The shortest distance between two points is always a straight line in Euclidean geometry".
This is what we have learned in our school geometry classes. We have learned in school, where the figures are two-dimensional and represented on a flat surface like a notebook sheet.
Image Source: https://theconversation.com/sublime-design-the-geodesic-dome-30196
In real life, the shortest distance is a curve called a geodesic. That's because our planet is not flat! Thus, Euclidean geometry is not used, but Riemannian geometry.
Geodesics are the curves on the surface that make turns but
A geodesic
This is the concept that flight planners use to chart airplane routes to save time and fuel. Practically, in most cases, a geodesic is the shortest curve that joins two points.
This effect has interesting implications; for example, when you fly on an airplane, the path it takes to go from one destination to another does not follow a "straight line", as many people imagine. It follows the “curvature” of the Earth, making small adjustments in travel direction to cover the shortest possible stretch. If the plane were simply "in a straight line", it would end up traveling a longer trajectory than it does when following the land curvature.
Look at the image below, it shows a planar distance in orange and the geodesic distance of that planar distance in blue. The maximum deviation of the geodesic from the planar line is near 2,000 Km and the difference in length is 644 Km.
