Rotation, Translation, and Reflection in the Plane

Date: 12/11/2005 at 14:05:18
From: Taylor
Subject: what do these math terms mean?

Dear Dr. Math,

Can you please help me?  I don't know what these terms mean.

  rotate
  translate
  reflect

I know they have to do with geometry but I don't know how exactly. 
Thanks for your help!

Date: 12/11/2005 at 22:59:38
From: Doctor Peterson
Subject: Re: what do these math terms mean?

Hi, Taylor.

There are three ways you can move an object around in a plane without
changing its shape or size.  (The technical term for this is "isometry".)

A rotation means turning it, like

  +---+
  |   |
  |   |                 +-----------+
  |   |        ----->   |           |
  |   +---+             |   +-------+
  |       |             |   |
  +-------+             +---+

Imagine sticking a pin in the object and spinning it by some amount.

A translation just slides it, without turning:

  +---+
  |   |
  |   |                 +---+
  |   |        ----->   |   |
  |   +---+             |   |
  |       |             |   |
  +-------+             |   +---+
                        |       |
                        +-------+

A reflection is like looking at it in a mirror; it flips the object over:

  +---+                     +---+
  |   |                     |   |
  |   |                     |   |
  |   |        ----->       |   |
  |   +---+             +---+   |
  |       |             |       |
  +-------+             +-------+

The "image" will be on the other side of the "mirror" line, the same
distance away:

  +---+      |      +---+
  |   |      |      |   |
  |   |      |      |   |
  |   |      |      |   |
  |   +---+  |  +---+   |
  |       |  |  |       |
  +-------+  |  +-------+

See these pages for the same ideas in different words:

  Translation, Reflection, Rotation
    http://mathforum.org/library/drmath/view/57839.html 

  Reflection and Rotation
    http://mathforum.org/library/drmath/view/62267.html 

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


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