coordinate transformations which is then given as equal to P P into this T this P is x y and 1 again. Remember we are talking on two dimensional aspects two dimensional objects two dimensional entities graphics features here we are adding third parameter here and this is the translation matrix which by virtue of this homogeneous coordinate transformation comes up in case of 2 D translation I repeat in case of 2 D translation this matrix happens to be of 3 by 3 which is not so in earlier case without these kind of homogeneous coordinate systems. So here T x and T y they represent the translational distances along two coordinates systems that is x and y. Secondly talking about scaling P dash new coordinates will be equal to P into SV multiply again here and in this case the scaling matrix S would be given by S x and S y put diagonally this way and 1 as a new augmented part of this matrix scaling matrix. So this together will give us in homogeneous coordinate systems the scaling operation where in S x and S y represent the scaling factors along x and y axis we are talking 2 D but here due to augmentation we have third parameter added in the matrix. Third about rotation again same P dash is equal to P into R and here P dash is as usual indicated here P as routinely put up here and R the rotation matrix would come up something like this wherein again third row and third column is a part of augmentation towards new algorithm thus multiplication of this would give us rotation of a point for new point P dash. These are the 2 common references of the books one is CADCAM by MP Grover and second is CADCAM Mastering by Ibrahim Zaid. Thank you.