Math 5336     Spring 2004
Sphere figures

Introduction

At present (March 23, 2004) the figures shown here all were generated by Matlab. It may be possible in the near future to supplement them with an interactive figure in which you can rotate the sphere to see what happens (or what doesn't happen) when you move a spherical triangle around. (This would be similar to the polyhedron and algebraic surface pictures linked to other parts of my homepage. But we'll have to see how much labor is involved in that sort of project.

Contents

Figure 1: Great circles

    

Back to the top

Figure 2: Sphere rotation

Actually, this figure pertains mainly to the proof of Theorem 3 in §7.1 of the text -- or at least to the version of the proof that was presented in class. The point is that the sphere can be rotated so that the great circle arc joining two given points of the sphere is mapped to a portion of a meridian. By calculating an integral (as in the text) we can show that a portion of a meridian (of 180 degrees or less) is the shortest path joining its endpoints and lying on the surface of the sphere. Therefore, any other great circle arc (of 180 degrees or less) has the same property since it can be rotated onto a meridian.

    

Back to the top

Figure 3: A spherical triangle

Let   A, B, C   be three points on the sphere, such that no great circle contains all three. In particular, this implies that no two are antipodal. Then these three points, together with the geodesics joining pairs of them form a spherical triangle. (By Theorem 3 of §7.1, the geodesic joining two non-antipodal points is a great circle arc of less than 180 degrees.)

Back to the top

Figure 4: a lune

Let  P  and  Q  be two antipodal points, and let  m  and  n  be great semicircles whose endpoints are  P  and  Q.  The region between the semicircular arcs  m  and  n,  as shown in the figure below is called a lune. More specifically, if    is the measure of the angle formed by  m  and  n  at  P,  then we can refer to the lune that we have constructed as the lune of measure    anchored at  P  and Q. 

Back to the top

Comments and questions to:  roberts@math.umn.edu

Back to  to the class homepage.