**Prerequisite – Basic types of 2-D Transformation :**

**Composite Transformation :**

As the name suggests itself Composition, here we combine two or more transformations into one single transformation that is equivalent to the transformations that are performed one after one over a 2-D object.

**Example :**

Consider we have a 2-D object on which we first apply transformation **T _{1}**

**(2-D matrix condition)**and then we apply transformation

**T**

_{2}**(2-D matrix condition)**over the 2-D object and the object get transformed, the very equivalent effect over the 2-D object we can obtain by multiplying

**T**

_{1 }& T_{2}**(2-D matrix conditions)**with each other and then applying the

**T**with the coordinates of the 2-D image to get the transformed final image.

_{12 }(resultant of T_{1 }X T_{2})**Problem :**

Consider we have a square **O(0, 0), B(4, 0), C(4, 4), D(0, 4)** on which we first apply **T1(scaling transformation) given scaling factor is Sx=Sy=0.5 **and then we apply **T2(rotatation transformation in clockwise direction)** it by 90^{*}(angle), in last we perform **T3(reflection transformation about origin).**

**Ans :** The square O, A, C, D looks like :

**First, we perform scaling transformation over a 2-D object :**

**Representation of scaling condition :**

**For coordinate O(0, 0) :**

**For coordinate B(4, 0) :**

**For coordinate C(4, 4) :**

**For coordinate D(0, 4) :**

**2-D object after scaling :**

***Now, we’ll perform rotation transformation in clockwise-direction on Fig.2 by 90**^{θ}**:**

**The condition of rotation transformation of 2-D object about origin is :**

**For coordinate O(0, 0) :**

**For coordinate B(2, 0) :**

**For coordinate C(2, 2) :**

**For coordinate D(0, 2) :**

**2-D object after rotating about origin by 90 ^{*} angle :**

***Now, we’ll perform third last operation on Fig.3, by reflecting it about origin :**

The conditon of reflecting an object about origin is

**For coordinate O(0, 0) :**

**For coordinate B'(0, 0) :**

**For coordinate C'(0, 0) :**

**For coordinate D'(0, 0) :**

**The final 2-D object after reflecting about origin, we get :**

**Note :** The above finale result of **Fig.4**, that we get after applying all transformation one after one in a serial manner. We could also get the same result by combining all the transformation 2-D matrix conditions together and multiplying each other and get a resultant of multiplication(R). Then, applying that 2D-resultant matrix(R) at each coordinate of the given square(above). So, you will get the same result as you have in **Fig.4.**

**Solution using Composite transformation :**

*First we multiplied 2-D matrix conditions of **Scaling transformation** with **Rotation transformation :**

*Now, we multiplied Resultant 2-D matrix(R_{1}) with the third last given Reflecting condition of transformation(R_{2}) to get Resultant(R) :

Now, we’ll applied the Resultant(R) of 2d-matrix at each coordinate of the given object (square) to get the final transformed or modified object.

**First transformed coordinate O(0, 0) is :**

**Second, transformed coordinate B'(4, 0) is :**

**Third transformed coordinate C'(4, 4) is :**

**Fourth transformed coordinate D'(0, 4) is :**

**The final result of the transformed object that you get would be same as above :**