The only way I can see to do this involves calculus. Is that fair game for you?
Question: What is the average distance between two points on the surface of a sphere with a radius r?
I came up with this question while thinking "What is the average distance of any point on earth from me" and cannot find out HOW I would go about figuring this out. Any guidance is greatly appreciated
EDIT: Revise the question above to "What is the average distance between two points on the surface of a sphere with a radius R by traveling directly in a straight line from point A to B" I apologize for the original wording.
I think I have an answer . . .
What is the average distance between two points
on the surface of a sphere with a radius ?
With no loss of generalization, let one point be at the North Pole of the sphere.
. . Let be the South Pole.
Let the other point be
Consider the great circle through and
It could pass through Greenwich Mean Time
. . (discovered by the Armenian explorer, Prime Meridian).
Code:A * o * * * P * o * * * * * * - - - - * E * O r * * * * * * o * * * * B
Point can be anywhere on the arc
Since half the points are above the Equator and half are below,
. . the average distance would be: .
Therefore, average distance .
. . just as Haytham predicted . . .
Let's say the circle is the unit circle in the complex plane, and we want the average distance between and when (we will have to multiply the final answer by ).
This equals: . However, , so that and so the final answer is the value of , which I guess you know how to compute.