This week, we’ll look at another type of geometry, namely spherical geometry. Quite simply, this is the geometry of a sphere. Here, a sphere is a set of points equidistant from a given center. In other words, throughout this post, you should imagine only a surface, with nothing inside it — think of the rind of an orange, without any of the slices. I began to describe spherical geometry in my original post, What is a Geometry?, so I’ll briefly summarize the ideas in that post first. It wouldn’t hurt to review it…
From a Euclidean standpoint, there are points on the sphere, but obviously no straight lines. In spherical geometry, we define a Point to be a pair of antipodal points on the sphere, and a Line to be a great circle on the sphere. This results in two nice theorems of spherical geometry: any two distinct Lines determine (intersect in) a single Point, and any two distinct points determine a single Line.
This is a departure from Euclidean geometry, for this means that there are no parallel Lines in spherical geometry, since distinct Lines always intersect. But there is something more going on here.
Consider the statement “Any two distinct Lines determine a single Point.” Now perform the following simple replacement: change the occurrence of “Line” to “Point,” and vice versa. This gives the statement “Any two distinct Points determine a single Line,” and is called the dual of the original statement.
Thus, we have the situation that some statement and its dual are both true. Now if this is true of some set of statements in spherical geometry — that the dual of each statement is true — and we derive a new result from this set of statements, then the following remarkable thing happens. Since the dual of any statement we used is true, then the dual of the new result must also be true! Just replace each statement used to derive the new result with its dual, and you get the dual of the new result as a true statement as well.
This is the principle of duality in mathematics, and is a very important concept. We will encounter it again when we investigate projective geometry.
Triangles and trigonometry are different on the sphere, as well. A spherical triangle is composed of three arcs of great circles, as in the image below.
If this were Earth, you could imagine starting at the North Pole, following 0 degrees of longitude to the Equator (shown in yellow), follow the Equator to 30 degrees east longitude, then follow this line of longitude back to the North Pole.
On the sphere, though, sides of a triangle are actually angles. Sure, you could measure the length of the arcs given the radius of the sphere, but that’s not as useful. Consider the choice of units on the Earth — kilometers or miles? We don’t want important geometrical results to depend on the choice of units. So a side is specified by the angle subtended by the arc at the center of the sphere. This makes sense since the sides are arcs of great circles, and the center of any great circle is the center of the sphere.
The angles between the sides are angles, too! A great circle is just the intersection of a plane passing through the origin and the sphere. So the angle between any two sides is defined to be the angle between the planes which contain them.
Really nothing from Euclidean trigonometry is valid on the sphere. For example, if you look at the triangle above, you can see that the angles between the sides are 30, 90, and 90 degrees. These angles add to 210! In fact, the three angles of a triangle always add up to more than 180 degrees on a sphere. (You may notice that the sides of this triangle are also 30, 90, and 90 degrees, but this is just a coincidence. The sides and angles are usually different.)
If A, B, and C are the angles of a spherical triangle, it turns out that the area of the triangle is proportional to A + B + C – 180. This means that the smaller a triangle is, the closer the angle sum is to 180 degrees.
There is no Pythagorean Theorem on the sphere, either. In addition, if a, b, and c are the sides opposite angles A, B, and C, respectively, then we have formulas like
One interesting consequence of the second of these formulas is this. If you know the angles of a triangle, you can determine the sides. You can’t do this in Euclidean trigonometry, since the triangles may be similar, but of different sizes. In other words, there are no similar triangles on the sphere. Spherical triangles are just congruent, or not. You can’t have two different triangles with the same angles.
For now, we’ve considered our sphere as being embedded in a Euclidean space. The definition of this surface is easy: just choose a point as the center of your sphere, and then find all points which are a given, fixed distance — the radius of the sphere — from that point. Sounds easy enough.
But can you imagine a sphere without thinking of the three-dimensional space around it? Or put another way, imagine you were a tiny ant, on a sphere of radius 1,000,000 km. That’s over 150 times the radius of the Earth! How would you know you were actually on the surface of a sphere? If you were that small and the sphere were that large, it would seem awfully flat to you….
So how could you determine the sphere was curved? This is a question for differential geometry, which among other things, is about the geometry of a surface without any reference to a space it’s embedded in. This is called the surface’s intrinsic geometry.
As an example of looking at the intrinsic geometry of the sphere, consider Lines. Now you can’t say they’re great circles any more, since this relies on thinking of a sphere as being embedded in three-dimensional space; in other words, its extrinsic geometry. You need the concept of a geodesic — in other words, the idea of a “shortest path.”
So if you’re the ant crawling between two points on a sphere, and you wanted to take the shortest path, you would have to follow a great circle arc. So it is possible to define Lines only using properties of the surface itself. But the mathematics to do this is really extremely challenging.
Lots of new ideas here — but we’ve just scratched the surface of a study of spherical geometry. You can see how very different spherical geometry is from both Euclidean and taxicab geometry. Hopefully you’re well on your way to wrapping your head around our original question, What is a Geometry?….