Question

Assuming that the earth is a perfect sphere with radius 6378 kilometers, what is the expected straight line distance through the earth (in km) between 2 points that are chosen uniformly on the surface of the earth?

Answer 1

I feel as if this question is really simple, but I know it's not.

Answer 2

http://math.stackexchange.com/questions/167932/mean-distance-between-2-points-within-a-sphere

Answer 3

I think this is the same problem as what the distance is between any two points on a circle.

Answer 4

Wait, is this a probability question?

Answer 5

I actually have no idea

Answer 6

Use expected value I suppose.

Answer 7

All I know is, it'll involve some geometry lol

Answer 8

Well, first come up with the probability distribution.

Answer 9

We can have the first point at the top, and then all that matters is the second point.

Even if the first point were not at the top, we would be able to rotate the sphere in a way tat it is at the top, without changing distances.

Answer 10

can't you use some symmetry to say that the points will be separated by an angle of pi/2?

Answer 11

like wrt to the centre of the earth?

Answer 12

so just would be \(\sqrt{2}R_e\)

Answer 13

This problem is actually somewhat difficult.

Answer 14

^

Answer 15

I am thinking that spherical coords might be the best way to create a random uniform probability distribution for this...

Answer 16

oh wait. I have done this before. It's not what I said I don't think

Answer 17

I found a solution someone wrote

Answer 18

Oh?

Answer 19

Anyway, If you have \(\phi \) and \(\theta\), then the straight line distance would be....

Answer 20

"Fix a point to find the average distance of the other points from it. Take the point as A and centre O. Consider another point B on the surface. Let <AOB= theta. AB = 2sin(theta/2) with band radius sin theta."

Answer 21

I don't really know much about probability integrals but. there's sort of this cone of distance that has an equal probability of being picked. Which is really just a circumference following the outside of the sphere.