Geodesics and how someone can end up in New Orleans by traveling East from the westernmost point in Alaska?
In: Planetary Science
Could you review your question? If you travel with a true heading of 090° (easterly) from anywhere in Alaska, you will not end up in New Orleans. I do not think you would even cross the Canadian border if you travelled east from anywhere in Alaska.
Sure, you could travel by rhumb line or great circle from Alaska to New Orleans. You will have an easterly component to your course, but will not be travelling dead easterly.
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.
If you assume the Earth is a sphere, then if you head in any direction going in a straight line, you will travel along a great circle, which is a circle the same size as the equator. Which of those circles you follow will depend on your starting point and your starting direction. But the fact that you follow such a circle is pretty intuitive. Consider a smaller ball that’s perfectly symmetric, and if you leave any point you’re going to go around an equator of the ball and back where you started.
If you look at a globe, you’ll see that none of the lines of constant latitude are great circles, except the equator. They’re all smaller circles, all the way down to a single point at the poles. So it should be clear that if you set off traveling east from anywhere, and keep going in a straight line, those constant latitude circles can’t be paths you will follow. In particular, if you start at the North Pole, you’ll go all the way around through the south pole, regardless of which direction you go, so you won’t stay on the same latitude at all.
So then, to figure out the path from any starting point, you can imagine the starting point being a “pole” of the globe. No matter which direction you go, you’ll end up following a great circle that goes through the antipode of your starting point, which has the same latitude but south instead of North. So you’ll have that whole range of latitudes on your trip.
A geodesic is a minimal path between two points. Minimal in this case can be thought of as least time assuming a constant velocity. This is equivalent to the shortest distance between two points.
On a piece of paper this is a straight line. On a cylinder it is a helical path kind of like a spring. On a sphere, it is a great circle, equivalent to a line traced by a plane and the sphere’s surface, passing through the sphere center and the start and end destinations.
The earth is an ellipsoid and the geodesic is not a great circle, but for short distances it is approximated by one.
If you do not do that you end up at a point on the coast of the Gulf of Mexico about 300 km east of New Orleans, 68 km east of the border with Texas.
It is also the westernmost point in Alaska not the westernmost point of Alaska, that point is on Attu island and the same path puts you west of California out in the ocean, the shortest distance to land is about 580km. If you say the westernmost point is the one closest to the 180-degree latitude but east of the line you end up on another island in the Aleutian Islands. A line from it will be west of the line from Attua
The confusing part is that you only travel due east (90 degrees) at the starting point, and you travel at 152 degrees at the end, If you travel due east all the time you end up at Husdons Bay where Southampton Island is. Due east all the time is a curved ling on Earth’s surface unless you start at the equator.
A straight line on the earth’s surface is a great circle, that is a circle you draw on earth where the center is at the center of the earth,. Put a rubber band on a globe and stretch it so it is straight all the time so it does not slip off, you have now created a great circle, create one that starts due east at that point and you end up close to the Mississippi-Texas border.
A great circle round like that will go around Earth and you end up where you started, It will go over South America, and pass a bit north of Antarctica. It passes over China and Russia before you end up where you started. You will see the part with just a rubber band and a globe.
Imagine you are standing on the surface of a small globe, 50 metres away from the North Pole. You take out a compass and start walking due east. Before long, you notice that in order to continue travelling east, you must bear left (*to complete the constant-latitude circle*). If you do not, you will “fall” southward, following a great circle path.
While the amount of bearing left diminishes closer to the equator, it is still true that one must bear left to keep travelling east. If one does not bear left, one’s heading will change from east to southeast.
Google Earth is a great way to see this.
Westernmost point of Alaska is in the Aleutian islands, 51~53 degrees latitude. If you maintain a heading of due east you won’t hit any other US state.
However, that path would require you consistently curve left a bit to stay on that line. the only line of latitude that’s a great circle is the equator. If you start somewhere else, facing due east, and don’t turn as you travel, you’ll cross the equator when you’re 1/4 the way around the planet, and be at the same latitude in the opposite hemisphere when you’re halfway around the planet(at that point you’d be facing East again)
Get a ball and a couple rubber bands that can fit on it, put one on so it divides the ball in half, this is the equator, now start the second rubber band higher up, and parallel, like you were at Alaska and going East, now continue that path such that it divides the ball in half again.
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