Explanation of Orientation in Transformations

Date: 05/24/2007 at 08:20:15
From: Laura
Subject: What is orientation?

We've been learning about transformations; translations, reflections, 
rotations, and dilations, and my teacher keeps saying that a 
translation, a rotation, and a dilation preserve the orientation. 

I'm confused as to what orientation is.  I think it has to do with
naming the triangle.  My teacher has mentioned about naming them
clockwise or counter-clockwise.  I've looked for it in our book, but
it's not explained.  Could you help?  

Date: 05/24/2007 at 22:43:29
From: Doctor Peterson
Subject: Re: What is orientation?

Hi, Laura.

Imagine making a triangle out of paper that is, say, white on one side
and black on the other.  Avoid making it isosceles.  Here's mine:

  A                       
   \   \                  
    \       \             
     \           \        
      C---------------B   

Now if you slide it around (translate it) or turn it (rotate it), or
look at it through a magnifying glass (dilate it), it will still
have the white side up, and if you go from A to B to C you will still
be going clockwise.

But now reflect it, say in a vertical line.  You can only do that with
your model by flipping it over (so the black side is now up):

  A                       |                       A
   \   \                  |                  /XXX/
    \       \             |             /XXXXXXX/
     \           \        |        /XXXXXXXXXXX/
      C---------------B   |   B---------------C

We can see in two ways that its orientation is different: we had to
turn it over; and when we go from A to B to C now, we are going
counterclockwise.  And that difference is what we call "orientation".

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/