 Hello everyone, welcome to the lecture series on computer graphics 2D reflection. At the end of this session, the students will be able to define 2D reflection, represent 2D reflection matrix, solve problem based on 2D reflection in computer graphics. In this video lecture, I will be discussing about two-dimensional reflection in computer graphics. We will discuss and understand how to represent a two-dimensional reflection matrix and finally, we will conclude with some practice problems based on two-dimensional reflection. We know that in computer graphics, transformation is a process of modifying and repositioning the existing graphic. When a transformation takes place on a 2D plane, it is called as two-dimensional transformation. Whereas when it takes place in a three-dimension plane, it is called as 3D transformation. Transformation in computer graphics are broadly classified as translation, rotation, scaling, reflection and shear. In this video lecture, we will be focusing on two-dimension reflection. So, what is reflection? That is, what is a two-dimensional reflection? The two-dimensional reflection is a mirror image of the original object. In the reflection process, the size of the object does not change. Reflection is a kind of rotation where the angle of rotation is 180 degree. The reflected object is always formed on the other side of the mirror. The size of the reflected object is same as the size of the original object. Moving ahead, we can represent reflection by using four ways. Reflection along x-axis, reflection along y-axis, reflection perpendicular to xy plane, reflection along with the line. However, in this video lecture, we will be focusing on the first two types of reflection. That is, reflection along x-axis and reflection along y-axis. So, consider a point object O that has to be reflected in a two-dimensional plane. Let the initial coordinates of the object O be x-old and y-old. New coordinates of the reflected object O after reflection are x-new and y-new. So, reflection on x-axis. We now understand this, as seen in the given diagram. Here, you have the original graphic primitive triangle ABC and you see here it being reflected along the x-axis. So, this reflection is achieved by using the following reflection equation. x-new is equal to x-old, whereas y-new is equal to minus y-old. Now, we see that here when the object is being reflected, the x-coordinates remain the same whereas the y-coordinates fall on the negative side of the y-axis. In the matrix form, the above reflection equation is represented as x-new, y-new is equal to 1, 0, 0, minus 1 into x-old, y-old. So, this is the matrix equation for reflection along x-axis. To simplify the problem solving, we represent this reflection matrix using homogeneous coordinates where a matrix is represented by three by three matrix. Reflection along x-axis is represented as given over here, x-new, y-new, 1 is equal to 1, 0, 0, 0, minus 1, 0, 0, 0, 1 into x-old, y-old, 1. So, this is the homogeneous coordinate representation of reflection matrix along x-axis. I hope you have understood the concept of two-dimensional reflection and the reflection that happens along the x-axis. Now, we understand the reflection on y-axis. As seen in the diagram over here, this is the original object and the reflected object is triangle that is ABC reflected you have A dash, B dash and C dash. So, this reflection is achieved by using the following reflection equation for y-axis that is x-new is equal to minus x-old. So, you can see over here as the reflection is along y-axis, the y-coordinates remain the same whereas the x-one turn on to the negative side. y-old is the same and as the y-new. So, in matrix form the above reflection equation is represented as x-new, y-new is equal to minus 1, 0, 0, 1 into x-old, y-old. So, this reflection matrix is a 2 by 2 matrix along the y-axis. In the similar way as we have seen the homogeneous representation for y-axis that is reflection along x-axis the reflection along y-axis is also represented using a 3 by 3 matrix as given here x-new, y-new, 1 minus 1, 0, 0, 0, 0, 0, 0, 1 into x-old, y-old, 1. So, this represents the reflection matrix along y-axis using the homogeneous co-ordinates representation. I hope you have understood the two-dimensional reflection along x-axis and y-axis and its representation using a 2 by 2 matrix equation and a 3 by 3 homogeneous co-ordinate representation. Now, we move ahead with the practice problems based on 2D reflection. So, given a triangle with coordinate points A, 3, 4, B, 6, 4 and C, 5, 6, we apply the reflection on the x-axis and obtain new coordinates of the object. So, here we are given a triangle ABC and you are expected to apply the reflection along the x-axis. So, by applying the equation to the old triangle coordinates ABC, the reflection has to be taken on the x-axis. So, we apply the formula here. Now, we get x-new is equal to for the coordinates A, 3, 4, the x-new turns out to be 3 and y-new is equal to minus old that is minus 4. So, the new coordinates for corner A after reflection are 3, minus 4. Now, for the next point of the triangle that is B, 6, 4, let the new coordinates of corner B after reflection P, x-new, y-new. Applying reflection equation along the x-axis, we have x-new is equal to x-old and that is 6 and y-new is equal to minus y-old that is minus 4. Thus, the new coordinates of corner P after reflection are 6, minus 4. Now, for coordinate C that is 5, 6, the new coordinates of corner C after reflection are x-new, y-new. Applying the reflection equation we have x-new is equal to x-old is equal to 5 and y-new is equal to minus y-old that is minus 6. Thus, the new coordinates of corner C after reflection are 5, minus 6. Now, we get the new coordinates of the triangle after reflection as shown in the diagram. This is the original coordinates that is ABC and this is the mirror that is the reflected image where you have all the y-coordinates on the negative side because this is a mirror image where you have applied reflection along the x-axis. Now, I hope you have got how to apply the equation for obtaining a reflection along a given axis. We have solved the problem now for x-axis. I want you all to pause the video for some time and solve this problem. Here, the problem that is being given is for again a triangle with coordinate points A 3, 4, B 6, 4, C 5, 6 and here you are expected to apply the reflection along y-axis and obtain the new coordinates. The previous example that we solved was for reflection along x-axis. In order to assure that you have understood, I have taken the reflection problem for y-axis. So, pause the video for some time and solve this question. So, on applying the equation for reflection along the y-axis, we see this is the original object that is ABC and when we apply the reflection equations to the points we get A is equal to minus 3, 4, y value remains the same. For B point we get minus 6, 4 and for C we get minus 5, 6. So, here we can observe that when we are obtaining the original object and the reflected object along this y-axis, x-coordinates have become negative. That was what was represented in the equation. So, this is the graphical representation of the same problem, original object and reflected object. So, when you solve a particular problem for reflection, it is very essential that you understand all about which axis is that being reflected and based on that you get the mirror image. Now dear students, I assume that you have understood the concept of two-dimensional reflection, reflection along x-axis and reflection along y-axis and how to solve and represent the given problem. So, these are the references. Thank you for your patient listening. Thank you.