# Shortest Paths Between Points on Maps and Why They Often Appear Curved

Drawing the shortest path between two points on the map is often non-intuitive. We have always been taught that the shortest distance between two points is a straight line. So the natural tendency would be to draw a straight line between points A and B and assume that to be the shortest path. Most of the shortest path is actually represented by a curved line on the map.

Half of the problem lies in the fact that earth is not flat but maps are. Though slightly oblong, for the purpose of our discussion earth can be thought of as a sphere. Maps are projections of the spherical earth on flat surfaces. Of course this involves mathematical formulas but it can be likened to cutting a basketball at the seams so that it can lie flat and then stretching the seams until they meet.

The other half of the problem is that the orientation of the projection is completely arbitrary as far as a sphere is concerned. Sure, it makes sense to always view the map directly above the equator with east / west directions horizontally and north / south directions vertically. But from a sphere point of view, the axis of rotation of earth which determinates east, west, north and south is irrelevant.

Before discussing map projection and viewing angle further, let us examine the shortest distance between two points on a sphere in general. First of all straight lines do not exist on the surface of a sphere. The only way to obtain a straight line between two points on the surface is by penetrating the sphere surface when connecting the points together. So the shortest distance on the surface of a sphere is the path that is the least curved. The least curved path lies along what is referred to as a great circle. Here is the definition of a great circle as found on the web: "A circle on the surface of a sphere that lies in a plane passing through the sphere's center".

Traveling from point A to point B along any circle other than the great circle would not be the shortest path. This is actually pretty easy to visualize. Just picture two circles of different sizes intersecting at two points. The distance between the 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 concerned with the short path between the points on each circle.

Now let's examine what causes a path on earth to show up as a straight line on a map. The path between two points on earth shows up as a straight line if the plane formed by the path and the straight line between the two points (through the surface) is parallel to the viewing angle. The equator shows up as a straight line because any two points on the equator form a plane that is parallel to the viewing angle. Same is true about the parallels of latitude or lines of constant latitude. The horizontal lines that are sometimes displayed on the map (often every 15 degrees) are examples of parallels of latitude.

The equator is a great circle; all other parallels of latitude are not since their planes do not pass through the center of the earth. So the shortest path between two points on any parallel north or south of the equator would not be along the parallel itself and therefore would not show up as a straight line on the map.

In conclusion let's return to our assertion that the viewing angle and the map projection can cause the shortest path between two points to appear curved. 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 tilted vertically, then the equator would show up as a curved line. Similarly, the earth can be tilted and rotated so that any great circle would have a plane parallel to the viewing angle and therefore appear as a straight line. Therefore, the curved representation of any great circle is merely a byproduct of the viewing angle and the way the path is projected onto a flat surface.