Question

In Exercises 3-8, find the \({\bf{3}} \times {\bf{3}}\) matrices that produce the described composite 2D transformations, using homogenous coordinates.

Translate by \(\left( { - {\bf{2}},{\bf{3}}} \right)\), and then scale the x-coordinate by .8 and the y-coordinate by 1.2.

Short answer

\(\left[ {\begin{array}{*{20}{c}}{0.8}&0&{ - 1.6}\\0&{1.2}&{3.6}\\0&0&1\end{array}} \right]\)

Step-by-step solution

Find the matrix for translation

The matrix for translationby \(\left( { - 2,3} \right)\) is

\(\left[ {\begin{array}{*{20}{c}}1&0&{ - 2}\\0&1&3\\0&0&1\end{array}} \right]\).

Find the matrix for scaling

The matrix for scaling can be expressed as

\(\left[ {\begin{array}{*{20}{c}}{0.8}&0&0\\0&{1.2}&0\\0&0&1\end{array}} \right]\).

Find the combined matrix of transformation

The combined matrix for transformation can be expressed as

\(\left[ {\begin{array}{*{20}{c}}{0.8}&0&0\\0&{1.2}&0\\0&0&1\end{array}} \right]\left[ {\begin{array}{*{20}{c}}1&0&{ - 2}\\0&1&3\\0&0&1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{0.8}&0&{ - 1.6}\\0&{1.2}&{3.6}\\0&0&1\end{array}} \right]\).

So, the transformed matrix is \(\left[ {\begin{array}{*{20}{c}}{0.8}&0&{ - 1.6}\\0&{1.2}&{3.6}\\0&0&1\end{array}} \right]\).