In this section, we will delve into the concept of transformation matrices, which are fundamental tools in 3D mathematics for manipulating and transforming objects in three-dimensional space. Understanding transformation matrices is crucial for applications in computer graphics, robotics, and many other fields.

1. Introduction to Transformation Matrices
2. Types of Transformations
   • Translation
   • Rotation
   • Scaling
3. Homogeneous Coordinates
4. Constructing Transformation Matrices
   • Translation Matrix
   • Rotation Matrix
   • Scaling Matrix
5. Combining Transformations
6. Practical Examples
7. Exercises

Transformation matrices are used to perform linear transformations on vectors in 3D space. These transformations include translation, rotation, and scaling. A transformation matrix is a square matrix that, when multiplied by a vector, transforms the vector in a specific way.

Translation involves moving an object from one position to another in 3D space. The translation matrix for moving a point \((x, y, z)\) by \((t_x, t_y, t_z)\) is:

\[ T = \begin{bmatrix} 1 & 0 & 0 & t_x
0 & 1 & 0 & t_y
0 & 0 & 1 & t_z
0 & 0 & 0 & 1 \end{bmatrix} \]

Rotation involves rotating an object around an axis. The rotation matrices for rotating around the x, y, and z axes by an angle \(\theta\) are:

• Rotation around the x-axis:

\[ R_x(\theta) = \begin{bmatrix} 1 & 0 & 0 & 0
0 & \cos\theta & -\sin\theta & 0
0 & \sin\theta & \cos\theta & 0
0 & 0 & 0 & 1 \end{bmatrix} \]

• Rotation around the y-axis:

\[ R_y(\theta) = \begin{bmatrix} \cos\theta & 0 & \sin\theta & 0
0 & 1 & 0 & 0
-\sin\theta & 0 & \cos\theta & 0
0 & 0 & 0 & 1 \end{bmatrix} \]

• Rotation around the z-axis:

\[ R_z(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta & 0 & 0
\sin\theta & \cos\theta & 0 & 0
0 & 0 & 1 & 0
0 & 0 & 0 & 1 \end{bmatrix} \]

Scaling involves changing the size of an object. The scaling matrix for scaling by factors \(s_x\), \(s_y\), and \(s_z\) along the x, y, and z axes is:

\[ S = \begin{bmatrix} s_x & 0 & 0 & 0
0 & s_y & 0 & 0
0 & 0 & s_z & 0
0 & 0 & 0 & 1 \end{bmatrix} \]

To perform transformations using matrices, we use homogeneous coordinates. A point \((x, y, z)\) in 3D space is represented as \((x, y, z, 1)\) in homogeneous coordinates. This allows us to use a single matrix to perform multiple transformations.

To translate a point \((x, y, z)\) by \((t_x, t_y, t_z)\):

\[ T = \begin{bmatrix} 1 & 0 & 0 & t_x
0 & 1 & 0 & t_y
0 & 0 & 1 & t_z
0 & 0 & 0 & 1 \end{bmatrix} \]

To rotate a point around the x, y, or z axis by an angle \(\theta\):

• Around the x-axis:

\[ R_x(\theta) = \begin{bmatrix} 1 & 0 & 0 & 0
0 & \cos\theta & -\sin\theta & 0
0 & \sin\theta & \cos\theta & 0
0 & 0 & 0 & 1 \end{bmatrix} \]

• Around the y-axis:

\[ R_y(\theta) = \begin{bmatrix} \cos\theta & 0 & \sin\theta & 0
0 & 1 & 0 & 0
-\sin\theta & 0 & \cos\theta & 0
0 & 0 & 0 & 1 \end{bmatrix} \]

• Around the z-axis:

\[ R_z(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta & 0 & 0
\sin\theta & \cos\theta & 0 & 0
0 & 0 & 1 & 0
0 & 0 & 0 & 1 \end{bmatrix} \]

To scale a point by factors \(s_x\), \(s_y\), and \(s_z\):

\[ S = \begin{bmatrix} s_x & 0 & 0 & 0
0 & s_y & 0 & 0
0 & 0 & s_z & 0
0 & 0 & 0 & 1 \end{bmatrix} \]

Transformations can be combined by multiplying their matrices. For example, to first rotate and then translate a point, you multiply the rotation matrix by the translation matrix:

\[ T \cdot R \]

Order matters in matrix multiplication, so the sequence of transformations is important.

Translate the point \((1, 2, 3)\) by \((4, 5, 6)\):

\[ T = \begin{bmatrix} 1 & 0 & 0 & 4
0 & 1 & 0 & 5
0 & 0 & 1 & 6
0 & 0 & 0 & 1 \end{bmatrix} \]

\[ \begin{bmatrix} 1 & 0 & 0 & 4
0 & 1 & 0 & 5
0 & 0 & 1 & 6
0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1
2
3
1 \end{bmatrix}

\begin{bmatrix} 5
7
9
1 \end{bmatrix} \]

Rotate the point \((1, 0, 0)\) around the z-axis by 90 degrees (\(\pi/2\) radians):

\[ R_z(\pi/2) = \begin{bmatrix} 0 & -1 & 0 & 0
1 & 0 & 0 & 0
0 & 0 & 1 & 0
0 & 0 & 0 & 1 \end{bmatrix} \]

\[ \begin{bmatrix} 0 & -1 & 0 & 0
1 & 0 & 0 & 0
0 & 0 & 1 & 0
0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1
0
0
1 \end{bmatrix}

\begin{bmatrix} 0
1
0
1 \end{bmatrix} \]

Translate the point \((2, 3, 4)\) by \((1, -1, 2)\). Write down the translation matrix and the resulting point.

Rotate the point \((0, 1, 0)\) around the x-axis by 90 degrees (\(\pi/2\) radians). Write down the rotation matrix and the resulting point.

Scale the point \((1, 2, 3)\) by factors \(2, 3, 4\) along the x, y, and z axes. Write down the scaling matrix and the resulting point.

Solution 1: Translation

Translation matrix:

\[ T = \begin{bmatrix} 1 & 0 & 0 & 1
0 & 1 & 0 & -1
0 & 0 & 1 & 2
0 & 0 & 0 & 1 \end{bmatrix} \]

Resulting point:

\[ \begin{bmatrix} 1 & 0 & 0 & 1
0 & 1 & 0 & -1
0 & 0 & 1 & 2
0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 2
3
4
1 \end{bmatrix}

\begin{bmatrix} 3
2
6
1 \end{bmatrix} \]

Solution 2: Rotation

Rotation matrix around the x-axis by 90 degrees:

\[ R_x(\pi/2) = \begin{bmatrix} 1 & 0 & 0 & 0
0 & 0 & -1 & 0
0 & 1 & 0 & 0
0 & 0 & 0 & 1 \end{bmatrix} \]

Resulting point:

\[ \begin{bmatrix} 1 & 0 & 0 & 0
0 & 0 & -1 & 0
0 & 1 & 0 & 0
0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 0
1
0
1 \end{bmatrix}

\begin{bmatrix} 0
0
1
1 \end{bmatrix} \]

Solution 3: Scaling

Scaling matrix:

\[ S = \begin{bmatrix} 2 & 0 & 0 & 0
0 & 3 & 0 & 0
0 & 0 & 4 & 0
0 & 0 & 0 & 1 \end{bmatrix} \]

Resulting point:

\[ \begin{bmatrix} 2 & 0 & 0 & 0
0 & 3 & 0 & 0
0 & 0 & 4 & 0
0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1
2
3
1 \end{bmatrix}

\begin{bmatrix} 2
6
12
1 \end{bmatrix} \]

In this section, we explored transformation matrices and their role in translating, rotating, and scaling objects in 3D space. We learned how to construct these matrices and apply them to points using homogeneous coordinates. Understanding these concepts is essential for manipulating 3D graphics and forms the foundation for more advanced topics in 3D mathematics.