Another spherical problem

Suppose the Earth is a perfect sphere (of unit radius). Starting somewhere other than the poles, you travel in a zigzag, where you start by travelling in a straight line which is initially due west, then turn round and go on a path which is initially due east, and so on. You make infinitely many turns, but the total distance you travel is finite. Under what conditions will you end up on the equator?

Originally posted by Acolyte
Suppose the Earth is a perfect sphere (of unit radius). Starting somewhere other than the poles, you travel in a zigzag, where you start by travelling in a straight line which is initially due west, then turn round and go on a path which is initially due east, and so on. You make infinitely many turns, but the total distance you travel is finite. Under what conditions will you end up on the equator?

Wait, how far around do I turn? If I'm initially heading west, do I then turn in the opposite direction and go back the way I came (east) or do I go east from the way I am currently facing (ie north from the point of view that defined west in my first walk)?

Directions don't change, besides you'll fall of the Earth if you go too far

Originally posted by royalchicken
Wait, how far around do I turn? If I'm initially heading west, do I then turn in the opposite direction and go back the way I came (east) or do I go east from the way I am currently facing (ie north from the point of view that defined west in my first walk)?

By going along the straight line that's initially west I mean as follows: draw the circle of latitude you're currently on (a path going west); draw the tangent line (or great circle) to it where you are; travel along that line in an anticlockwise direction, as viewed from the North Pole. Alternatively, you get your compass out, make sure you're facing due west, then walk in a straight line. Similarly for east.

East is east - it doesn't matter which way you are facing. However, it is not the case in general that you'll be going back the way you came, as the 'lines' of latitude aren't really lines.