David A. wrote: > > I've seen a lot about spherical geometry and wether it's non-Euclidean > or not, and I don't want to ressurect the whole debate. I just have > one question. One of Euclid's postulates was that any two points > determine a line. If you take the globe, and pick two points on the > same line of latitude, do they determine a line. I had thought that > lines on a sphere were defined as great circles. > Thanks, > David
Yes, they do determine a great circle (line). To find the great circle, intersect the plane that passes through the two points and the center of the globe with the surface of the globe. Of course, in most cases this great circle will be different from the common line of latitude (that is, except when the points are on the equator.)
By the way, the shortest distance between two points on the globe lies on a great circle, so this shows that the shortest distance between two points on the same line of latitutde is generally NOT on along the common line of latitude. The shortest route from New York to Tokyo passes close to the north pole, I believe.