Comment by GMoromisato 14 hours ago
Is there a metric for distance on the surface of a sphere? I imagine it's not one of the metrics in the family in the article, but in such a metric wouldn't Pi be less than 3.14?
[I dropped my physics major in college in favor of computer science, mostly because I couldn't handle the math, so I acknowledge that this could be a stupid/non-sensical question.]
That makes perfect sense!
I guess my point is that Pi is only a minimum in the selected family of metrics that the article examines. There are plenty of other metrics where Pi is as small or as big as you want.
Yes. The angle subtended by the two points at the center of the sphere. It's the angular displacement made to go from one point to the other along the great circle joining the two points on the surface.
The great circle is the one that passes through those points and has the center of the sphere as it's center.
There is a local metric. The value of length/2/r depends on how big the circle is:
Imagine the Earth is a sphere. You make circles centered in the north pole:
* If the circle is tiny, the Earth is almost flat and you get almost pi.
* If the circle is the equator, you have to walk 1/4 of length the circle from the pole to the equator, so the result is 4/2=2
* If the circle is so big that you walked almost to the south pole, the result is almost 0.