> The shortest path between two points is a straight line
uh?
uh?
Also look at Manhattan/taxicab geometry. When you can only move along one axis at a time, you usually can't move in a straight line between two points.
Both of these situations have common applications in the real world. For example, in mapping and navigation.
https://www.quora.com/How-can-I-draw-a-triangle-with-three-9...
Also, taxicab geometry: https://en.m.wikipedia.org/wiki/Taxicab_geometry
It doesn't mean that there is a viable route on that line or that that line is the preferred one.
That one is really bad, imo
https://courseware.e-education.psu.edu/projection/chapter9.h...
For regional maps it doesn't particularly matter, the map will be in a projection that has relatively minimal error, but a straight line in most projected coordinate systems will not be the shortest path between those points.
On the surface of a sphere, the shortest distance between two points is along a great circle route (which is the equivalent of a line in spherical geometry). Polar map projections preserve straight lines as great circle routes, but, e.g., Mercator projections do not.
If you're looking at a 5 km × 5 km topo map, the difference isn't particularly significant. If you're looking at a map of Europe or the United States, it is (note that many airline routes seem to follow curved lines--it's because the shortest distance is that curved line).
Even on maps it is true.