In this section, we will explore three fundamental types of linear transformations in 3D space: rotations, translations, and scalings. These transformations are essential for manipulating objects in 3D graphics and are widely used in computer graphics, animation, and various simulations.

Rotation is a transformation that turns every point of an object around a fixed axis by a certain angle. In 3D space, rotations can be more complex than in 2D because they can occur around any of the three principal axes (x, y, z) or any arbitrary axis.

The rotation of a point in 3D space can be represented using rotation matrices. Here are the rotation matrices for rotating around the principal axes:

• Rotation around the x-axis by an angle θ:

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

• Rotation around the y-axis by an angle θ:

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

• Rotation around the z-axis by an angle θ:

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

To rotate a point \( P = (x, y, z) \) around the z-axis by 90 degrees (π/2 radians):

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

\[ P' = R_z\left(\frac{\pi}{2}\right) \cdot \begin{bmatrix} x
y
z \end{bmatrix} = \begin{bmatrix} 0 & -1 & 0
1 & 0 & 0
0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} x
y
z \end{bmatrix} = \begin{bmatrix} -y
x
z \end{bmatrix} \]

Translation is a transformation that moves every point of an object by the same distance in a given direction. It is represented by adding a translation vector to the coordinates of each point.

In homogeneous coordinates, translation can be represented using a 4x4 matrix:

\[ T(tx, ty, tz) = \begin{bmatrix} 1 & 0 & 0 & tx
0 & 1 & 0 & ty
0 & 0 & 1 & tz
0 & 0 & 0 & 1 \end{bmatrix} \]

To translate a point \( P = (x, y, z) \) by \( (tx, ty, tz) \):

\[ T(tx, ty, tz) = \begin{bmatrix} 1 & 0 & 0 & tx
0 & 1 & 0 & ty
0 & 0 & 1 & tz
0 & 0 & 0 & 1 \end{bmatrix} \]

\[ P' = T(tx, ty, tz) \cdot \begin{bmatrix} x
y
z
1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & tx
0 & 1 & 0 & ty
0 & 0 & 1 & tz
0 & 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} x
y
z
1 \end{bmatrix} = \begin{bmatrix} x + tx
y + ty
z + tz
1 \end{bmatrix} \]

Scaling is a transformation that enlarges or diminishes objects. It is defined by scaling factors along the x, y, and z axes.

The scaling transformation can be represented using a 4x4 matrix in homogeneous coordinates:

\[ S(sx, sy, sz) = \begin{bmatrix} sx & 0 & 0 & 0
0 & sy & 0 & 0
0 & 0 & sz & 0
0 & 0 & 0 & 1 \end{bmatrix} \]

To scale a point \( P = (x, y, z) \) by \( (sx, sy, sz) \):

\[ S(sx, sy, sz) = \begin{bmatrix} sx & 0 & 0 & 0
0 & sy & 0 & 0
0 & 0 & sz & 0
0 & 0 & 0 & 1 \end{bmatrix} \]

\[ P' = S(sx, sy, sz) \cdot \begin{bmatrix} x
y
z
1 \end{bmatrix} = \begin{bmatrix} sx & 0 & 0 & 0
0 & sy & 0 & 0
0 & 0 & sz & 0
0 & 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} x
y
z
1 \end{bmatrix} = \begin{bmatrix} sx \cdot x
sy \cdot y
sz \cdot z
1 \end{bmatrix} \]

Rotate the point \( P = (1, 2, 3) \) around the y-axis by 45 degrees.

Solution:

\[ R_y\left(\frac{\pi}{4}\right) = \begin{bmatrix} \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2}
0 & 1 & 0
-\frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \end{bmatrix} \]

\[ P' = R_y\left(\frac{\pi}{4}\right) \cdot \begin{bmatrix} 1
2
3 \end{bmatrix} = \begin{bmatrix} \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2}
0 & 1 & 0
-\frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \end{bmatrix} \cdot \begin{bmatrix} 1
2
3 \end{bmatrix} = \begin{bmatrix} 2.828
2
1.414 \end{bmatrix} \]

Translate the point \( P = (4, -3, 2) \) by \( (1, 2, -1) \).

Solution:

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

\[ P' = T(1, 2, -1) \cdot \begin{bmatrix} 4
-3
2
1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 1
0 & 1 & 0 & 2
0 & 0 & 1 & -1
0 & 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 4
-3
2
1 \end{bmatrix} = \begin{bmatrix} 5
-1
1
1 \end{bmatrix} \]

Scale the point \( P = (2, 3, 4) \) by \( (2, 0.5, 3) \).

Solution:

\[ S(2, 0.5, 3) = \begin{bmatrix} 2 & 0 & 0 & 0
0 & 0.5 & 0 & 0
0 & 0 & 3 & 0
0 & 0 & 0 & 1 \end{bmatrix} \]

\[ P' = S(2, 0.5, 3) \cdot \begin{bmatrix} 2
3
4
1 \end{bmatrix} = \begin{bmatrix} 2 & 0 & 0 & 0
0 & 0.5 & 0 & 0
0 & 0 & 3 & 0
0 & 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 2
3
4
1 \end{bmatrix} = \begin{bmatrix} 4
1.5
12
1 \end{bmatrix} \]

In this section, we have learned about three fundamental transformations in 3D space: rotations, translations, and scalings. We have explored their mathematical representations using matrices and provided practical examples and exercises to reinforce the concepts. Understanding these transformations is crucial for manipulating 3D objects in various applications, including computer graphics, animation, and simulations.