Question

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

Reflect points through the x-axis, and then rotate \({\bf{30}}^\circ \) about the origin.

Short answer

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

Step-by-step solution

Find the matrix for reflection

The reflection matrix can be written as

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

Find the matrix for rotation

Thematrix for rotation is

\(\left[ {\begin{array}{*{20}{c}}{\cos 30^\circ }&{ - \sin 30^\circ }&0\\{\sin 30^\circ }&{\cos 30^\circ }&0\\0&0&1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{\frac{{\sqrt 3 }}{2}}&{ - \frac{1}{2}}&0\\{\frac{1}{2}}&{\frac{{\sqrt 3 }}{2}}&0\\0&0&1\end{array}} \right]\).

Find the combined matrix of transformation

The combined matrix for transformation can be expressed as

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

So, the transformed matrix is \(\left[ {\begin{array}{*{20}{c}}{\frac{{\sqrt 3 }}{2}}&{\frac{1}{2}}&0\\{\frac{1}{2}}&{ - \frac{{\sqrt 3 }}{2}}&0\\0&0&1\end{array}} \right]\).