Consider the following statements regarding \(2𝐷\) transforms in computer…
2019
Consider the following statements regarding \(2𝐷\) transforms in computer graphics:
\(S1: \: \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\) is a \(2×2\) matrix that reflects (mirrors) only \(2𝐷\) point about the X-axis.
\(𝑆_2\) : A \(2×2\) matrix which mirrors any \(2𝐷\) point about the \(𝑋\)-axis, is a rotation matrix.
What can you say about the statements \(𝑆_1\) and \(𝑆_2\)?
- A.
Both
\(𝑆_1\)and\(𝑆_2\)are true - B.
Only
\(𝑆_1\)is true - C.
Only
\(𝑆_2\)is true - D.
Both
\(𝑆_1\)and\(𝑆_2\)are false
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Correct answer: B
Answer: Only S1 is true.
Explanation:
Why the first statement is true: The matrix [[1, 0], [0, -1]] applied to a point (x, y) gives (x, -y). That mapping reflects points across the X-axis.
Why the second statement is false: Reflection matrices like [[1, 0], [0, -1]] are orthogonal but have determinant -1. A 2×2 rotation matrix has the form [[cosθ, -sinθ], [sinθ, cosθ]] and always has determinant +1. Because the mirror about the X-axis has determinant -1, it cannot be a rotation.
Key numeric check: det([[1, 0], [0, -1]]) = -1, confirming it is a reflection (an improper orthogonal transformation), not a proper rotation.
Conclusion: Statement S1 correctly describes a reflection about the X-axis; statement S2 is incorrect because reflections are not rotations.