A puzzle to puzzle you

You are the only survivor of a plane crash. You are standing on flat
ground. There is nothing around you: no trees, no houses, nothing at
all. You try to find out where you are. You leave your start walking
and go one km south. You don't see anything at all. So you go one km
west. Still nothing. So you go one km north and find you are back to
where you started, at the plane crash. How is this possible?

A variation on this puzzle is that you are again lost. This time you go
one km north and then one km south, but you are not where you started.

What is going on? How can this be?

Whenever we think of directions, we think of Earth as being flat. This
is typically true, since the Earth is so large. However, we live on the
surface of a sphere (almost spherical). The latitudes and longitudes
that tell distances on Earth are perpendicular to each other only near
the Equator.

So there are places on Earth where longitudes are not parallel such as
at the poles (first version of the puzzle depends on this).

This is more easily understood with the help of a figure. In the first
case, the person clearly crashed at the North Pole. He walked south
along one longitude, as shown, then walked west until he reached another
longitude 1 km away, then simply walked back north to the north pole.

In the second case, he must have started at the point marked "2" on the
figure. In he was on the Equator, for instance, he would have been very
far from /near to his starting point, as can be seen from the dotted
path starting on the Equator in the figure.