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September 13th, 2018, 07:33 PM   #1
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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!!
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September 13th, 2018, 07:54 PM   #2
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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 $\pi$
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September 13th, 2018, 09:17 PM   #3
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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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