 Hello everybody, I am Dr. Keshav Valase from Mechanical Engineering Department of Valchian Institute of Technology, SolarPool. In this session, we will be discussing about the mathematical treatment that is some examples of 2D geometric transformations which we have discussed in earlier video. At the end of this session, viewers are expected to understand the basic concepts of three fundamental types of geometric transformations that is translation, scaling and rotation as well as mathematical treatments in dealing with these transformations, how the operations are carried out for a given data. Particularly, we are focusing on two-dimensional geometric transformations. Just to have a review of what we have discussed in earlier video, 2D geometric transformations are nothing but the operations which are carried out to modify the existing configuration of the object so as to alter its shape, size, position and orientation. Now, as a matter of example, a point in a Baxial system is taken as 1 by 2 matrix whereas, a line in Baxial system is taken as 2 by 2 matrix. Thus, any object that we take up in this competitive graphics or CAD-GAM for geometric transformations, they are perceived in the form of a matrix and these geometric transformations are carried out on the points which are used to define that object. It is a basic fundamental rule, the conceptual aspect that is carried out in geometric transformations. These are the three transformations we will talk about with mathematical examples with the given data. At this point, I expect the viewers to think of some image, some object like maybe a line or a triangle and for this object, think of how these geometric transformations will have the effect on that object. Let us talk firstly the example for translation as we have already discussed. In this translation as a geometric transformation, we are repositioning the given object. The example given here is to translate a line passing through two points. These two points given are 1, 1 and 2, 4. Line is passing through these two points and we are supposed to translate this line by a distance of two units along x direction and three units along y direction. Now we already discussed this relationship in earlier video for the point. The same we will now apply for a line wherein 2 by 2 matrix we need for the representation of a line. A line passes through two points, hence conveniently a line is represented as a 2 by 2 matrix. So if L is the original line given, L dash is a translated line and T is a translation matrix then we have L dash that is a translated line is equal to L given or original line plus translation matrix. Remember in translation we add the matrices thus with the given data L the line can be represented as this 2 by 2 matrix of two points point 1 will be put up as 1, 1 here in first row and second point will be put up in second row 2, 4 and next to this is a translation matrix T which is represented as 2, 3 and 2, 3. This 2 is the translation distance along x direction and this 3 is the translation distance along y direction. As we are adding the matrices here these translation distances along x and y we are supposed to repeat for both the rows. So this is the way we represent translation matrix for a line which is represented as a 2 by 2 matrix and thus we are supposed to find out what is L dash that is new translated line. Now with the relation we have L dash is equal to L plus T this L is put up as a given line in 2 by 2 matrix form. This T as a translation matrix is put up as 2 by 2 form adding these two matrices we get L dash that is translated line we get these two points in the form of 2 by 2 matrix. If you see this figure here, here this is the original line passing through two points 1, 1 and 2, 4 whereas this line is the translated line which is passing through 3, 4 and 4, 7. So these are the points 3, 4 and 4, 7 we have calculated. So this is the original line this is the translated line. Coming to second type scaling wherein we either enlarge the object or reduce the object. Now here again the same example is given same data is given wherein a line is passing through 2 points 1, 1 and 2, 4 and we are expected to use a scaling factor of 2 along both the axis that is along x and y. Scaling factor along both the axis is given to be same that is 2. Formula here to be used is L dash is equal to L into S wherein L is the original line L dash is the line after scaling and S is the scaling matrix. This L line as usual we have represented as 2 by 2 matrix of 1, 1 and 2, 4 these two points. S the scaling matrix needs to be represented as 2, 0 and 0, 2. In first row these two represent the scaling factor along x direction as we are multiplying it here for the sake of algorithm this coordinate this point in the matrix needs to be 0 whereas here this 2 is the scaling factor along y axis. So with this representation for the given data we have to find out new line after scaling. Taking to the formula again we have L dash is equal to L into S we are multiplying here in translation we add the matrices but in scaling we multiply the matrices. So this L dash is equal to this L is the given line in the form of 2 by 2 matrix into S, S is the scaling factor T x and T y or these are scaling factors along x and y directions. After multiplication we get these coordinates 2 to 4 it coming back to this representation here. This is the original line given and this one is the scaled line after scaling the scaling factor is given 2 so you can compare here against this line this line is almost doubled. So this is point 2, 2 as we have got it here 2, 2 and this point is 4, 8 which is here 4 and 8. The point to be noted here is after scaling this original line to this position this point is shifted to this point this point does not remain here if you need this line to be doubled with this point fixed at this location you are required to add up one more geometric transformation of translating this line from new line from this point back to this point that we are not talking here that is called as concatenation which we are not talking here. Coming to third one rotation here a given object we rotate about origin through an angle of theta wherein here plus theta as we have discussed earlier rotation is along clockwise direction counter clockwise direction whereas minus theta it is clockwise same example is given for the point 1, 1, 2, 4 the line passing through 2 points 1, 1, 2, 4 but the angle given is plus 30 theta is plus 30 so plus 30 means we have to rotate through counter clockwise direction L dash again as usual line after rotation L is the original line R is the rotation matrix. So this is given line and R the rotation matrix is in terms of cos theta sin theta given here as discussed in earlier video with the formula we know this L dash we get with the multiplication of L into R. So this is L this is rotation matrix R and the product of these 2 comes up as it is given here. So this point 0.366 and 1.366 this point comes up here after rotation and second point comes up at this position this is the original line this is the rotated line. These are the 2 books we have referred Grover and Ibrahim Zaid. Thank you.