298 CHAPTER 8 GEOMETRY FOR AMERICANS
symmetry is important.
The great German mathematician Felix Klein introduced this "transformational"
viewpoint in 18 72, when he proposed that studying the algebraic properties of geo
metric transformations would shed new light on geometry itself (and by "geometry,"
he did only mean Euclidean geometry). This seemingly innocent suggestion changed
mathematics-and not just geometry-profoundly.
This section is a very brief survey of some of the most useful transformations in
Euclidean geometry. We will start with rigid motions that leave distances invariant,
and then move on to more exotic transformations that leave angles alone, but drasti
cally change everything else. In the interest of time, we will omit the proofs of some
statements, because we are anxious that you begin applying the concepts as soon as
possible. We leave the proofs, some of them rather challenging, to you.Rigid Motions and Vectors
A transformation is a mapping, i.e., a function, from the points of the plane to itself.
Transformations can take many forms and can be defined in many ways, using, for
example, coordinates, complex numbers, matrices, or words. The rigid motions are
those transformations that preserve length. In other words, if a rigid motion takes an
arbitrary point X to the point X', then A'B' = AB for all points A, B. There are four
types of rigid motions: translations, reflections, glide reflections, and rotations. In
other words, the composition of any two of these four transformations must be one of
the four types. This is not at all obvious; we will prove parts of it in this section.Translations
A translation moves each point in the plane by a fixed vector. Here are some equivalent
notations for the same translation T:- Cartesian coordinates: T(x,y) = (x +2,y - l).
• Complex numbers: T(z) = z +a, where a = (^2) - i.
- Vectors: T (1) = 1 + ii, where ii is the vector with magnitude v's and direction
(J = -arctan(1/2). - Words: T moves every point two units to the right and one unit down.
We may also omit the T ( .) notation, writing z f---> z + a, etc.
Generally, the vector form, or rather the vectorial point of view, is most useful.
Think of translations as dy namic entities, in other words, as actual "motions." Re
member that vectors have no fixed starting point; they are relative motions. In any
vector investigation, one must be aware of or carefully choose the location of the ori
gin. For example, suppose the vertices of a triangle are the points A, B, C. We can
think of A, B, C as vectors as well, but with respect to an origin O. This allows us to
use vector notation for positions; we thus have a flexible notation where we can move
between position and relative motion whenever convenient.
Hence A,li,c mean the motions from the origin 0 to A,B,C, respectively, shown
with dashed lines in the figure below (on the left). We can also write, for example
oA=A.