Geodesics and how someone can end up in New Orleans by traveling East from the westernmost point in Alaska?
In: Planetary Science
Straight lines can be kinda unintuitively defined when the surface the line is drawn on isn’t flat. But there’s a test you can do with a ribbon or thin strip of paper. Try it on a flat surface first like a table, counter, desk… Lie the strip of paper down. Imagine you are a being walking along that line the paper draws. Now imagine taking a left-hand turn while halfway down the paper. Try to make the strip follow the path you would walk. It will not lie flat. Even a slight bend in trajectory will force the ribbon off the table at certain places.
When a path is not straight on a 1-dimensional line, it’s hard to tell, but because the ribbon/strip has some finite width, the edge on the inside of the curve has less distance to travel than the outside of the curve. But both sides of the paper are the same length, so the one going a shorter distance has too much paper, causing it to bunch up and not lie flat.
The same is true on any curved surface. If you have a globe or basketball you can try it there too. And something really cool about using the paper is that it allows bending to conform to the surface of the sphere, but not laterally like a person making a left/right-hand turn on the surface.
While east/west is a valid test to run. I actually prefer the north/south test instead because it’s easier to visualize if you don’t have a globe. You can picture two people standing at the equator, both facing north. (If you do have a globe, or even a basketball you can use two strips of paper to represent their paths. Bonus points if you can use one of the circumference lines on the basketball as an equator or draw one on along with either using the inflating nipple or otherwise marking the poles 90 degrees from the equator.) both paths start parallel to each other. Both are walking due north. But you know that no matter where you are on the globe, if you walk due north, you must arrive at the north pole. So both of these lines that appear to start parallel actually slowly creep closer together until they converge at the pole. With the 2 ribbons, you can see this very clearly.
If you do happen to have a globe, you can now try this with a strip of paper starting somewhere in Alaska and pointing directly east at the start of its journey. It will only lie flat if you allow it to curve southward. If you try to make it stay directly east, the top (north) side will bunch up off the globe.
Actually, there’s another fun experiment you can do. If you have one of those retractable badge holders, it works great by itself. But if you have any piece of string, you can just manually hold tension on it. By pulling on either end, you are taking up any available slack until the string automatically follows the shortest path to get from one hand to the other. You can try laying a loop in the string, but as soon as you tug at the ends, the loop must disappear because there is more string than is necessary to bridge the gap between your hands.
If you let the string lie against a globe while you do this, it will also naturally follow a geodesic. The benefit of the paper is two-fold (heh, fold): you get a nice visual representation if you don’t do it right and the line isn’t perfectly straight as the paper will not lie flat, and you also can just let it rest naturally instead of having to hold tension on it like you would with the string. But the nice benefit of the string is understanding why planes follow geodesics.
This part is pretty tricky if you don’t have a retractable badge string (actually, it’s no walk in the park even if you do have one, but it’s at least a bit easier.) I recommend anchoring the badge reel with your knee against the globe. Pull the end out with one hand and it will form a geodesic, naturally as it follows the globe’s curvature. Now take your free hand and try to deflect the string in the middle. Say, push it an inch north or south. You will see that more string unspools from the badge clip meaning any vehicle traveling that path would have traveled further than if you’d just left the string naturally take the shortest path. And if you then release the middle, it will spring back to the geodesic while the spool pulls back in the extra slack.
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