Consider the matrix \(M=\begin{bmatrix} 2 & 0 & 2 \\ 0 & 1 & 1 \\ 0 & 0 & 1…
2018
Consider the matrix \(M=\begin{bmatrix} 2 & 0 & 2 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}\) representing a set of planar (2D) geometric transformations in homogeneous coordinates. Which of the following statements about the matrix M is True?
- A.
M represents first, a scaling of vector (2, 1) followed by translation of vector (1, 1)
- B.
M represents first, a translation of vector (1, 1) followed by scaling of vector (2, 1)
- C.
M represents first, a scaling of vector (3, 1) followed by shearing of parameters (−1, 1)
- D.
M represents first, a shearing of parameters (−1, 1) followed by scaling of vector (3, 1)
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Correct answer: A
Answer: The matrix represents a scaling by (2, 1) followed by a translation by (2, 1). None of the provided statements exactly matches this.
Linear part (upper-left 2×2): A = [2 0; 0 1], which scales x by 2 and y by 1 (no rotation or shear).
Translation vector (top-right entries): t = (2, 1). These are the amounts added after the linear transform.
Order of operations: the homogeneous action is x' = A x + t, so the matrix applies the linear scaling first and then the translation.
Check with an example: applying the matrix to point (1,0) gives (2*1 + 2, 1*0 + 1) = (4,1), showing the translation in x is 2 (not 1).
Why the provided statements fail:
The statement that the matrix is scaling by (2,1) followed by translation by (1,1) is close but wrong because the translation entries are (2,1), not (1,1).
The statement that it translates by (1,1) then scales by (2,1) is wrong because the order is reversed: the matrix applies the linear scaling first and then translation. Also the translation vector is (2,1), not (1,1).
Statements claiming scaling by (3,1) or the presence of shear are incorrect because the linear part is diagonal [2 0; 0 1], so there is no shear and the x-scaling is 2 (not 3).
Conclusion: The correct interpretation is a non-uniform scale by factors 2 in x and 1 in y, followed by a translation by (2,1).