In this section, we will explore the concept of composing multiple linear transformations to achieve complex transformations in 3D space. Understanding how to combine transformations is crucial for tasks such as animating objects, applying multiple effects, and manipulating 3D models in computer graphics.

1. Linear Transformation: A function that maps vectors to vectors in a linear manner, preserving vector addition and scalar multiplication.
2. Transformation Matrix: A matrix that represents a linear transformation.
3. Composition of Transformations: The process of applying multiple transformations in sequence.

The composition of two transformations \( T_1 \) and \( T_2 \) is a new transformation \( T \) such that \( T(\mathbf{v}) = T_2(T_1(\mathbf{v})) \) for any vector \( \mathbf{v} \).

If \( T_1 \) is represented by matrix \( A \) and \( T_2 \) is represented by matrix \( B \), then the composition \( T \) is represented by the matrix product \( B \cdot A \).

The order of multiplication matters. \( B \cdot A \) means that \( A \) is applied first, followed by \( B \).

Consider two transformations:

• \( T_1 \): Scaling by a factor of 2, represented by matrix \( A \)
• \( T_2 \): Rotation by 90 degrees, represented by matrix \( B \)

The matrices are: \[ A = \begin{bmatrix} 2 & 0
0 & 2 \end{bmatrix} \] \[ B = \begin{bmatrix} 0 & -1
1 & 0 \end{bmatrix} \]

The composition \( T \) is represented by: \[ T = B \cdot A = \begin{bmatrix} 0 & -1
1 & 0 \end{bmatrix} \cdot \begin{bmatrix} 2 & 0
0 & 2 \end{bmatrix} = \begin{bmatrix} 0 & -2
2 & 0 \end{bmatrix} \]

1. Translation: Move a point by vector \( \mathbf{t} = (3, 4) \)
2. Rotation: Rotate the point by 45 degrees

Translation matrix \( T \): \[ T = \begin{bmatrix} 1 & 0 & 3
0 & 1 & 4
0 & 0 & 1 \end{bmatrix} \]

Rotation matrix \( R \): \[ R = \begin{bmatrix} \cos(45^\circ) & -\sin(45^\circ) & 0
\sin(45^\circ) & \cos(45^\circ) & 0
0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & 0
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0
0 & 0 & 1 \end{bmatrix} \]

The composition \( C \) is: \[ C = R \cdot T = \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & 0
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0
0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 & 3
0 & 1 & 4
0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 7\frac{\sqrt{2}}{2}
0 & 0 & 1 \end{bmatrix} \]

1. Scaling: Scale by a factor of 2
2. Translation: Move by vector \( \mathbf{t} = (1, 2) \)

Scaling matrix \( S \): \[ S = \begin{bmatrix} 2 & 0 & 0
0 & 2 & 0
0 & 0 & 1 \end{bmatrix} \]

Translation matrix \( T \): \[ T = \begin{bmatrix} 1 & 0 & 1
0 & 1 & 2
0 & 0 & 1 \end{bmatrix} \]

The composition \( C \) is: \[ C = T \cdot S = \begin{bmatrix} 1 & 0 & 1
0 & 1 & 2
0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 2 & 0 & 0
0 & 2 & 0
0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 2 & 0 & 1
0 & 2 & 2
0 & 0 & 1 \end{bmatrix} \]

Given two rotation matrices: \[ R_1 = \begin{bmatrix} \cos(30^\circ) & -\sin(30^\circ) & 0
\sin(30^\circ) & \cos(30^\circ) & 0
0 & 0 & 1 \end{bmatrix} \] \[ R_2 = \begin{bmatrix} \cos(60^\circ) & -\sin(60^\circ) & 0
\sin(60^\circ) & \cos(60^\circ) & 0
0 & 0 & 1 \end{bmatrix} \]

Find the composition \( R = R_2 \cdot R_1 \).

\[ R = \begin{bmatrix} \cos(60^\circ) & -\sin(60^\circ) & 0
\sin(60^\circ) & \cos(60^\circ) & 0
0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} \cos(30^\circ) & -\sin(30^\circ) & 0
\sin(30^\circ) & \cos(30^\circ) & 0
0 & 0 & 1 \end{bmatrix} \]

\[ R = \begin{bmatrix} \frac{1}{2} & -\frac{\sqrt{3}}{2} & 0
\frac{\sqrt{3}}{2} & \frac{1}{2} & 0
0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} & 0
\frac{1}{2} & \frac{\sqrt{3}}{2} & 0
0 & 0 & 1 \end{bmatrix} \]

\[ R = \begin{bmatrix} \frac{1}{2} \cdot \frac{\sqrt{3}}{2} + -\frac{\sqrt{3}}{2} \cdot \frac{1}{2} & \frac{1}{2} \cdot -\frac{1}{2} + -\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} & 0
\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{1}{2} \cdot \frac{1}{2} & \frac{\sqrt{3}}{2} \cdot -\frac{1}{2} + \frac{1}{2} \cdot \frac{\sqrt{3}}{2} & 0
0 & 0 & 1 \end{bmatrix} \]

\[ R = \begin{bmatrix} \frac{3}{4} - \frac{1}{4} & -\frac{1}{4} - \frac{3}{4} & 0
\frac{3}{4} + \frac{1}{4} & -\frac{\sqrt{3}}{4} + \frac{\sqrt{3}}{4} & 0
0 & 0 & 1 \end{bmatrix} \]

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

Given a scaling matrix \( S \) and a rotation matrix \( R \): \[ S = \begin{bmatrix} 3 & 0 & 0
0 & 3 & 0
0 & 0 & 1 \end{bmatrix} \] \[ R = \begin{bmatrix} \cos(45^\circ) & -\sin(45^\circ) & 0
\sin(45^\circ) & \cos(45^\circ) & 0
0 & 0 & 1 \end{bmatrix} \]

Find the composition \( C = R \cdot S \).

\[ C = \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & 0
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0
0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 3 & 0 & 0
0 & 3 & 0
0 & 0 & 1 \end{bmatrix} \]

\[ C = \begin{bmatrix} \frac{\sqrt{2}}{2} \cdot 3 & -\frac{\sqrt{2}}{2} \cdot 3 & 0
\frac{\sqrt{2}}{2} \cdot 3 & \frac{\sqrt{2}}{2} \cdot 3 & 0
0 & 0 & 1 \end{bmatrix} \]

\[ C = \begin{bmatrix} \frac{3\sqrt{2}}{2} & -\frac{3\sqrt{2}}{2} & 0
\frac{3\sqrt{2}}{2} & \frac{3\sqrt{2}}{2} & 0
0 & 0 & 1 \end{bmatrix} \]

In this section, we have learned about the composition of transformations, which is essential for combining multiple linear transformations into a single operation. We explored the matrix representation of composed transformations and worked through practical examples to solidify our understanding. By mastering these concepts, you will be able to manipulate 3D objects more effectively in various applications, including computer graphics and animation.