Geodesics - from intuition to equations
The simplest kind of geometry, taught in schools, is the so called Euclidean geometry - named after an ancient Greek mathematician, Euclid, who described its basics in the 4th century BC in his "Elements". It is based on the notions of points, straight lines and planes and it seems to correspond perfectly to our everyday experiences with various shapes. However, we can notice problems for which Euclidean geometry is insufficient even in our immediate surroundings.
Let's imagine, for example, that we are airline pilots and our task is to fly as quickly as possible from Warsaw, Poland to San Francisco. We take a world map and knowing from Euclidean geometry that a straight line is the shortest path between two points, we draw such a line from Warsaw to San Francisco. We're getting ready to depart and fly along the course we plotted... but fortunately, our navigator friend tells us that we fell into a trap.
The trap is that the surface of the Earth isn't flat! The map we used to plot our straight line course is just a projection of a surface that is close to spherical in reality. Because of that, the red line on the map below is not the shortest path - the purple line is: