The shortest route on a map

Here are two cities: Anchorage in Alaska -- And Helsinki, in Finland. What is the shortest route between the two? Let's draw a straight line on the map. Surely, that has to be the shortest distance. Or is there another route that is even shorter?

Look at a globe instead! The cities are located equally far north of the equator - on the same latitude - but on the other side of the earth from each other. But... look at this: From Anchorage, it's not that far to the North pole. And from the North pole, it's not very far to Helsinki.

If we turn the globe and look at it from above, we see that this path is shorter. If we were to draw this path, which crosses the North pole, on a map, it would look -- like this. Despite being the shortest route, it looks like a curve on the map. How can that path be shorter, when we draw a much longer line? This has to do with the fact that the surface is more stretched out on the map, the closer we get to the poles.

Look at these lines, going from west to east. On the map, they appear to be the same length. On the globe, we see that, the actual distance is shorter, the closer we get to the North pole. This length measured left to right on the map... ... represents a shorter distance on the earth's surface, the further away from the equator we place it.

Let's look at the trip between Anchorage and Helsinki again. On the map, these two sections of the two routes seem to be the same length. But is it like that in reality as well? Look at the globe. This section of the trip, up here, close to the pole, is a lot shorter in reality.

In order to get from Anchorage to Helsinki, we need to travel across about half the width of the map. But when we do it up here -- ... the total distance travelled will be much shorter than crossing the Atlantic. On a flat map, a straight line in most cases, does not show the shortest distance between two points.