Question

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

Rotate points through \({\bf{45}}^\circ \) about the point \(\left( {{\bf{3}},{\bf{7}}} \right)\).

Short answer

\(\left[ {\begin{array}{*{20}{c}}{\frac{{\sqrt 2 }}{2}}&{ - \frac{{\sqrt 2 }}{2}}&{3 + 2\sqrt 2 }\\{\frac{{\sqrt 2 }}{2}}&{\frac{{\sqrt 2 }}{2}}&{7 - 5\sqrt 2 }\\0&0&1\end{array}} \right]\)

Step-by-step solution

Find the matrix for translation

The point \(\left( {3,7} \right)\) must be shifted back to the origin.

The translation matrix is

\(\left[ {\begin{array}{*{20}{c}}1&0&{ - 3}\\0&1&{ - 7}\\0&0&1\end{array}} \right]\).

Find the matrix for rotation

The matrix for rotation by \(45^\circ \) is

\(\left[ {\begin{array}{*{20}{c}}{\cos 45^\circ }&{ - \sin 45^\circ }&0\\{\sin 45^\circ }&{\cos 45^\circ }&0\\0&0&1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{\frac{1}{{\sqrt 2 }}}&{ - \frac{1}{{\sqrt 2 }}}&0\\{\frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}}&0\\0&0&1\end{array}} \right]\).

Find the matrix for translation

The point must again be shifted back to \(\left( {3,7} \right)\). The translation matrix is

\(\left[ {\begin{array}{*{20}{c}}1&0&3\\0&1&7\\0&0&1\end{array}} \right]\).

Find the combined matrix of transformation

The combined matrix for transformation can be expressed as

\(\left[ {\begin{array}{*{20}{c}}1&0&3\\0&1&7\\0&0&1\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{\frac{1}{{\sqrt 2 }}}&{ - \frac{1}{{\sqrt 2 }}}&0\\{\frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}}&0\\0&0&1\end{array}} \right]\left[ {\begin{array}{*{20}{c}}1&0&{ - 3}\\0&1&{ - 7}\\0&0&1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{\frac{{\sqrt 2 }}{2}}&{ - \frac{{\sqrt 2 }}{2}}&{3 + 2\sqrt 2 }\\{\frac{{\sqrt 2 }}{2}}&{\frac{{\sqrt 2 }}{2}}&{7 - 5\sqrt 2 }\\0&0&1\end{array}} \right]\).

So, the transformed matrix is \(\left[ {\begin{array}{*{20}{c}}{\frac{{\sqrt 2 }}{2}}&{ - \frac{{\sqrt 2 }}{2}}&{3 + 2\sqrt 2 }\\{\frac{{\sqrt 2 }}{2}}&{\frac{{\sqrt 2 }}{2}}&{7 - 5\sqrt 2 }\\0&0&1\end{array}} \right]\).