 Hello everyone, myself Professor Prithish Chittay, working as assistant professor in mechanical engineering department, Valchan history of technology, Solapur. In the first session, today we will discuss two dimensional geometrical transformations. What are the learning outcomes? So, the student will be able to explain the different types of geometrical transformations like translation, scaling and the rotation. Also, they can solve the problems or the numericals based on the 2D geometrical transformations. These are the contents. First we will discuss what is meant by 2D geometrical transformations, why it is needed. After that we will see what are the different types of 2D geometrical transformations that is the translation, rotation and scaling. Also, the equation of the translation, rotation and scaling. So, what is meant by geometrical transformation? Geometrical transformation is required. So, whenever we are doing the modeling or suppose maybe the drafting and suppose maybe the one of the part is not observing properly to the user. So, we have to do some process. We have to provide some operation like, we have to move the model, we have to move the geometrical entity or maybe we have to rotate the geometrical entity. For example, maybe the triangle or maybe the rectangle or maybe the circle, any geometrical entity or maybe we have to zoom in or the zoom out so that we can observe the geometrical entity very properly. So, it is defined as the operation that modifies the size, shape, position and the orientation with respect to its current configuration. Now, what is the in general equation for the 2D geometrical transformation? So, P dash is equal to L of P, where P dash is the modified image after the transformation. The transformation may be the translation, scaling and the rotation. And the P is the original image. What are the different types of 2D geometrical transformation? The first one is we will see the translation, rotation and scaling. Now think about this question, relate about types of transformation to change in the behavior of the original image. For example, translation, what is mean by translation in general is to change in the position of the original image, so that there will be the modified image after the moving of the object. So that we can observe the particular image or the geometrical entity very easily. Think regarding this question. Now translation, first we will see the translation, we will go through the figure, here P xy is the original point, P dash x dash y dash is the modified point after the translation. So we will take the example of the triangle, where a b c triangle a b c is our original triangle and after that whenever we are providing the translation transformation, we will get the a dash b dash c dash triangle a dash b dash c dash. So what is mean by translation? Sometimes whenever we are performing the geometrical modeling, we have to move a particular object or maybe the model from one coordinate to the another coordinate so that we can see very easily the particular modified object. There will not be any change in the dimension because we are only moving the or translating the object or the geometrical entity. So we can say that it consists of moving an object from one coordinate to the another coordinate without changing its dimension which is very important without changing its dimension. The in general equation for the translation P dash is equal to P plus t, where t is the translation matrix. We will go ahead, suppose we are having the xy coordinates, so x dash y dash is equal to xy plus tx ty, tx ty is translation matrix. So here P x dash y dash is equal to P xy plus tx ty. So P x dash y dash are the modified coordinates, P xy are the original coordinates and tx ty is our translation matrix. Similarly for the triangle ABC, so a dash x a dash y b dash x b dash y and c dash x c dash y is equal to this will be our modified triangle that is a triangle a dash b dash c dash is equal to a x a y b x b y c x c y. This will be our original triangle that is triangle ABC plus tx ty triangle sorry the matrix tx ty this will be our translation matrix. We will move forward to the rotational transformation, rotational geometrical transformation, we will go through the figure first here P xy is our original coordinate, now we have to rotate this P xy through the angle theta which is having the radius r, so that we can get the new coordinate which is called as P dash x dash y dash. So P is our original coordinate and P dash is our modified coordinate, so why to rotate actually maybe a geometrical entity or maybe the point, so that similarly like the translation so that we can see a geometrical entity very easily. In the sum of the geometrical 3D geometrical model we cannot see a particular surfaces, we have to rotate that particular surface so that we can observe very easily that particular surface. Now it consists of taking an object from one coordinate to the another coordinate at particular angle where theta is the rotational angle, now we will go through the a derivation here we have to find out the rotational matrix here, so x is equal to r cos phi and y is equal to r sin phi, so x dash is equal to r cos theta plus phi, we are knowing that cos theta plus phi is equal to r cos theta cos phi minus r sin theta sin phi, so from the equation 1 x dash is equal to x cos theta minus y sin theta because we are keeping the values of the x and y, x is equal to r cos phi and y is equal to r sin phi, so we can get x dash is equal to x cos theta minus y sin theta, similarly y dash is equal to r into sin theta plus phi, r into sin theta plus phi, so sin theta plus phi r into sin theta cos phi plus r cos theta sin phi keeping the values of x and y here, so y dash is equal to x sin theta plus y cos theta, combining together 2 and 3 we will get the particular matrix, so x dash y dash is equal to x y, so x dash y dash will be our original coordinates sorry the modified coordinates, x y will be our original coordinates and cos theta minus sin theta sin theta cos theta, so cos theta sin theta minus sin theta cos theta that will be our rotational matrix, so cos theta sin theta that is minus sin theta cos theta that will be our rotational matrix scaling, now here we are having S x and S y the scaling coordinates or we can call the scaling distances from the 0 0 point triangle A B C is our original triangle, triangle A dash B dash C dash our modified triangle, now why scaling is required, so scaling is required whenever you want to zoom in or zoom out the particular geometrical model, so whenever you are dealing with the complex problems like the engine or maybe the chassis, we have to zoom in or zoom out so that we can go into the particular surface or maybe the solid edge or maybe a node very easily, we have to zoom in and zoom out, there are the different modeling softwares like Katia or maybe the solid version with the help of which we can zoom in and zoom out, but manually if you want to take the scaling in the drawing sheet we can zoom in and zoom out, but with the help of Katia software which is very easily we can done, so scaling is a type of transformation to zoom in and zoom out out of the object or the drawing, so original coordinates are multiplied uniformly by the scaling factor S x and S y in the direction x and y respectively, we will see the equation, so x dash is equal to x into S x and y dash is equal to y into S y very simple equation, so A dash x A dash y is equal to A x A y into S x 0 and 0 S y for the triangle ABC, so this will be our modified triangle that is triangle A dash B dash C dash, so A dash x A dash y B dash x B dash y C dash x C dash y is equal to triangle ABC that is A x A y B x B y C x C y into S x 0 0 S y and this will be our scaling matrix, always remember in the translation matrix there will be the addition, in the scaling and the rotational matrix there will be the multiplication, these are the references, thank you.