My Math Forum  

Go Back   My Math Forum > High School Math Forum > Geometry

Geometry Geometry Math Forum

LinkBack Thread Tools Display Modes
September 13th, 2018, 07:33 PM   #1
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!!
lilycatherinee is offline  
September 13th, 2018, 07:54 PM   #2
Senior Member
romsek's Avatar
Joined: Sep 2015
From: USA

Posts: 2,647
Thanks: 1476

suppose we have a point $(x,y)$

Reflecting it would be accomplished by pre-multiplying by the matrix

$ref_x = \begin{pmatrix}-1 &0 \\ 0 &1\end{pmatrix}$


$ref_y = \begin{pmatrix}1 &0 \\ 0 &-1\end{pmatrix}$

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 $\pi$
romsek is offline  
September 13th, 2018, 09:17 PM   #3
Global Moderator
Joined: Dec 2006

Posts: 21,124
Thanks: 2332

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.
skipjack is offline  

  My Math Forum > High School Math Forum > Geometry

geometric, transformations

Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
Transformations u17159BR Trigonometry 2 May 31st, 2018 05:19 PM
transformations laxus95 Linear Algebra 2 December 11th, 2013 06:40 PM
exp - log transformations reto11 Applied Math 1 October 18th, 2010 11:08 PM
Transformations Adrian Algebra 10 July 7th, 2010 05:40 PM
Transformations math8553 Linear Algebra 1 December 2nd, 2008 06:34 PM

Copyright © 2019 My Math Forum. All rights reserved.