It is well known that the shortest path between two points on the surface of the earth (assumed to be a perfect sphere of radius 6400 km) is a great circle route. To get this, draw the circle (on the surface of the earth) that goes through the two points which is centred at the centre of the earth. The shorter arc of the circle between the two points is the shortest distance. In the figure below, the solid circle represents the earth (assumed to be a sphere), with centre O. There are two points X and Y on the surface (whose positions can be specified by a latitude and longitude). The dotted line represents a circle on the surface of the earth, with centre O. The length of the arc of the circle between X and Y is the great circle distance, which is the shortest distance between the two points while travelling on the surface. A traveler wishes to accomplish a complex itinerary, going from point to point. Each point is specified by its latitude in degrees north or south, and its longitude in degrees East or West. The objective is to calculate the total distance travelled by the traveler as he goes from point to point in order, if he travels by the shortest distance between any two points.
Answers
An interesting topic in 3-dimensional geometry is Earth geometry. The Earth is very close to a sphere (ball) shape, with an average radius of \displaystyle{6371}\ \text{km}6371 km. (It's actually a bit flat at the poles, but only by a small amount).
Earth geometry is a special case of spherical geometry. When we measure distances that a boat or aircraft travels between any 2 places on the Earth, we do not use straight line distances, since we need to go around the curve of the Earth from one place to another. (Think about the direct or straight-line distance between London and Sydney, through the Earth. That's going to be a lot less than the distance a plane flies around the surface of the Earth.)
Let's start with an example. What distance does a plane fly between Beijin