Final answer:
To find the 3x3 transformation matrix for a composite transformation in homogeneous coordinates, one must calculate both the rotation matrix and reflection matrix, then multiply them to achieve the composite matrix.
Step-by-step explanation:
The student is tasked with finding a 3x3 transformation matrix that performs a specific composite transformation in 2D space. This involves applying a sequence of transformations using homogeneous coordinates—first a rotation and then a reflection. To build this composite transformation matrix, we need to determine the rotation matrix and the reflection matrix separately, and then multiply them to obtain the composite matrix.
Rotation Matrix
For a counterclockwise rotation by an angle θ, the rotation matrix R in homogeneous coordinates is:
R =
[
cos(θ) -sin(θ) 0
sin(θ) cos(θ) 0
0 0 1
]
Reflection Matrix
If the reflection is through the x-axis, the reflection matrix M is:
M =
[
1 0 0
0 -1 0
0 0 1
]
The final composite matrix C is obtained by multiplying the reflection matrix by the rotation matrix: C = M * R.