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Reflect the polygon whose vertices are A(-1,0), B(0, -2), C(1,0) and D(0,2) about the line x =2 using homogeneous co-ordinates.

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).

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