Orientation Preserving and Reversing Isometries of the Plane

We know some facts in general about what it takes to define an isometry uniquely. The most important general theorems are these:

1. If ABC and A'B'C' are congruent triangles, there is exactly one isometry T so that T(A) = A', T(B) = B', T(B) = B'.
2. If ABC and A'B'C' are congruent triangles, there is a set of 1, 2 or 3 line reflections, so that the isometry T above is the composition of these line reflections. (Note: The line reflections are not determined uniquely; there are many choices of line reflection that will work.)
3. An isometry is either the composition of two line reflections or else the composition of three line reflections but not both. The isometries of the first type are called the orientation-preserving isometries and those of the latter type are called orientation-reversing isometries.
4. If an isometry is the product of an even number of line reflections, it is orientation-preserving. If it is the product of an odd number of line reflections, it is orientation-reversing. Note: R also falls into the orientation reversing category.
5. The orientation-preserving isometries are the identity I, the translations and the rotations. Note: I is the product of two line reflections since I = RR for any line reflection,
6. The orientation-reversing isometries are the line reflections and the glide reflections. Note: Any line reflection R = RRR, so it is a product of 3 line reflections.

Orientation of Compositions of Isometries

From the facts above about isometries, it is immediate that one can deduce the orientation of a product ST of isometries from the orientation of the factors S and T, simply by counting the number of line reflections and checking for even or odd.

For example, the composition TU of a glide reflection T with a rotation U is the product of 5 line reflections, 3 from T and 2 from U. Since 5 is odd, TU is an orientation reversing transformation and thus a glide reflection or a line reflection. TU cannot be a rotation or a translation or the identity. This is very important information, and it is easy to see as 2+3! Of course if on wants to know precisely which transformation TU is, that is much more detailed work. But a lot can be said very fast.

This multiplication table shows all the cases.

	Orientation Preserving U	Orientation Reversing V
Orientation Preserving S	Orientation Preserving SU	Orientation Reversing SV
Orientation Reversing T	Orientation Reversing TU	Orientation Preserving TV

Tell what possible kinds of isometries could be these: (a) the product of two glide reflections (b) the product of a line reflection and a translation.