The first time I got on the plane, I was going to Istanbul from New York. I was nervous, and I was checking everything about my flight. When I checked the flight route, I saw that the plane was going to make a curve. I started thinking about that. Why was the plane not going straight and saving some fuel? Instead of going through the United Kingdom, we could go through Italy and Portugal.
My flight was not the only plane that behaved this way. If you also check any flight route, you will see a curved line. And probably you would say: “The shortest distance has to be a straight line, and these flight routes don’t look straight.
So, why are these planes going in such interesting directions, making arcs? The answer is hiding behind the geometry. And probably, planes are choosing the shortest distance. In other words, planes take straight lines because they need to save money.
What does it mean for a line to be straight, by the way? A line is straight if it takes the shortest path if between any two points on a line. On a flat surface, it is easy to draw a straight line. But what about spheres?
The Earth is a sphere. So I can use a globe to simulate the Earth. My purpose is to find any straight lines on Earth or a sphere. First of all, there are two types of lines; longitudes that run in a north-south direction from pole to pole and latitudes that run in an east-west direction. Secondly, since these latitudes and longitudes are bendy, it is reasonable to think that they are not straight.
To find the shortest path on a flat surface, I can use a taut string. If I pull the string tight, that shows me a straight line. A string is an excellent apparatus to work on a globe since it is flexible.
Now I can choose any two points on my globe, and I pull my string taut between these points. Surprisingly, my string lines up perfectly with the longitude. That should work on any two points on a longitude on the Earth. Thus all longitudes take the shortest path between any two points on a longitude.
I can start working on latitudes by choosing the most popular latitude, the equator. The first point is the Galapagos Islands, and the second point is somewhere in Africa. So, when I pull my string taut again, it hugs the equator. So the equator is a straight line!
OK, what about other latitudes? If I go up north and go back to the flight route that I mentioned to you at the beginning. When I pull my string taut, my latitude changes, it is my flight route!
Now we can see why all these airplanes are going on curvy routes. They’re taking the shortest distance so they can save fuel.
The other interesting thing about longitudes and latitudes is that any longitude cuts the sphere correctly in half, creating a great circle; however, there is only one straight latitude, the equator. So the equator is the only latitude that counts as a straight line on spherical geometry.
There are no parallel lines in spherical geometry because; if you have two different lines cutting the sphere in half, they will eventually intersect no matter where I put these points. And if lines intersect, they can’t be parallel. So there are no parallel lines in spherical geometry.