# Geodesics

What’s the shortest way to travel from London Heathrow airport (LHR) to JFK International airport (JFK)?

You’d think it would be a straight line like this:

But you’d be wrong. The shortest distance between LHR and JFK is actually a curve:

This is because the “shortest distance between two points is a straight line” rule only applies to a flat surface: normally when you use a map it covers such a small area that the curvature of the Earth isn’t noticeable, but on a map of the world it becomes important.

Because a map takes the curved spherical surface of the Earth and maps it onto a flat surface no map can accurately show the whole world – every possible map projection is cursed to distort the size and/or shape of countries and their location relative to each other. The only accurate representation is a true globe. The surface of the Earth is non-Euclidean and thus the rules of geometry that you’re used to don’t apply: parallel lines will eventually meet (e.g. lines of longitude meeting at the poles) and the angles inside a triangle can add up to more than 180°.

The shortest distance between two points, regardless of geometry is called a geodesic: on  flat 2-dimensional planes geodesics are straight lines and on the surface of 3-dimensional spheres geodesics are curved.