Adventures in Non-Euclidean Geometry

My prior post on the shape of the universe got me thinking about spheres.  Spheres are difficult shapes to wrap your head around.  Luckily, we don’t really have to worry about them that often.  Unless, that is, you happen to be a pilot.

Most of our interaction with the concept of the Earth is in the form of a map.  Maps literally warp our view of the Earth, though.  They are a Euclidean representation of a non-Euclidean space.  Three dimensions projected onto a two dimensional object.  This works fine for extremely small sections of the globe like if you wanted to travel from one state to an adjoining state, just like Newtonian physics is fine for constant gravity and much slower than light velocities.  For larger sections of the globe, though, this will lead you to make very poor travelling decisions if you are using a map.  Those straight lines on the map are not straight on a globe.  No lines are straight on a globe.

How do pilots get around this?  Enter the Great Circle.  The math behind a great circle calculation is pretty complicated so let’s ignore the math for another day and wrap our head around the idea of a great circle.

In order to find the shortest path from point A to point B on a globe, you must find the great circle solution.  The great circle solution is simply the one circle that you can draw around the globe that does two things.  First, it must go through both point A and Point B.  And second, the circle must bisect the exact center of the globe.  That circle will have a unique property.  It’s diameter will always be the diameter of the globe.

The most obvious example of a great circle is the Equator.  If you wanted to travel the shortest distance between two points on the Equator, you’d always travel along the Equator to get there.  That is the only latitude line on Earth that can be a great circle solution.

Less obvious of an example is travelling from the North Pole to anywhere south.  No matter where you want to travel to, your great circle solution will travel through the South Pole.  This means that you will always travel along one of the longitudinal lines to get to your destination.  That’s right, every line of longitude is a great circle solution.

Now that you have two points on a sphere and you know the sphere’s diameter, you have everything that you need to figure out the great circle route.  Now all you have to do is the math.