Given two direct isometries, say the rotation RP,ϕ and the translation TAB, we know that their product RP,ϕ TAB is a direct isometry, and we may even suspect that it is a rotation RQ,θ. The question is, what is Q and what is θ? This chapter will describe some of the ways that we can determine the values of the parameters that describe the result, in this case, Q and θ.
We first need to clear up some possible ambiguities about directed angles.
In Example 8.3.2, we showed that the product of two reflections in nonparallel lines is a rotation whose center is the intersection point of the two lines and whose angle of rotation is twice the angle from the first line to the second line; that is,
where Q = l ∩ m and where α is twice the directed angle from l to m. If l and m are rays, as in (a) below, there is no ambiguity in the meaning of the phrase “the directed angle from l to m.” We can think of it as being 36° or −324°, since we identify the angles of θ and θ + 360n for all integers n.8
When we talk about the directed angle from one line to another, however, there are other possible interpretations. In (b) above, in addition to 36° and −324°, the directed angle from l to m can be legitimately interpreted as 216° or −144°. Does ...