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spherical_geometry

# spherical_geometry - Math402 Spherical Geometry Spherical...

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Unformatted text preview: Math402 Spherical Geometry Spherical geometry is the geometry on the surface of a sphere (eg. the Earth or a ball). In this model we interpret point to mean a point on the ﬁxed sphere, and straight line to mean a great circle on the sphere. Recall that great circles are the largest possible circles on the sphere. In other words, a great circle is cut (on the sphere) by a plane passing through the center. For example, on the Earth lines of longitude are great circles (cut by various vertical planes through the center). On the other hand, among lines of latitude (cut by horizontal planes) only the equator is a great circle. EfﬁeQ‘E C i gal: lag f: 2,. ﬁgs 9% aisle"; :étéilifié \E' net a gear: Circle: {tag oi; malaria) caveat Ci vale, {éfg’twi’lﬁtf} Of course there are other (inclined) great circles on the sphere (see the picture below). The ones which are “vertical” or “horizontal” are just easier to draw (so try to use those ﬁrst in the problems). The reason we declare great (but not other) circles to be “straight lines” is that they behave like straight lines: o a person walking on the sphere without turning left or right would walk on a great circle (to convince yourself imagine you start on the North Pole); 0 (arcs of) great circles are shortest distances between points on the sphere (so a rubber band would lie along a great circle on a smooth sphere). Having deﬁned points and lines we can do geometry. For example, the following picture shows a spherical triangle with vertices A, B, and C. Be careful — don’t use small circles as sides of triangles ~ those circles are not “straight”. To measure an angle on the sphere we just look at it very closely, so that it looks like an angle on the plane. For example, a line of longitude and the equator are perpendicular (form the right angle). ...
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