UGC NET CSE | July 2018 | Part 2 | Question: 10

Question

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)

Answer 1

multiply given matrix by following   3x1 matrix

[

x

y

1

]     we will get  x'=  2x+2 , y'=y+1  , 1=1   which is  (x+1)2 + (y+1)1  i.e translation of 1,1 followed by scaling of 2 and 1

hence option 2 is the right answer

 

 

Answer 2

Answer should be A. M represents first, a scaling of vector (2, 1) followed by translation of vector (1,1)

Matrix for scaling vector (2,1)=

2	0	0
0	1	0
0	0	1

Matrix for translation vector (1,1)=

1	0	1
0	1	1
0	0	1

Multiplying both we get the resultant vector

2	0	2
0	1	1
0	0	1

 

Comments

Ans is B option in official Key.

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But this solution seems to be perfect so I guess ans in official key might be wrong!!

 

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multiplication would be performed from right to left.

Thus answer would be 'B'

Answer 3

Scaling matrix is given by
X   0    0                                                                                                 
0   Y    0                        
0   0    1      
whereas Translation matrix is given by
1  0  X
 0  1  Y  
0  0  1

So if we apply scaling first and then translation, we won't end up with the elements which we require in the 3rd column of the given matrix.

Hence translation of vector (1,1) i.e    1  0  1

                                                                  0  1  1

                                                                  0  0  1

and then applying scaling vector (2,1) on translated vector gives      2  0  2

                                                                                                                        0  1  1

                                                                                                                        0  0  1

Ans is option B.

Comments

@ApoorvaRaikar Can u explain little bit more on this.

like after scaling how there is 2 at (0,2) position

thank you!