 I am Dr. Keshav Valase from Valchian Institute of Technology, Solapur. In today's session, we will be talking about 3D geometric transformations. In earlier video sessions, we have discussed about 2D and in this session we will be talking about 3D geometric transformations. Regarding learning outcomes, at the end of this session, the viewers are expected to understand about explaining different types of 3D geometric transformations. More particularly about translation, scaling and rotation. This is we are talking in 3 dimensional aspects. At the same time, viewers are expected to solve the problems in 3D geometric transformations. Let us revise few things as basics from 2D geometric transformations which are covered in earlier videos. Geometric transformations are basically nothing but the operations which are used to modify the given objects shape, size, its position and orientation with respect to its current position or current configuration. As we have discussed earlier, a point normally we represent in back cell system with x y coordinates which we perceive as 1 by 2 matrix and a line we perceive as a 2 by 2 matrix, a line passing through 2 coordinates x 1, y 1 and x 2, y 2 we perceive it as a 2 by 2 matrix. So, this is the basic fundamental aspect of how we represent points and lines and similarly other features in geometric transformations. Coming to the basics of conceptual aspect on transformations, these geometric transformations are basically carried out on the points which represent the given object. So, while we are transforming any object, conceptually we are transforming points which represent the object. What I mean is if you are talking about a line, then in transforming a line we are basically transforming these 2 points which represent the line and then that line as a graphics feature is redrawn at a new position of the points with its basic algorithm of representation of line. Thus again to repeat here, geometric transformations are basically carried out on the points which are used to represent the objects and again at that new position objects are redrawn. In this session, we will be elaborating on 3D translation, 3D scaling and 3D rotation. These are 3 commonly used geometric transformations, there are many more transformations required but these 3 are primarily and very widely used. So, we will be talking, we will be focusing on these 3 types of transformations in 3D. At this point I expect the viewers to think about any objects in 3 dimensions and imagine the changes due to transformations. You can imagine any object in 3D and how these transformations would affect that particular given object, just think for a while. Coming to first transformation of 3D translation, here as indicated here in the figure P is initial position, original position of the given point with the x, y, z coordinates in three dimensional representation. P dash is the new translated position of the point, we are talking first on the point here and this T as a matrix would give you the translational distances. What do we do in this is in 3D translation, we are moving an object say here a point from one position to another position where we are not altering the dimensions of the object. We know the relationship P dash is the new translated position of a point, the point translated that is equal to P the coordinates of the matrix representing the original point the position of original point and T is the translation matrix. Now here P would be x, y, z original dimensions, T the translation matrix is a T x, T y, T z where in these three T x, T y and T z they represent the distances of translation along x, y and z axis. Thus with this basic relationship for translation, we have this relation where in this is P dash which is x dash, y dash and z dash representing this particular point is equal to x, y, z the original coordinates of the point 3D we are talking so x, y and z three points three coordinates of position this point P in its original status and we are adding to that these three parameters which are increments the distances of translation along x, y and z axis. So these are in all in translation in 3D case we have three points coordinates for translated new position as well as old position and the distances coming to 3D scaling here either we reduce the object or enlarge the object in scaling and the scaling factor along all the axis x, y, z it may or may not be same. So S x, S y and S z if these three we take as the scaling factors along x, y and z axis respectively then we use these three scaling factors to reduce or increase or reduce or enlarge the given object as it is given in this figure. Mathematically in matrix form we can put the relation as given here P dash is equal to P into S in case of scaling we multiply two matrices. So this P the given point its matrix we multiply with this scaling matrix. Scaling matrix happens to be with S x, S y and S z as the scaling factors put up along the diagonal of 3 by 3 matrix rest all values is 0. We are talking on 3D scaling. So multiplication of this matrix with the point original point will give us new position of the point after scaling. Coming to rotation as indicated in the figure the given object can be rotated through the certain angle say theta about origin. Necessarily the rotation of the points representing the object happens about the origin plus theta if the angle is given then the rotation is counter clockwise. If that angle of rotation is minus theta the rotation happens in clockwise direction. Coming to the mathematical representation as we know again in P dash the new coordinate is equal to P old coordinate into R R is a rotational matrix. Now here this R z R y and R x these three are the matrices to be used for a rotation about z y and x axis respectively. So here we see this particular matrix if we multiply here we get the rotation about z axis. If we use this particular matrix R y it gives a rotation about y axis and if you are using this third matrix R x that gives us the rotation about x axis. Thus whichever the rotations we need we can have the combination of those one, two or three requirements of these and ultimately get the rotation in three dimensions for the given object. Now these are the references wherein this first three are from some sites wherein we have used the schematic representations and this fourth and fifth they represent two books of Gadkamp by MP Grover and Ibrahim Zaid. Thank you.