Consider the following statements regarding \(2š·\) transforms in computerā¦
2019
Consider the following statements regardingĀ \(2š·\)Ā transforms in computer graphics:
\(S1: \: \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\) is aĀ \(2Ć2\) matrix that reflects (mirrors) onlyĀ \(2š·\) point about the X-axis.
\(š_2\)Ā :Ā AĀ \(2Ć2\)Ā matrix which mirrors anyĀ \(2š·\) point about theĀ \(š\)-axis, is a rotation matrix.
What can you say about the statementsĀ \(š_1\)Ā andĀ \(š_2\)?
Answer: B. Only \(š_1\) is true ā 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 mappingā¦
- A.
BothĀ
\(š_1\)Ā andĀ\(š_2\)Ā are true - B.
OnlyĀ
\(š_1\)is true - C.
OnlyĀ
\(š_2\)is true - D.
BothĀ
\(š_1\)andĀ\(š_2\)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 2Ć2 rotation matrix has the form [[cosĪø, -sinĪø], [sinĪø, cosĪø]] 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.