Compositions of isometries:

An isometry is a transformation of the plane that preserves lengths.

Isometries come in one of 3 or 4 types

• Translation (specified by a vector)
• Rotation (around a point by and angle)
• Reflection (across a line)
• Glide Reflection (specified by a vector--translation by the vector and then reflected across the line of the vector)

The last (glide reflection) is clearly a composition of two of the others, but it is added to the list because of the key result

So, every composition of two isometries can be expressed as a single isometry from the above list.

In class, we discovered that

• the composition of two translations is another translation. (This is a basic relationship. This relationship is always true for every pair of vectors)

Additionally, we found the relationship between the tranformations that

• the vector describing the translation that is the composition, is the sum of the other vectors describing the other two translations. (this makes the relationship specific).

In investigating glide reflections, we also discovered that, for a reflection across a line, and a translation by a vector

• The composition of the translation and then a reflection is the same as the reflection and then the translation if and only if the vector is parallel to the reflection line. (this is a condition for when this relationship is true)

Your task is to find relationships:

• Are compositions of the transformations ever the same when the order is reversed?
• What one transformation is the same as the composition of two transformations?
• Under what circumstances are those above relationships true? (Always? When the vector and line are parallel? When?)
• Be specific about the transformation that is the composition. (How long is the vector? In what direction? What is the angle? Where is the rotation point? Where is the reflection line?

Here are some situations to experiment with:

• A reflection and then a rotation (created with Sketchpad, and saved as a web page)
• Two reflections (first download Geogebra, and then download and open this file). Alternately, you can try this web page that has it embedded, but it may take a long time to load.
  • The original flag is B (green), which is reflected over e to get B' (gray), which is in turn reflected over f to get B'' (red).
  • Does the composition ever give the same result as reflecting across the lines in the opposite order, and if so, when?
  • Is the composition the same as the result y of a single translation? (Always? Under what conditions? How is the translation vector related to the reflection lines?)
  • Is the composition the same as the result y of a single rotation? (Always? Under what conditions? How are the rotation point and angle related to the reflection lines?)
  • Is the composition the same as the result y of a single reflection? (Always? Under what conditions? How is the reflection line related to the other two reflection lines?)
  • Is the composition the same as the result y of a single glide reflection? (Always? Under what conditions? How is the glide reflection line and vector related to the other two reflection lines?)
    If any of 2-5 seem to be correct, the next questions are: is this always true, and if not, under what circumstances is this true?
    
• Two rotations: the geogebra file. If you want this as a stand-alone web page, send me a request.

When you think you've found something, share it on the D2L discussion board. If you want another geogebra file showing a different composition of functions, send me an e-mail.