To reflect the polygon with vertices A(−1,0), B(0,−2)), C(1,0), and D(0,2) about the line x=2x = 2 using homogeneous coordinates, we can follow these steps:
Convert the reflection line to a more convenient form: The line x=2 is vertical, but reflecting about this line directly in homogeneous coordinates is not straightforward. Instead, we can use a sequence of transformations:
Translate the line x=2 to the y-axis (i.e., x=0).
Perform the reflection about the y-axis.
Translate the line back to x=2.
Construct the translation matrix: To translate x=2 to x=0x (move everything 2 units to the left), we use the translation matrix:
Construct the reflection matrix: Reflecting about the y-axis can be done using:
Construct the inverse translation matrix: Translate back by 2 units to the right to return the line to x=2:
Combine the transformations: The combined transformation matrix M is given by:
Now we can apply translation matrix to each vertex of the given figure.
So the reflected polygon has vertices A′(5,0), B′(4,−2), C′(3,0), and D′(4,2).