What are the properties of a sphere?
There is an entire field of spherical geometry. One amazing thing is that the parallel lines on the sphere are the great circles I mentioned and all of them intersect. So unlike the plane, all parallels on a sphere intersect. In fact, for a given geodesic, there is no parallel geodsesic. Also, a triangle on a sphere does not have an interior angle total of 180 degrees as it does in the plane. It is more than that! So any given area of the sphere that contains a triangle cannot be mapped to the plane without distortion of angles or distances. This is a problem with map making.
