Imagine in front of you a sheet of paper with a simple X-Y coordinate system drawn on it. 
Now take a compass and draw a line from the (0,0) point to the (10,0) point which 
crosses the (5,1) point. This line now gets the name x' (x prime).
If you now drew a curve above x' from the point (0,1) to the point (10,1) through the point 
(5,2) you'd have a curve in your original coordinate system.

If you now imagine that this curve describing a plane traveling a certain distance, where it has to remain
at constant height for fuel economy and air current reasons, you can see that the planes path is not a 
straight line relative to the original coordinate system on our page. 
If we compare the curve to our alternative axis x' though, both our measurment axis and our path are in parallel.
Meaning that if we imagine ourselves standing on x' and watching the plane fly over our head, it would feel like the plane was flying 
on a straight path at every position on x'.

The idea, that the shortest paths in coordinate systems based on spheres are not straight, comes from the way 
these paths look when the spheres surface is projected on a flat surface.
Similarly to the differences between our straight x-axis and our newly drawn curved axis x'.

Coordinate systems need to fit the spaces they describe and then switching between them can create unintuitive 
projection problems. But a plane can't bore a hole trough the planets crust to take the shortest euclidian path between
Beijing and LA, which is why a sphere based coordinate system is used instead.
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