3 ms·

> geodesics are not straight lines, they're curves! If you are working in the 3D Euclidean space in which the Earth is embedded, yes. But not if you're workin

by pdonis 2y ago

> geodesics are not straight lines, they're curves! If you are working in the 3D Euclidean space in which the Earth is embedded, yes. But not if you're working within the 2D non-Euclidean space of the Earth's surface alone. Within that space great circles, geodesics, satisfy all of the geometric properties of "straight lines". In other words, the geometric properties that define "straight lines" do not actually pick out the precise concept that Euclid thought they did. They pick out his intended straight lines in Euclidean space, but they also pick out great circles on 2-spheres, and hyperbolas on a hyperboloidal 2-surface. This discovery was made in the 1800s by several different people, and it led, among other things, to the theory of General Relativity in physics, so it's not something that can just be put aside.

lxgr 2y ago
But I live in Euclidean space, not on a 2-sphere!
- pdonis 2y ago
  > I live in Euclidean space, not on a 2-sphere! You live in both. In your daily life, you're restricted to the Earth's surface, and if someone asks you to walk in a straight line, you will walk in a straight line (geodesic/great circle) along the Earth's surface. You're not going to walk off into space, and you're not going to constantly be turning to maintain constant latitude. Over short distances the difference is negligible, but the article is not restricting you to short distances.
  - lxgr 2y ago
    If someone asks me to walk in a straight line unassisted by navigational aids, I'd probably end up walking in a medium-sized circle of (optimistically) a few kilometers' diameter. For everything beyond that, I'd need a compass or GPS and map, and then... I'd ask them to clarify if they meant a rhumb line, great circle line, or something else :)
    - pdonis 2y ago
      > If someone asks me to walk in a straight line unassisted by navigational aids, I'd probably end up walking in a medium-sized circle of (optimistically) a few kilometers' diameter. Why? If you just put one foot in front of the other, you will naturally walk along a geodesic--a great circle.
      - lxgr 2y ago
        No, it’ll really be a not-so-great circle :) https://www.sciencedaily.com/releases/2009/08/090820123927.htm https://www.sciencedaily.com/releases/2009/08/090820123927.h...

by pdonis 2y ago

lxgr 2y ago

But I live in Euclidean space, not on a 2-sphere!

pdonis 2y ago
> I live in Euclidean space, not on a 2-sphere! You live in both. In your daily life, you're restricted to the Earth's surface, and if someone asks you to walk in a straight line, you will walk in a straight line (geodesic/great circle) along the Earth's surface. You're not going to walk off into space, and you're not going to constantly be turning to maintain constant latitude. Over short distances the difference is negligible, but the article is not restricting you to short distances.
- lxgr 2y ago
  If someone asks me to walk in a straight line unassisted by navigational aids, I'd probably end up walking in a medium-sized circle of (optimistically) a few kilometers' diameter. For everything beyond that, I'd need a compass or GPS and map, and then... I'd ask them to clarify if they meant a rhumb line, great circle line, or something else :)
  - pdonis 2y ago
    > If someone asks me to walk in a straight line unassisted by navigational aids, I'd probably end up walking in a medium-sized circle of (optimistically) a few kilometers' diameter. Why? If you just put one foot in front of the other, you will naturally walk along a geodesic--a great circle.
    - lxgr 2y ago
      No, it’ll really be a not-so-great circle :) https://www.sciencedaily.com/releases/2009/08/090820123927.htm https://www.sciencedaily.com/releases/2009/08/090820123927.h...