(i) The composite transformation matrix

(ii) The new coordinates of rectangle.

Solution:

Steps:

1. Translate Point P(3,2) to origin O(0,0)

2. Rotate the object by $ \theta= + 30^o$

3. Inverse translation of Point P to its original position P(3,2) from origin O(0,0)

Step 1:

Translation Matrix is given as:

$$\left[\mathrm{T}_{\mathrm{r}}\right]=\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {\mathrm{tx}} & {\text { ty }} & {1}\end{array}\right] \quad \mathrm{tx}=-3 \quad \& \ ty =-2$$

$$\therefore[\mathrm{Tr}]=\left[\begin{array}{ccc}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {-3} & {-2} & {1}\end{array}\right]$$

Step 2:

Note: CCw is taken as +ve angle and CW is taken as –ve angle. Rotation Matrix is given as,

$$\begin{array}{lll}{[\mathrm{R}]} & {=\left[\begin{array}{ccc}{\cos \theta} & {\sin \theta} & {0} \\ {-\sin \theta} & {\cos \theta} & {0} \\ {0} & {0} & {1}\end{array}\right]}\end{array}$$

$$\theta = +30^0$$

$$[\mathrm{R}]=\left[\begin{array}{ccc}{\cos 30} & {\sin 30} & {0} \\ {-\sin 30} & {\cos 30} & {0} \\ {0} & {0} & {1}\end{array}\right]=\left[\begin{array}{ccc}{0.866} & {0.5} & {0} \\ {-0.5} & {0.866} & {0} \\ {0} & {0} & {1}\end{array}\right]$$

Step 3:

$$[\mathrm{Tr}]^{-1}=\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {\mathrm{tx}} & {\text { ty }} & {1}\end{array}\right]=\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {3} & {2} & {1}\end{array}\right]$$

Now,

i. The composite Transformation Matrix / Affine Matrix

$$[\mathrm{T}] \quad=\left[\operatorname{Tr}[\mathrm{R}][\mathrm{Tr}]^{-1}\right.$$

$$=\left[\begin{array}{ccc}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {-3} & {-2} & {1}\end{array}\right]\left[\begin{array}{ccc}{0.866} & {0.5} & {0} \\ {-0.5} & {0.866} & {0} \\ {0} & {0} & {1}\end{array}\right]\left[\begin{array}{ccc}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {3} & {2} & {1}\end{array}\right]$$

$$=\left[\begin{array}{ccc}{0.866} & {0.5} & {0} \\ {-0.5} & {0.866} & {0} \\ {1.4} & {-1.23} & {1}\end{array}\right]$$

ii. The new coordinates of rectangle

$$ \left[\mathrm{X}^{\prime}\right]=[\mathrm{X}][\mathrm{T}] $$

$$\left[\begin{array}{l}{\mathrm{A}^{\prime}} \\ {\mathrm{B}^{\prime}} \\ {\mathrm{C}^{\prime}}\\{\mathrm{D}^{\prime}}\end{array}\right]=\left[\begin{array}{l}{\mathrm{A}} \\ {\mathrm{B}} \\ {\mathrm{C}}\\{\mathrm{D}}\end{array}\right][\mathrm{T}]$$

$$=\left[\begin{array}{ccc}{1} & {1} & {1} \\ {2} & {1} & {1} \\ {2} & {3} & {1} \\ {1} & {3} & {1}\end{array}\right]\left[\begin{array}{ccc}{0.866} & {0.5} & {0} \\ {-0.5} & {0.866} & {0} \\ {1.4} & {-1.23} & {1}\end{array}\right]$$

$$=\left[\begin{array}{llll}{1.77} & {0.136} & {1} \\ {2.63} & {0.636} & {1} \\ {1.63} & {2.368} & {1} \\ {0.77} & {1.868} & {1}\end{array}\right]$$

Hence, new coordinates of rectangle are:

$$\mathrm{A}^{\prime}=(1.77,0.136)$$ $$\mathrm{B}^{\prime}=(2.63,0.636)$$ $$\mathrm{C}^{\prime}=(1.63,2.368)$$ $$\mathrm{D}^{\prime}=(0.77,1.868)$$