Drawing the shortest path between two points on a map is often non-intuitive. We have always been taught that the shortest distance between two points is a straight line. Thus, the natural tendency would be to draw a straight line between points A and B and consider it to be the shortest. Most of the shortest path is actually represented by a curved line on the map.
Half the problem is that the land is not flat, but maps. Although slightly oblong, for the purposes of our discussion, land can be considered as a sphere. Maps are the projections of a spherical earth onto flat surfaces. Of course, this includes mathematical formulas, but it can be compared to cutting basketball at the seams so that it can lie flat and then stretch the seams until they meet.
The other half of the problem is that the orientation of the projection is completely arbitrary with respect to the sphere. Of course, it makes sense to always look at the map directly above the equator with the east / west direction horizontally and north / south vertically. But from the point of view of the sphere, the axis of rotation of the Earth, which defines the east, west, north and south, does not matter.
Before discussing the map projection and viewing angle, consider the shortest distance between two points on the sphere as a whole. First of all, straight lines do not exist on the surface of a sphere. The only way to get a straight line between two points on the surface is to penetrate the surface of the sphere when connecting the points. Thus, the shortest distance on the surface of a sphere is the path that is least curved. The least curved path lies along what is called the great circle. Here is the definition of a large circle found on the Internet: "A circle on the surface of a sphere that lies in a plane passing through the center of the sphere."
Going from point A to point B in any circle other than the large circle would not be the shortest path. It is actually quite easy to visualize. Just draw two circles of different sizes intersecting at two points. The distance between two points along the smaller circle would be more curved and, therefore, longer than the path along the larger circle. Of course, it should be understood that we are only interested in the short path between the points on each circle.
Now consider what leads to the fact that the path on Earth appears as a straight line on the map. The path between two points on earth is displayed as a straight line if the plane forms a trajectory, and a straight line between two points (across the surface) is parallel to the viewing angle. The equator is displayed as a straight line, because any two points on the equator form a plane parallel to the viewing angle. The same is true for parallels of latitude or lines of constant latitude. Horizontal lines, which are sometimes displayed on the map (often every 15 degrees), are examples of parallels of latitude.
The equator is a big circle; all other latitude parallels are not because their planes do not pass through the center of the earth. Thus, the shortest path between two points on any parallel north or south of the equator will not be along the parallel itself and therefore will not be displayed as a straight line on the map.
In conclusion, let us return to our statement that the viewing angle and the projection of the map can lead to the fact that the shortest path between two points will be bent. The equator appears to be a straight line, because the map has a viewing angle that is directly above the equator and, therefore, parallel to its plane. If the earth is rotated vertically, the equator will be displayed as a curved line. Similarly, the earth can be tilted and rotated so that any large circle has a plane parallel to the viewing angle, and therefore appears as a straight line. Therefore, the curved view of any large circle is simply a by-product of the viewing angle and the path along which the path is projected onto a flat surface.
