Two points within the Arctic Circle are chosen at random, using a uniform distribution over the entire area. That is, any region of a given area is as likely as any other region with that area, to receive a given point.
What is the expected value of their great circle distance from each other?
Assume:
-
The Earth is a perfect sphere with radius 3,958.8 miles.
-
The Arctic Circle is located at 66.55° North.
Both calculus answers and simulation answers are welcome. As we are using approximations here, especially about the perfect sphericity of the Earth, the exactness of calculus is not really needed.
Part 2:
... and how about two points on the whole Earth?
If I tweak the parameters of the random point generator from my solution to part 2 then we can get a system that stays exactly in the Arctic Circle.
Phi at the Arctic Circle equals 90-66.55=23.45 degrees, and cos(23.45 degrees)=0.9174. Then random points within the Arctic Circle can be generated by:
Theta = random (0,2pi)
Phi = arccos(random (0.9174,1))
The arc of the great circle subtended by (Theta1, Phi1) and (Theta2, Phi2) can be calculated as
arccos( sin(Phi1)*sin(Phi2)*cos(Theta1-Theta2) + cos(Phi1)*cos(Phi2) ).
Theta1-Theta2 can be replaced with just ThetaD, but that's still three variables.
The triple integral of that looks nasty.
Part 1 thoughts
