Question

Consider the following geometric 2D transformations: D, a dilation (in which x-coordinates and y-coordinates are scaled by the same factor); R, a rotation; and T a translation. Does D commute with R? That is, is \(D\left( {R\left( {\bf{x}} \right)} \right) = R\left( {D\left( {\bf{x}} \right)} \right)\)for all \({\bf{x}}\) in \({\mathbb{R}^{\bf{2}}}\)? Does D commute with T? Does R commute with T?

Short answer

D commutes with R, and matrices D and R do not commute with T.

Step-by-step solution

Write the transformation matrix for dilation, rotation, and translation

Let the transformation matrices in homogenous coordinates for dilation, rotation and translation be

\(D = \left[ {\begin{array}{*{20}{c}}s&0&0\\0&s&0\\0&0&1\end{array}} \right]\), \(R = \left[ {\begin{array}{*{20}{c}}{\cos \phi }&{ - \sin \phi }&0\\{\sin \phi }&{\cos \phi }&0\\0&0&1\end{array}} \right]\), and \(T = \left[ {\begin{array}{*{20}{c}}1&0&h\\0&1&k\\0&0&1\end{array}} \right]\).

Compute the product of transformation matrices

Calculate the value of \(DR\).

\(DR = \left[ {\begin{array}{*{20}{c}}{s\cos \phi }&{ - s\sin \phi }&0\\{s\sin \phi }&{s\cos \phi }&0\\0&0&1\end{array}} \right]\)

Calculate the value of \(RD\).

\[RD = \left[ {\begin{array}{*{20}{c}}{s\cos \phi }&{ - s\sin \phi }&0\\{s\sin \phi }&{s\cos \phi }&0\\0&0&1\end{array}} \right]\]

Calculate the value of \(DT\).

\(DT = \left[ {\begin{array}{*{20}{c}}s&0&{sh}\\0&s&{sk}\\0&0&1\end{array}} \right]\)

Calculate the value of \(TD\).

\(TD = \left[ {\begin{array}{*{20}{c}}s&0&h\\0&s&k\\0&0&1\end{array}} \right]\)

Calculate the value of \(RT\).

\(RT = \left[ {\begin{array}{*{20}{c}}{\cos \phi }&{ - \sin \phi }&{h\cos \phi - k\sin \phi }\\{\sin \phi }&{\cos \phi }&{h\sin \phi + k\cos \phi }\\0&0&1\end{array}} \right]\)

Calculate the value of \(TR\).

\(TR = \left[ {\begin{array}{*{20}{c}}{\cos \varphi }&{ - \sin \varphi }&h\\{\sin \varphi }&{\cos \varphi }&k\\0&0&1\end{array}} \right]\)

Check the commutation of transformation matrices

For the transformation matrices, \(DR = RD\).

So, D commutes with R.

\(DT \ne TD\)

So, D does not commute with T.

And

\(RT \ne TR\)

So, R does not commute with T.

Hence, D commutes with R, and matrices D and R do not commute with T.