Introduction to Spherical Geometry

Student will learn about lines and angles and how to measure them in spherical geometry. We will start to compare the spherical and plane geometries.

The first new geometry we will look at is not actually new at all. We will look at the geometry of the sphere. It has been studied for thousands of years because of the need for accurate navigation. The subject was essentially finished by the beginning of the 20th century by the publication of Todhunter’s book on the subject (which is available through Project Gutenberg). The first question to be answered about geometry on a sphere is: what is a straight line? There aren’t any on a sphere! The primary property of a straight line (segment) is that it is the shortest distance between two points. So pick two points on a sphere. What is the shortest path between them?

Definition: a geodesic is a path between two points of shortest length.

We can rotate the sphere so that one of the points is the north pole. Then, as long as the other point is not the south pole, the shortest distance along the sphere is obvsiouly to go due south. We are, from now on, going to rule out pairs of antipodal points such as the north and south poles, because there are infinitely many geodesics between them. Otherwise, just as in geometry for the plane, each two (non-antipodal) points determine a unique geodesic. In the plane, this is the line segment between them. On the sphere, as we have just seen, it is the arc of a great circle that they determine. Side note: This is why airplanes fly weird-looking routes. For instance, flying Helsinki to Tokyo looks like it should mostly cross through Russia on a map of Europe. Instead, though, it flies almost over the north pole, taking the great circle route, saving hundreds of miles versus flying on a constant bearing. Curves of constant bearing (e.g., due southwest) are called loxodromes or rhumb lines. Any two (non-antipodal) points together with the center of the sphere determine a plane, and that plane intersects the sphere in the great circle that is determined by these two points. Since we can’t move along a line to get the shortest distance, at least we can stay in one single plane. Also, the sphere’s center together with the two points determines an angle. Since the points are not antipodes, the angle is less than 180◦. An important thing to note is that the distance along the great circle between these points is directly proportional to this angle. In fact, it is exactly the angle times the radius of the sphere. Unless otherwise stated, we will only consider spherical geometry on a sphere of radius one, so that distances can be measured by angles from the center of the sphere. These is, of course, another kind of angle to consider. That is the angle between two “lines.” How can we measure this? There are two ways that give the same answer. First is to take the tangent lines to the two great circles at the vertex of the angle. The other is to find the dihedral angle between the planes of the great circles. At this point, let’s compare spherical geometry with plane geometry. What is the same, what is different?

1 • In both geometries, two points (usually) determine a line.

• In both, you have angles beween lines. However, in plane geometry, lines that meet do so exactly ones. In spherical, they do so twice, at antipodal points.

• There are no parallel lines in spherical geometry!

Let’s try looking at triangles. Any three non-collinear points determine a triangle in both geometries. What do we know about plane triangles?

• The lengths of the sides obey the triangle inequality, and the angles add to 180◦.

• Aside from that, there is nothing else we can really tell about the triangle given just two sides, or given one side and one angle.

• Given two angles, we know the third, and can identiy the triangle up to similarity.

• Given two angles and a side, or two sides and the included angle, or all three sides, we know the triangle up to congruence. Given two sides and a non-included angle, there might be zero, one, or two triangles that work (ambiguous case of the law of sines).

• Given enough of the angles and sides, we can use trigonometry—specifically the laws of sines and cosines—to “solve” the triangle.

• We can also use trig to determine the area of triangles.

Right away we will notice things are quite weird for spherical triangles. Because all lines meet and there are no parallels you can do things like create triangles with two right angles. In fact, there are a whole range of triangles with two right angles, so knowing two angles isn’t enough to guarantee similarity—if there even exist similar triangles! And, in fact, every triangle in spherical geometry has total angle measure exceeding 180◦. So we’re going to need to take a much closer look.

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