34 1. Various Ways of Representing Surfaces and Examples x y z dxy dyz dxz Ix Iy z1 z2 dxz dyz 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. Lecture 5 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 : R2 → R2, 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

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