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 September 13th, 2018, 06:33 PM #1 Newbie   Joined: Aug 2018 From: United States Posts: 2 Thanks: 0 geometric transformations Matt wants to reflect a shape over the x-axis and then reflect it over the y-axis. Cathy says that this would be the same as rotating the shape 180 degrees. Do you agree or disagree with Cathy? Justify your answer and use matrix transformations as part of your justification. thanks for any help!!
 September 13th, 2018, 06:54 PM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,100 Thanks: 1093 suppose we have a point $(x,y)$ Reflecting it would be accomplished by pre-multiplying by the matrix $ref_x = \begin{pmatrix}-1 &0 \\ 0 &1\end{pmatrix}$ similarly $ref_y = \begin{pmatrix}1 &0 \\ 0 &-1\end{pmatrix}$ multiplying these two reflections matrices $ref_y ref_x = \begin{pmatrix}-1 &0 \\ 0 &-1 \end{pmatrix}$ a rotation matrix of $\pi$ radians is given by $rot_\pi = \begin{pmatrix}\cos(\pi) &\sin(\pi) \\ -\sin(\pi) &\cos(\pi) \end{pmatrix} =\begin{pmatrix}-1 &0 \\ 0 &-1 \end{pmatrix}$ and we see that the two reflections are indeed equivalent to the single rotation by $\pi$
 September 13th, 2018, 08:17 PM #3 Global Moderator   Joined: Dec 2006 Posts: 19,542 Thanks: 1751 Your $ref_x$, $ref_y$ and $rot$ are usually denoted by $\operatorname{Ref_y}$, $\operatorname{Ref_x}$ and $\operatorname{Rot}$ respectively. If the point $(x,~ y)$ is written as a row vector, use post-multiplication. If the point $(x,~ y)$ is written as a column vector, use pre-multiplication.

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