Conjugation, or how to move a motion
Background: Consider these two questions.
In each case, we want to "move" the rotation, which is a motion itself, to some new location by means of a second isometry. The way this is done is called conjugation. You have seen conjugation many times, but you may not have articulated it formally, as we will do now.
Experiments with translating a rotation
In a new sketch, draw a point P, a point Q and a point A. Then let the isometry P45 be rotation with center P by 45 degrees; let Q45 be rotation with center Q by 45 degrees; let T be translation by vector PQ and S be the inverse of T, translation by vector QP.
Then A1 is related to A by rotation by 45 degrees by center Q.
Experiment with TP45
Thus a first guess as to how to form the isometry Q45 would be that Q45 is P45 followed by T.
Experiment with U = T P45 S.
We discovered something about the nature of T P45, but it is not the rotation around Q that we seek. So we will try another candidate.
Lets reexamine the original figure. What we want to do is start with point A and end up with point A1. How do we do this? The first step is to move from A to A (this can be done by S); then rotate by P45 to get A1, and move from A1 to A1 by T. Thus the process is a three-step procedure, namely T P45 S. Lets call this triple composition U.
Note on Hiding Labels: If the labels are cluttering up the landscape, you can hide them all. First, select all these objects. Then go to the Display menu. If it only says Show Labels, do not despair. Choose that. Now immediately go back to the menu, without changing the selection. Now it will say Hide Labels. So choose that.
Summary and extension of results: Conjugation
The isometry U is an example of a general principle. Let F and J be any isometries. Let K be the inverse of J. Then let G = JFK.
G = JFK is called the conjugation of F by J.
Then G is an isometry, since it is a composition of isometries. But G has the same nature as F. So if F is a rotation, so is G. If F is a reflection, so is G.
One way to see this is to look at a point A and its images, B = F(A) and C = F(C). Then if A = J(A), B = J(B), C = J(C). Then G(A) = JFK(J(A)) = JF(A) = J(B) = B. Likewise, G(B) = C.
Thus, for example, if A is a fixed point of F, then A is a fixed point of G. A consequence of this is that if F has exactly one fixed point (which happens if and only if F is a rotation), so does G. So G is a rotation if and only if F is a rotation.
F has two fixed points M and N, if and only if F is reflection in the line MN. Thus G is then reflection in line J(M) J(N).
Likewise, if A, B, C are collinear, then A, B, C is collinear. But A, B, C is collinear for any choice of A only when F is a translation, so F is a translation if and only if G is a translation.
This only leaves glide reflections. We can conclude that F is a glide reflection if and only if G is a glide reflection by process of elimination, or we can observe a feature that is preserved. For instance, F is a glide reflection if and only if there is exactly one invariant line m that is mapped into itself by F. We can show that J(m) is mapped by G into itself.
More Conjugation Experiments
Now that we have stepped through two experiments, we will outline the next experiments and let you fill in the details.
Conjugating a Rotation by a Line Reflection
This time you can vary the angle of rotation by dragging F, so you can experiment with different angles. If the angle of R is t, what is the angle of V? How does this differ from what happened with U? Can you see a reason for this?
Conjugating a Translation by a Line Reflection
What kind of isometry is the composition WT? How is it related to TW? What can you say about the direction of these isometries?
What kind of isometry is the composition W^(-1)T? [We use W^(-1) for the inverse of W.] How is it related to T^(-1)W? What can you say about the direction of these isometries?
Note: This particular combination is a key element in understanding wallpaper groups than contain line reflections or glide reflections. It is discussed as a theorem in Bix.
More conjugation experiments.
There are many more possibilities to investigate. Here are some to try if you have time. Try to guess what you will get before trying the experiment.