Rigid Motions in the Plane

We'll just stick with rigid motions in the plane.

• Two figures are congruent if there is an isometry taking one to the other.
• A translation is a shift and is specified by a direction and distance.
• A rotation is, well, a rotation and is specified by its center and an angle.
• A reflection is specified by a line of reflection, a.k.a a mirror.
• The product, or composition, of two isometries is the result of applying one and then the other in order.
• One more isometry will be important, a glide reflection, a special product of a reflection and a translation along the line of reflection. This produces a "footprint" pattern.

See the GSP sketch Isometries for an interactive version of this image (if you're at a Macintosh with Geometer's Sketchpad installed, that is):

Theorem 0: The only isometries of the plane are combinations of translations, rotations, and reflections. More specifically, if two figures are congruent, you can transform one into the other first by reflecting (if necessary), then rotating (if necessary), finally translating (if necessary).

Q2 : Suppose a two copies of a figure are slapped down in the plane. Specify a way to find a reflection, a rotation, and a translation whose product takes one copy of the figure to the other. Here are two examples to work on with pencil and paper. Remember--- you might not need all three operations!

hmk I. Demonstrate each of the following five theorems using Geometer's Sketchpad

Before you begin, think about how to go about this! In particular, what elements of the sketch should be arbitrarily placed and what elements need to depend on other elements? After the sketch is made, pay some attention to clean presentation. Aim the sketches towards the effective teaching of your own students.

Theorem 1 The product of any two reflections in the plane is a translation or rotation. In particular, if the mirror lines ...... then their product is a translation by vector ........; if the mirror lines ...... their product is a rotation through angle .... with center ...... : begin with something like the following:

Theorem 2 Any translation is the product of two carefully chosen reflections. begin with a motif and some line segment representing the translation. construct a pair of reflections whose product is the translation.

Theorem 3 Any rotation in the plane is the product of reflections. begin with a motif and something representing an angle and center of rotation. construct a pair of reflections whose product is rotation.

Theorem 4 Any transformation is the product of just three reflections! Recall theorem 0. Thus the theorem could be restated: any product of a reflection, a rotation and a translation is equivalent to the product of three carefully chosen reflections.
A "starter sketch" has been provided. Theorem 4 starter
Test your ideas on these examples:

Theorem 5 Given a pair of isometries A and B, the product AB is not necessarily the same as the product BA. That is, find an example and illustrate with a sketch.

Once you've made your sketches and drawn some conclusions...

  Chaim Goodman-Strauss
  Dept. Mathematics
  Univ. Arkansas
  Fayetteville, AR 72701
  strauss@comp.uark.edu
  501-575-6332