Lecture 5 35

x = Ix

y

z

Iy

Iz

Rotation—one fixed point

x

y

z

Ix

Iy

Iz

Translation—no fixed points

Figure 1.21. Orientation preserving isometries.

orientation is reversed. So for two pairs of distinct points a, b and

a , b such that the distances between a and b and between a and b

coincide, there are exactly two isometries which map a to a and b to

b ; one of these will be orientation preserving, the other orientation

reversing.

Passing to algebraic descriptions, notice that any isometry I must

carry lines to lines, since as we saw last time, three points in the plane

are collinear iff the triangle inequality becomes degenerate. Thus

it is an aﬃne map—that is, a composition of a linear map and a

translation—so it may be written as I : x → Ax + b, where b ∈

R2

and A is a 2 × 2 matrix. In fact, A must be orthogonal, which means

that we can write things in terms of the complex plane C and get (in

the orientation preserving case) I : z → az + b, where a, b ∈ C and

|a| = 1. In the orientation reversing case, we have I : z → a¯ z + b.

Using the preceding discussion, we can now classify any isometry

of the Euclidean plane as belonging to one of four types, depending

on whether it preserves or reverses orientation, and whether or not it

has a fixed point.

Case 1 : An orientation preserving isometry which possesses a

fixed point is a rotation. Let x be the fixed point, Ix = x. Fix

another point y; both y and Iy lie on a circle of radius d(x, y) around

x. The rotation about x which takes y to Iy satisfies these criteria,

which are enough to uniquely determine I given that it preserves

orientation; hence I is exactly this rotation.