34 1. Various Ways of Representing Surfaces and Examples
Figure 1.20. Images of three points determine an isometry.
A similar observation holds on the sphere, where the triangle
inequality becomes degenerate for the triple (x, y, z) iff y lies along
the shorter arc of the great circle connecting x and z. So in both
these cases, degeneracy occurs when the points lie along a geodesic;
this suggests that in general, a characteristic property of a geodesic
is the relation d(x, z) = d(x, y) + d(y, z) whenever y lies between two
points x and z which are suﬃciently close along the curve.
a. Isometries of the Euclidean plane. There are three ways to
describe and study isometries of the Euclidean plane: synthetic; as
aﬃne maps in two real dimensions; and as aﬃne maps in one complex
dimension. The last two methods are closely related. We begin with
observations using the traditional synthetic approach.
If we fix three non-collinear points in R2 and want to describe
the location of a fourth, it is enough to know its distance from each
of the first three. This may readily be seen from the fact that three
circles whose centres are not collinear intersect in at most one point.
As a consequence of this, an isometry of R2 is completely de-
termined by its action on three non-collinear points. In fact, if we
have an isometry I :
and three such points x, y, z, as in
Figure 1.20, the choice of Ix constrains Iy to lie on the circle with
centre Ix and radius d(x, y), and once we have chosen Iy, there are
only two possibilities for Iz; one (z1) corresponds to the case where
I preserves orientation, the other (z2) corresponds to the case where