Question
Download Solution PDFWhich homogeneous 2D matrix transforms the figure (a) on the left side to the figure (b) on the right?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFTo transform figure on left side into figure on right side, two things must be done in this.
1) It requires anticlockwise rotation by 90 degree.
2) It requires translation along x-axis and y-axis.
In case of 2-D, rotation is done by:
\(P = \;\left[ {\begin{array}{*{20}{c}} {\cos \theta }&{ - \;sin\theta }\\ {sin\theta }&{cos\theta } \end{array}} \right]\)
In case of 3-D rotation, also rotation is with respect to 2 on y-axis
\(P = \;\left[ {\begin{array}{*{20}{c}} {\cos 90}&{ - sin90}&0\\ {sin90}&{cos90}&0\\ 0&0&1 \end{array}} \right] = \;\left[ {\begin{array}{*{20}{c}} 0&{ - 1}&0\\ 1&0&0\\ 0&0&1 \end{array}} \right]\)
So, final matrix after rotation = \(\left[ {\begin{array}{*{20}{c}} 1&{ - 2}&0\\ 1&0&0\\ 0&0&1 \end{array}} \right]\)
Next step is translation, along x-axis translation factor is 6 units and along y- axis translation factor is 1 unit.
Translation matrix = \(\left[ {\begin{array}{*{20}{c}} 1&0&{tx}\\ 0&1&{ty}\\ 0&0&1 \end{array}} \right]\)
By putting tx = 6 and ty = 1
Final matrix will be = \(\left[ {\begin{array}{*{20}{c}} 1&{ - 2}&{tx}\\ 1&0&{ty}\\ 0&0&1 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0&{ - 2}&6\\ 1&0&1\\ 0&0&1 \end{array}} \right]\)Last updated on Jun 6, 2025
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