Just read somewhere that the shortest path between two locations is an arc and not a straight line! eli5 how an arc is the shortest path for planes.
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Grab a large ball and pick two points on it. The shortest path *is* a straight line, but that goes straight through the ball. The shortest path along the surface of the ball is an arc because the surface is curved.
If you look at a flat map of the Earth, it isn’t easy to find the shortest path over long distances, because the flat map doesn’t accurately capture the curvature of the Earth. Imagine you have half a lime and you’ve scooped out the inside; now, try to make the peel flat.
It’s an arc on a 2D map because those maps distort the path. The earth is a sphere, there is no way to make a 2D picture of it that has no oddities.
On a globe it’s always a straight line when viewed from directly above.
Well, technically the shortest distance between two points on a globe is a straight line. But that straight line goes *through the middle of the earth* which is not something planes can do.
So the next best path is to travel along the surface of the sphere, in a straight line. But when you project that sphere (globe) onto a 2D surface, the path looks like it’s curved.
It’s still a straight line, but you’re thinking about a geodesic.
In curved space, the straight line actually forms an arc when you view it on a 2D map
https://gisgeography.com/great-circle-geodesic-line-shortest-flight-path/
All the geometry you’re taught in school is Euclidean and assumes that everything is flat, this isn’t true for a lot of applications
In a flat open field if you need to get to a point 200 meters north and 300 meters west then the shortest path is a straight line between them that is 360 meters long
But what if you’re in a nicely gridded city and need to get to a point 2 blocks north and 3 blocks west? You can’t travel diagonally and thus need to cover 5 blocks not 3.6 blocks. This is [Taxicab or Manhattan Geometry](https://en.wikipedia.org/wiki/Taxicab_geometry)
For measuring on a sphere, you similarly can’t just take the straight distance between the two points because that would go through the sphere so you need to follow the surface. By following the surface your line is inherently curved in the same way our taxi cab can only travel north/south or east/west but not north east.
The arc is also super exaggerated when you look at it on a map because we have taken the much smaller polar regions and stretched them to be as wide as the equatorial regions. If you hold a string on a globe is looks a lot closer to the “straight” line you’re expecting and not the big sweeping arc that google maps shows you
In very simple terms: The Earth is not flat. Go on to one of the flight tracking apps and see the routes planes take. They look odd on a two-dimensional map, but not on a globe.
On the surface of a globe, the shortest distance between two points is the arc of what is known as a *great circle*. A great circle is a circle with the largest possible circumference on the surface the globe. For example, the equator is a great circle, but the Artic Circle is not.
However, whenever you turn a globe into a 2D map, you are going to end up with some distortions. And the most common projections distort it in such a way that these “straight lines” look curved on these flat maps.
There is a type or projection which conserves the the “straightness” of great circles, but it’s quite odd.
https://en.wikipedia.org/wiki/Gnomonic_projection
Maybe not exactly what you were asking, but shortest can have 2 different uses. The shortest distance has been elegantly covered in other answered. There’s also the shortest time. For that, our hero is typically the “brachistochrone curve”. But that already assumes some things. Namely, the object is in a force field which tries to continuously and uniformly accelerates it downward, and the object is not propelled by anything else. In the real world, there is friction and wind resistance. However, a straight ramp will still lose to some sort of curve, it’s just not as simple the exact geometry of the curve (though arguably, the actual roots used to make the word “brachistochrone” imply that it is the shortest time path regardless of circumstances. In that case, for an object travelling at a constant speed, the brachistochrone curve is a straight line).
The main thing to consider is the worst case scenario. The path has a very very slight angle until just before it gets to point B, and then curves sharply downward. The ball or cart or whatever is making the journey will creep along ever so slowly for the vast majority of the journey, and will accelerate rapidly at the end. A straight line path will let the object accelerate uniformly through the entire journey. But even better is a curve that falls more steeply at the start, then flattens out a bit as it approaches the target. This way, it accelerates rapidly at the start, having a higher speed as it travels the flatter section. This higher speed makes up for the slightly longer path… To an extent. If you deform the curve too much, suddenly the extra length is not worth the extra speed.
The arc is the shortest path between two locations on earth (on its surface) because the earth isn’t a flat plane.
On an ideal flat plane the shortest path between two points is straight line.
In the real world you live on a globe that only comes close to being flat at very small scales.
On the globe the shortest distance (on the surface) between two points follows a great circle. A great circle is one the divided the globe into two equal halves.
you can draw a great circle through any two points on a sphere so that the shortest path between those two points lies on the great circle.
Depending on the scale and your map projection it may even look like a straight line or it might very much not look like one.
Technically you could always have the shortest distance be a straight line by tunneling though the earth instead of along it surface, but that doesn’t really work for most airplanes, ships or cars, that people use to travel between points on this globe.
Straight lines don’t exist on the surface of a sphere. The closest you can get is a great circle- a circle with the largest possible diameter. If you have a globe and a piece of string you can try pulling the string tight between two points and observing the path of the line over the earth. When a globe is “”unrolled” into a flat map the great circle arc can sometimes be distorted into a shape that seems counterintuitive.
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