 Hello everybody, I am Dr. Keshav Ualesi from Mechanical Engineering Department of Walton Institute of Technology, Sulapur. In this session, we will be discussing about basics of geometric transformations pertaining to CADCAM subject particularly. At the end of this session, viewers are expected to understand the basic concept of what is geometric transformations and in particular basic three types of transformations which are very commonly used that is translation, scaling and rotation. To define geometric transformation, these are the operations carried out to modify the existing object, its configuration, its orientation, its shape, size, position. So as to meet the desired or required design aspects for the designer or the modular. Now these matrix transformations they are carried out in different sequential orders on the objects so that ultimately from basic graphics primitives using these transformations combining different primitives or graphics elements with the help of these transformations the designer can ultimately get what he wants. Let us concentrate on representation of some of the graphics features in case of two dimension that is by excel system. The simplest or the basic graphic element is a point we all know like a point we represent in by excel system as x, y indicating x and y coordinates of that particular point which is a graphics element the primitive it is also called as. Now in computer graphics this representation is perceived as a 1 by 2 matrix for example if a point is represented as 2, 5. So as you all know x is equal to 2 and y is equal to 5 in a by excel system but in CG this representation is perceived as a 1 by 2 matrix and this very basic concept of perceiving these graphics features in a matrix form is the main aspect of geometric transformations. Secondly we will talk about another graphics element that is line normally we represent a line passing through two points as it is given here l is equal to x 1, y 1, x 2, y 2 this is a line passing through two points this is perceived as a 2 by 2 matrix representation. So a point is perceived as a 1 by 2 matrix a line is perceived as 2 by 2 matrix. Next we will see one more element graphics feature primitive it is also called as that is a triangle, triangle we need three points. So x 1, y 1, x 2, y 2 and x 3, y 3 these may be say for example a, b and c three points their coordinates are perceived as a 3 by 2 matrix for a triangle thus any geometric transformation that we think of is fundamentally carried out on the points which are used to represent which are used to draw that particular graphics primitive. So basically for first time representation what are the algorithm we are using say for example to represent a triangle same is used after points of that feature are transformed that is a basic fundamental rule I repeat again points of any graphics feature are first transformed and then with the new points using the algorithm of representing that object that object is redrawn. In this session we will be focusing on these three very commonly used geometric transformations in case of two dimensional representations first is translation second is scaling and third is rotation at this point I would expect you all to just relate these discussions that we have so far made here to relate with some object you can imagine some object or any image and think for a while how these three transformations might be taking place just think for a while coming to first geometric transformation in case of two dimensions that is translation this is nothing but moving the object from one point to another point it is a repositioning but major thing is that without altering its dimensions we are just moving from one point to other point repositioning we can say now if you see here for point to be translated in back sail system how do we represent it mathematically. If p is the original point with x y coordinates and p dash is the translated point we are talking about a point p dash is the translated point and t is the translation matrix wherein t x and t y are given here which are distances of a translation along x and y directions ultimately we can represent the relationship for the translation of a point as p dash is equal to p plus t I repeat p dash is equal to p plus t here note that we are adding two matrices this is the transformation wherein we need to add two matrices secondly let us talk now on scaling in this case we either enlarge the given object or reduce in size now this enlargement or the reduction by enlarge it is equal in all the directions in all the axis that you consider. So, circle if you consider and if you scale it with the same scaling factor along x and y directions it will remain circle, but conceptually these scaling factors may be different in different directions. So, again talking about the circle if we scale a circle with the different scaling factors along x and y it would become an ellipse thus again thinking of a point p is the original point with x y coordinates p dash is the point after scaling and s is the scaling matrix this scaling matrix representation here happens to be in first row sx and 0 then second row 0 sy this sx and sy are the scaling factors along x and y directions. Now here this p dash we can represent as a p into s that is coordinates of new scaled point are equal to coordinates of given original point into s that is scaling matrix. This is a basic logic wherein we are multiplying these two matrices in earlier case of a translation we were adding the two matrices here we are multiplying two matrices in case of scaling. Now, coming to third geometric transformation that is rotation here points of the object we are rotating about the origin this is very important the rotation is about the origin by any given angle theta more important point here again is if the angle of rotation is plus theta that is positive then we are rotating the object in counter clockwise direction and if this angle is a negative we are rotating this object or a point along clockwise direction. Similarly, let us think of a point in back cell system for a rotation p as original point p dash is point after rotation and r here this r matrix is the rotation matrix which is represented as first row cos theta sin theta and second row minus sin theta cos theta. So, here again this relationship we have p dash is equal to p into r here also we are multiplying two matrices. Thus, the multiplication of these two matrices would give us the resultant matrix p dash that is the coordinates of rotated point. Thus, in these discussions to summarize first transformation of translation we are adding two matrices second and third that is scaling and rotation we are multiplying two matrices that is what is the basic aspect of geometric transformations for these three basic types. We have referred these two books one is CADCAM by MP Grover and second is the CADCAM mastering by Ibrahim Z. Thank you.