Determine the concatenated transformation matrix for translation by vector [1 1] followed by rotation of 45 degrees in 2D as shown in Figure 2 below.

1. Break Down the Problem

We need to find the concatenated transformation matrix for two transformations:

1. Translation by the vector $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$
2. Rotation by $45^\circ$

2. Relevant Concepts

The transformation matrices in 2D for translation and rotation are defined as follows:

• Translation Matrix $T$ for a vector $\begin{bmatrix} t_x \\ t_y \end{bmatrix}$: $T = \begin{bmatrix} 1 & 0 & t_x \\ 0 & 1 & t_y \\ 0 & 0 & 1 \end{bmatrix}$

• Rotation Matrix $R$ for an angle $\theta$ (in radians): $R = \begin{bmatrix} \cos(\theta) & -\sin(\theta) & 0 \\ \sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 1 \end{bmatrix}$

For $45^\circ$, we need to convert the angle to radians: $\theta = 45^\circ = \frac{\pi}{4} \text{ radians}$

3. Analysis and Detail

First, we will calculate the individual transformation matrices.

1. Translation Matrix for vector $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$: $T = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}$

2. Rotation Matrix for $45^\circ$ (in radians): $\cos\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$ Hence, the rotation matrix $R$ becomes: $R = \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & 0 \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ 0 & 0 & 1 \end{bmatrix}$

3. Concatenation of Matrices: The concatenated transformation is obtained by multiplying the translation matrix $T$ by the rotation matrix $R$: $TR = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix} \times \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & 0 \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ 0 & 0 & 1 \end{bmatrix}$

4. Verify and Summarize

Carrying out the matrix multiplication: $TR = \begin{bmatrix} 1 \cdot \frac{\sqrt{2}}{2} + 0 \cdot \frac{\sqrt{2}}{2} + 1 \cdot 0 & 1 \cdot -\frac{\sqrt{2}}{2} + 0 \cdot \frac{\sqrt{2}}{2} + 1 \cdot 0 & 1 \cdot 0 + 0 \cdot 0 + 1 \\ 0 \cdot \frac{\sqrt{2}}{2} + 1 \cdot \frac{\sqrt{2}}{2} + 1 \cdot 0 & 0 \cdot -\frac{\sqrt{2}}{2} + 1 \cdot \frac{\sqrt{2}}{2} + 1 \cdot 0 & 0 \cdot 0 + 1 \cdot 0 + 1 \\ 0 \cdot \frac{\sqrt{2}}{2} + 0 \cdot \frac{\sqrt{2}}{2} + 1 \cdot 0 & 0 \cdot -\frac{\sqrt{2}}{2} + 0 \cdot \frac{\sqrt{2}}{2} + 1 \cdot 0 & 1 \end{bmatrix}$

Calculating each entry:

1. First Row:

   • First column: $\frac{\sqrt{2}}{2}$
   • Second column: $-\frac{\sqrt{2}}{2}$
   • Third column: $1$

2. Second Row:

   • First column: $\frac{\sqrt{2}}{2}$
   • Second column: $\frac{\sqrt{2}}{2}$
   • Third column: $1$

3. Third Row:

   • First column: $0$
   • Second column: $0$
   • Third column: $1$

Thus, the final concatenated transformation matrix $TR$ is: $TR = \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & 1 \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 1 \\ 0 & 0 & 1 \end{bmatrix}$

Final Answer

The concatenated transformation matrix is: $\begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & 1 \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 1 \\ 0 & 0 & 1 \end{bmatrix}$