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**Example text**

48 (Above) Fig. 49 (Right) I 4 R Fig. 50 Fig. 51 THE P R O J E C T I V E PLANE 43 Fig. 52). Now the shortest curve, or geodesic, on a sphere joining two points is an arc of a great circle, whereas in the plane the geodesic between two points is the straight-line segment joining them. A gnomonic map of a hemisphere of the earth has the useful property of representing geodesies on the earth by straight lines on the map. 53 is a gnomonic map of the portion of the earth south of latitude — π/8. A gnomonic map is not conformai, and shapes on the earth are distorted.

In contrast, the edge a appearing twice on the map and twice in the edge equation can be approached from both sides. The unmatched edges b and c are called boundary edges, whereas the matched edge a is an interior edge. The boundary edges are grouped together into simple closed curves^ called boundary curves. Since b and c are both closed curves, each is a boundary curve. Our particular cylinder r = 2, |z| < 1 is inscribed in the torus (r — 2)2 + z 2 = 1 so that the boundary curves of the cylinder are on the torus.

123 denotes the triangle with vertices labeled 1,2,3. ) What is the topological nature of SI 2. Consider two Euclidean loci, one defined by the parametric equations x = sin «, y = (sin v)(2 + cos u), z = (cos v)(2 + cos u) where —π < u < π and —π<ν<π, and the second by x = sin w, y = sin i\ z = u2 + v2 where —π < u < π and —π<ν<π. One of these loci is topologically a torus and one is not. Which one is the torus? Explain why the other is not a torus. 49 EXERCISES 3. What is the topological nature of the Euclidean locus with the parametric equations x = u2 + r 2 , 2 y = (u2 + v2 DM, v, 2 where u + v < 1 ?