 Hello everyone, welcome to the third session of the Geometrical Transformation. In the third session we will discuss the concatenation method which is very important regarding the Geometrical Transformation. Myself, Professor Prithish Chitte, working as an Assistant Professor in Mechanical Engineering Department, Valchin Institute of Technology, Solapur, Learning Outcomes. The students will be able to explain the concept of concatenation method in 2D Geometrical Transformation. Also, they can solve the numerical based on the concatenation method. These are the contents. First, we will discuss the homogeneous 2D transformation. After that we will see the concatenation method in geometrical transformation and after that we will see the numerical based on the concatenation method. Now, the first point, homogeneous coordinates, two-dimensional homogeneous coordinate system. Why it is required? Why it is required the homogeneous coordinate for the concatenation method? This will be the prerequisite for the concatenation method. If you are not knowing the homogeneous coordinates, you can't understood any concatenation method in the geometrical transformation. Homogeneous coordinates are the 3 by 3 or 3 dimensional matrices. Now here, when it is required the homogeneous transformation or homogeneous coordinate transformation, whenever we are providing the multiple transformations, why multiple transformation? Because there will be the necessity of providing the multiple transformations like the combination of translation and scaling or maybe the combination of translation and rotation. Only with the help of translation, that is moving the object from one coordinate to the another coordinate, we are not getting the required result. But whenever we are combining the geometrical transformation like the translation and after that the rotation, we are acquiring the result. So it is necessary to have the homogeneous coordinate in the concatenation method and which will provide the cumulative effect. So first we will discuss the translation matrix in the homogeneous coordinates. The same equation, in general equation, the P dash is equal to P into t. This is the translation matrix but the translation matrix is different here. This is the 3 by 3 matrix. In the in general geometrical transformation, that is the translation that will be only the 2 by 2 matrix. But here this will be the 3 by 3 matrix where T x and T y are the translation factors or the distances. So here x dash, y dash are the modified coordinates, x y are the original coordinates and T x and T y are the translation distances along x and y axis respectively. Rotational homogeneous coordinates. So P dash is equal to P into r in general equation where r is the rotational matrix. So this will be our 3 by 3 rotational matrix cos theta sin theta minus sin theta cos theta plus here the identity matrix. We are adding the identity matrix here. So cos theta sin theta 0 minus sin theta cos theta 0 0 0 1 which will be our rotational matrix. This will be our 2 dimensional geometrical transformation for the rotational matrix. So x dash, y dash are the modified coordinates, x y will be our original coordinates and this will be our rotational matrix where theta is our angle of rotation. Scaling matrix in the homogeneous coordinates. 3 dash is equal to P into s, s is our scaling matrix where x dash, y dash is our modified coordinates, s x y is our original coordinates, s x 0 and 0 s y. These 2 by 2 matrix which is similar to our general scaling matrix in the geometrical transformation that will be 2D but here these are the homogeneous coordinates. So here s x 0 0, 0 s y 0, 0 0 1. So here we are adding the identity matrix here. So s x and s y which will be representing the scaling factors for the s in the direction of x and y axis respectively. Now we have discussed the homogeneous coordinates. Now we are knowing the homogeneous coordinates. Now we can understand the concatenation method in the geometrical transformation. So why the concatenation method is required? Why there is a need of the concatenation method? As we have discussed in the homogeneous coordinates. So whenever we want to provide the multiple transformations, multiple geometrical transformations like the combination of translation or the scaling or the translation and the rotation after that only listen very carefully after that only we can get the a particular required result that is the modified geometrical entity. And here the 2D geometrical transformation we are providing on the particular point now here the concatenation method we can do it on any other point and this process is called as the cumulative effect. Here there is a necessary of the reorientation of the object and this will be only possible with the help of multiple transformations carried out at a one time and this whole process is called as concatenation method in the geometrical transformation. Again I will repeat so concatenation method is providing the multiple transformation at one time so that we can get the required result. Let's think about this question. Can you correlate the two dimensional geometric transformations and the homogeneous transformations? Think about it. Now first we will see the case one that is combination of translation and rotation matrix with the help of concatenation method. Now we are having the original figure that is Pxy. Now first we have to provide the translation transformation now this will be our original image now it will move to the this particular coordinate after that we are not shopping here because we are using the concatenation method that is multiple transformation after translation we are rotating again that translated or the modified triangle now again we are rotating that particular triangle here by a particular degree. So this will be our the required result now we will go through the equation. So equation for the concatenation method for the combination of translation and rotation so P dash that will be our modified coordinates P our original coordinate T will be our translation rotational matrix and the this will be our the transpose of the or may be the inverse of the translation matrix. So this will be our equation so P dash into 1 0 0 0 0 1 minus XR minus YR 0 1 1 where XR and YR are our coordinates here XR and YR our translation coordinates or the distances cos theta sin theta minus cos theta sin theta 0 0 0 0 1 this will be our rotational matrix 1 0 0 0 0 0 1 and XR YR 1 this will be our translation matrix after that so here the combination of translation and scaling here we are translating the image first here this is our original triangle after that we are providing the scaling transformation after the translation we are providing the scaling transformation. So this will be our the required image we will see the equation. So P dash is equal to P into T into S into T inverse so this will be our a particular formula for the concatenation method with the help of the translation and the scaling. So P dash is equal to P into 1 0 0 0 0 1 minus XR minus YR SX 0 0 SY 0 1 and XR YR 1 this will be our equation we will see the example triangle ABC has its vertices 0 0 4 0 and 2 3 for the points PQR and it is to be translated by 4 unit in X direction and it is to be rotated in the counter clockwise about its point through the angle 90 degree we have to find out the new position for the triangle ABC. Now we are knowing the coordinates that is 0 0 4 0 and 2 3 here we have to use a 3 by 3 triangle we are knowing that is TX is equal to 4 and TY is equal to 2 these are the our translation distances now we can get the triangle P dash Q dash and R dash. So this will be our P dash Q dash and R dash after the translation transformation. So P dash into TX TY so P into TX TY so this will be our original triangle coordinates and this will be our translation matrix after that we are getting the P dash X Q dash X and Y dash sorry the R dash XY Z so 4 2 8 2 and 6 5 are the new coordinates of P dash Q dash and R dash respectively after that we have to rotate that triangle P dash Q dash R dash through the angle theta 90 degree through the new point 65 through the new point 65 very important we are knowing the formula. So triangle P dash Q dash R dash is equal to triangle P dash Q dash R dash into T into R into T inverse keeping the values here and we can get the new coordinates that is P double dash 9 3 Q double dash 9 7 and R double dash 65 after solving the different matrices. So 9 3 9 7 and 65 are the new coordinates of P double dash Q double dash and R double dash after the concatenation method these are the references.