How to Run the World with Spherical Geometry
Gems in STEM: An Intro to Spherical Geometry
Who run the world? Girls! (As Beyonce dictates.) But, I am personally not a good runner, so I want to know the shortest possible distance required to “run” the world. In particular, given any two points on the globe (which we assume is a sphere, for simplicity), I need to know what the shortest path between them is so I don’t have to run for too long. (Yeah, yeah, I should build my endurance, whatever, I’m not in the Olympics.)
While at first glance this question doesn’t seem so difficult, it’s a little harder than it looks! Unfortunately, we can’t just draw a flat, straight line between any two points because we are on a sphere (and I definitely cannot dig my way from one point to the next). So, how do we go about answering this question?
Luckily, we have some help! As you can probably guess, we are not the first ones to ask this question. It’s an important concept in navigation and exploration, and lies in the area of spherical geometry, which has now been studied for thousands of years (so we’re wayyy behind). Spherical geometry is pretty much what it sounds like: the geometry of the two-dimensional surface of a sphere (meaning we don’t really care about what is inside the sphere right now). This geometry provided explorers with the tools to successfully map the globe and astronomers to accurately track the paths of stars and planets. You also have spherical geometry to thank for your vacation–it’s key to planning out the airplane routes every day!
In order to begin answering this shortest distance question, we need to start from the ground up. Now, traditional geometry (aka Euclidean geometry) is built around two basic objects: points and lines. So, as with any other geometry, we have to ask a few core questions!
First off, in spherical geometry, what is a point defined to be? Still a normal point— just a place on the surface of the sphere. No worries there! Now, what is a straight line in this geometry? Turns out there is no such thing! Remember that we are operating on the surface of a sphere, so any line on the surface will follow the curve of the sphere. But, in Euclidean geometry, we use straight line segments to figure out the shortest distance between any two lines. If we have no straight lines, what do we do?!
Patience, young grasshopper. We can figure this out. To do this, let’s pick two random points on a sphere. Let’s quickly define a geodesic to be the path of shortest length between two points, i.e. exactly what we want to find. (We just define this for convenience, don’t be scared off by the word!) Now, we can rotate our sphere such that one of the points is the north pole (point P in the diagram below). Notice that this rotation doesn’t change any distances–all we’re doing is looking at the sphere from a different perspective.
Then, if the other point is not the south pole (the point P’), it is clear that the shortest path is just the southward arc from P to the second point. For example, if the second point is A, B, or C in the diagram above, you can see how to simply follow the curve of the sphere to get a direct, shortest-distance path from the first point P to the second point.
But why is the south pole P’ not included in this? Think about it for a minute. If you want, here’s a hint: how many geodesics are there between P and P’?
Answer to the hint: infinitely many! You can travel down in any direction from the top point P, and the path to P’ is the same length, so they’re technically all the shortest path. (In particular, each path length is equal to half the circumference of the circle with the same radius as the sphere.) Since there are infinitely many directions, there are infinitely many geodesics! We call these pairs of diametrically opposite points antipodal points. From now on, we’re just going to ignore these pairs because, as you can see, the question of the shortest distance path between them is not super interesting.
Excluding this case, we can see that any two points on a sphere have a unique geodesic! But, we haven’t really given a precise analog to a straight line yet. Well, if we look back at our example paths, we can see that each arc path lies on a circle with the same radius as the sphere, where the direction of the circle depends on where the second point is relative to the north pole (the first point). These circles are called great circles (and they are indeed great!). You can also think about them as the largest possible circle that can be drawn around a sphere. One example of a famous great circle is the Equator! Here’s a nice visualization of other great circles:
Let’s recap: we’ve learned that points are the same and that the equivalent of straight lines in Euclidean geometry is great circles in spherical geometry! So, to find the shortest distance path between two points on a sphere, first draw the great circle that these two points lie on. Then, the shortest distance path is the smallest curve (which lies on this great circle) connecting the two points. Thus, we’ve covered our ass…ets in spherical geometry by defining the basic tools. But what about other objects? Angles? Shapes? Oops, for those questions I’m going to leave you on a cliff-hanger that will be resolved in the next column (sorry not sorry)! In the meanwhile–despite the suspense–you can now efficiently run the world! If you’re fast enough, maybe you can sneakily grab a medal at the Olympics. Good luck with that!
Until next time! If you found this interesting, make sure to check out the next column! If you have any questions or comments, please email me at firstname.lastname@example.org.
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This column, Gems in STEM, is a place to learn about various STEM topics that I find exciting, and that I hope will excite you too! It will always be written to be fairly accessible, so you don’t have to worry about not having background knowledge. However, it does occasionally get more advanced towards the end. Thanks for reading!