A traveller starts out from the Earth's equator, heading exactly northeast. Undeterred by mountains, oceans and political boundaries, he continues on a northeasterly heading until he can go no further.

Where does he end up?

How far did he go?

How many times did he circumnavigate the earth? (For these purposes, this means travel through 360 degrees of longitude.)

(In reply to re(7): Doesn't matter. by TomM)

OK

Lets assume that the traveller is far enough North that the effect of the curvature of the Earth is small. Pick two concecutive crossings of the Prime Meridian. The pat the traveller followed between them is less than the circle of latitude of the earlier point and greater than the circle of latitude of the later point.

On a flat surface, all such circles would be at a fixed ratio from the next, and could be used to limit the length of the total path: if the series represented by the sum of the circles converges, then the pat must be less than the limit (For example if each circle were half the circumference of the previous one, the limit of the series would be twice the largest circle, and the length of the path must be less than that.

So the first problem is determining whether the series converges.

But even if it does converge, the Earths surface is not actually flat, even relatively close to the pole. The sequence of circles is not at a fixed ratio. Instead, atany point, the next larger circle is smaller than it would be on a flat surface, and the larger circles considerably less that they should be. Looked at from the other direction, the next smaller circle is larger than expected, and there are many more steps in the sequence. If we take this into account, will the series still converge?

Posted by TomM on 2002-09-26 15:33:50