The Art and Craft of Problem Solving

302 CHAPTER 8 GEOMETRY FOR AMERICANS

Rotations

A rotation with center C and angle 8 does just what you'd expect: rotates every point

counterclockwise by the angle 8 , about the center C. Rotations can be expressed in

matrix form, using trigonometry, but complex numbers are much better. Recall that

multiplying by e

i8

rotates a complex vector by the angle 8. Thus if c is the center, then

the rotation is the mapping z 1--+ c + (z -c )e

i8 •

In contrast to translations and reflections/glide reflections, each rotation has ex­

actly one fixed point, namely, the center.

Example 8.5.4 What transformation results from the composition of two reflections?

Solution: Let TI, T 2 denote reflections across the lines .e I, .e 2 , respectively. There

are three cases to consider.

• If.e 1 = .e 2 , then clearly the composition T 2 0 TI is the identity transformation.


• If.el II .e 2 , we have the following situation. TI takes X to X', and T 2 takes X' to
  X". It should be clear that the net result is a translation by the vector 2A]}, where
  A, B are points on .el, .e 2 respectively such that the line AB is perpendicular to
  both .el and .e 2. (In other words, AB is the translation that that takes line.el to
  line fz.)

x

• If.el and .e 2 intersect in a point C, then this point will be fixed by each of the
  reflections, and hence fixed by the composition. This suggests that the compo­
  sition is a rotation about the center C, and indeed a quick dose of angle chasing
  on the diagram below easily verifies that the composition is indeed a rotation
  about C, by 28 , where 8 is the angle from .e 1 to.e 2 (i.e., in the picture below, the
  angular direction is clockwise).

In summary, the composition of two reflections is either a translation (including the

identity translation) or a rotation. _

As we mentioned earlier, transformations shed light on a problem when they are

used selectively, i.e., when only part (presumably, the "symmetrical" part) of a diagram