Given two direct isometries, say the rotation **R**_{P,ϕ} and the translation **T**_{AB}, we know that their product **R**_{P,ϕ} **T**_{AB} is a direct isometry, and we may even suspect that it is a rotation **R**_{Q,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* + 360*n* 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 ...

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