this post was submitted on 07 Sep 2025
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I got this far on the Wikipedia and gave up:
"Locally straight" is just a mathsy way of saying "it's straight if you zoom in a bunch".
I went down a rabbit hole about globes and maps recently
Basically, to find the shortest distant between two places on a globe (a 'straight' line), imagine a hoop or circle round the earth that cuts it exactly in half, and rotate it until it passes through both places (still cutting it exactly in half)
That's a great circle.
There are 2d map projections that are built around this, but they only work when one of the locations is at the center of the map. So it could show the shortest distance from, say, London to anywhere with a straight line, but it wouldn't work for a route not including London
Another way to think about it is with elastic bands.
Imagine getting a globe and putting a pin in each place. One pin in the UK, and one in New Zealand. Now put an elastic band between those two pins so that it's tight. The elastic will be as short as possible, which is as straight a line as possible. But, since the globe is curved the elastic has to curve with it. So, that's your straight line on a curved surface.
If you wrap the elastic around the other side of the globe (you might need a bigger elastic), you can find the other half of the circle. It's the place where the elastic is at its tightest, but also where its evenly balanced between slipping to either side. For example, say you have a pin in California and another one in Japan. Both Japan and California are at about 30-40degrees north latitude. But, if you put an elastic starting in Japan and then going around the earth at 30 degrees north through China, Turkey, Spain, etc. when you let go the elastic will slip to the north until there's no tension anymore. To keep it from slipping you have to balance the tension so it doesn't slip to the north and doesn't slip to the south, so it's going flat around the whole globe. That makes the long half of the great circle.
In case anyone else finds visual guides to be helpful for this sort of thing, I made a graphic to accompany your words:
Ah, okay that makes more sense! Thanks!
Another way to say it, if you cut a sphere in half and both sides are equal, its a great circle. All lines of longitude and the equator are great circles.
All that and not even one rabbit.