 Hello everyone, welcome to the lecture series on 2D transformation that is 2D scaling. At the end of this session, the students will be able to define 2D scaling, represent 2D scaling matrix, solve problems based on 2D scaling in computer graphic. In the previous video lectures, we have discussed about 2D translation, two-dimensional rotation. However, in this video, I will be focusing on two-dimensional scaling in computer graphic, how to represent a scaling matrix using the scaling factor. Also, we will be seeing the representation of scaling matrix using homogeneous coordinate. And finally, we will conclude the video by solving some practice problem based on 2D scaling. In computer graphic, transformation is a process of modifying and repositioning the existing graphic. When this transformation takes place on a two-dimensional plane, it is called as 2D transformation, whereas when it takes place in three-dimension plane, it is called as 3D transformation. We see that transformation in computer graphics are broadly classified as translation, rotation, scaling, reflection and shear. In this video, we focus on scaling. Scaling is a process of modifying or altering the size of the object. I say once again, in scaling, there is modifying or alteration of the original size of the object, means what scaling may be used to increase or reduce the size of the object. Scaling subjects the coordinate points of the original object to a change. As you can see in the diagram, this is a square which is applied the scaling transformation on applying the scaling transformation. You can see there is a change in the dimension, that is change in the positive aspect, that is increase in the size of the original object. So this is called as scaling, that is there is a modification. However, we must note here when there is a modification, the square remains a square. It doesn't change to a circle. So these are called as rigid body transformation. On applying alteration or modification, the object preserves its original identity. Scaling factor determines whether the object size is to be increased or reduced. What do I mean by this? By this, if the scaling factor is greater than 1, then the object size is increased. If at all, if the scaling factor is less than 1, then the object size is reduced. So we say that the scaling factor determines whether the object size is to be increased or reduced. Now consider a point, object O, that has to be scaled in a two-dimensional plane. Let the initial coordinates of the object O be x-old and y-old. The scaling factors for x-axis is equal to sx and the scaling factor for y-axis is equal to sy. Thus, the new coordinates of the object O after scaling is equal to x-new and y-new. So as we have seen here, whenever we want to scale a particular object that is a graphic primitive, may be a circle, a square, a polygon, what is important, the scaling factor in x and y direction and what is the scaling size, okay? Now we see that for homogeneous coordinates, we have the representation matrix for scaling as a 3x3 matrix given as x-new, y-new, 1 is equal to sx00, 0sy0, 001 into x-old, y-old, 1. This is the way we represent the scaling for a given object. We simplify it using homogeneous coordinates so that it becomes convenient for us to operate or calculate a given object or a polygon. Now as you have seen the concept of scaling, it actually alters the original object, the dimensions in x and y direction. We have seen how to represent a scaling matrix using homogeneous coordinates. We now solve a practice problem based on 2D scaling. The problem is given a square with object coordinates point a, b, c, d that is a03, b33, c30, b00, apply scaling parameter 2 towards x-axis and 3 towards y-axis, obtain the new coordinates of the object. So what do we understand here that given an object that is abcd, okay, it is a square or a polygon you can say because it is not having uniform coordinates, here you have to apply the scaling parameter 2 in the x-axis and 3 using y-axis. So we solve the problem now. So the given older corner coordinates of the square are a03, d33, c30, d00, the scaling factor along x-axis is 2 and the scaling factor along y-axis is 3. Now using the stepwise approach for every coordinate we apply the formula. Let the new coordinates for corner a after scaling means what for each coordinates we are going to apply the scaling formula and then we are going to get the scaled answer for the square abcd that is the coordinates of the corner a after scaling will be x-nu and y-nu. So after applying the scaling equations we have x-nu is equal to x-old into sx that is 0 into 2 whereas that is gives you 0 whereas y-nu is equal to y-old into sx is equal to 3 into 3 that is 9 thus the new coordinates of corner a after scaling turn out to be 09. Now for the next coordinate b that is 3, 3. So the new coordinates for corner b after scaling are x-nu, y-nu. After applying the scaling equations we have x-nu is equal to x-old into sx that is 3 into 2 because 2 is the scaling factor in x direction we obtain 6 whereas y-nu is equal to y-old into sy that is 3 into 3 is equal to 9 thus the new coordinates of corner b after scaling is equal to 6, 9. Now for coordinate c we get 3, 0 so the new coordinates for corner c after scaling are x-nu, y-nu after applying scaling equations we have x-nu is equal to x-old into sx is equal to 3 into 2 that is 6, y-nu is equal to y-old into sy is equal to 0 into 3 is equal to 0 thus the new coordinates of corner c after scaling is equal to 6, 0. Now we go for the last coordinate that is 0, 0 so after applying the scaling equations we have x-nu is equal to s-old into xx that is 0 into 2, y-nu is equal to y-old into sy that is 0 into 3 that is 0, 0 thus the new coordinates of the corner d after scaling are 0, 0. Now we see that the new coordinates of the square after scaling a, z, r, a, 0, 9, b, 6, 9, c is equal to 0, 6, 0 and d is equal to 0. You can see in the given diagram over here this was the original square after scaling using 2 along the x-axis and 3 along the y-axis we have obtained this scaled square a, b, c, d with this new values. I hope by now you have understood the concept of 2D scaling how to represent a 2D scaling matrix using homogeneous coordinates and how to solve a problem. So I now request you all to pause the video for some time and answer this problem. A square object with coordinates point p, 1, 4, q, 4, 4, r, 4, 1 and t, 1, 1 is given. Apply the scaling factor 3 on x-axis and 4 on y-axis. Find out the new coordinates for the square. Once again pause the video, reflect on the formula and the concept and solve the given question. Thus the new coordinates on solving for the square after scaling are you can see over here p is equal to 3, 16, q is equal to 12, 16, r is equal to 12, 4, s is equal to 3, 4. So this was the diagram before scaling and this is what we have obtained p dash, q dash, r dash, s dash after scaling in the respective directions using scaling factor 3 in x direction and 4 using y direction that is y-axis. So these are the references. Thank you for your patient listening.