I was just doing some reading about elliptical geometry and came across a problem. I've read that this type of geometry is pretty much based off the 'rejection' of Euclid's fifth postulate where instead of having one parallel line, you don't have any and this can be pictured by two 'great circles' intersecting on a sphere. My question is why do these two lines have to be 'great circles'. I mean, it's possible for two lines to be drawn on the sphere without touching each other at all - i.e. making it parallel. So, I don't really understand the explanation in which my book / other internet sources are saying ... Thanks in advance.
"Straight lines" are the shortest paths between two points. On a plane, they are literally straight lines. On a sphere, you can show with a bit of calculus which escapes my memory that the shortest path between two points on a sphere is a section of a great circle. Hence, if you're considering the 5th postulate, you're still considering 'straight lines', but just not in flat space (or at least not with a flat metric). A slightly more physical way of thinking about it is to consider yourself on the Earth's surface. If you walk in what you think is a straight line, what path do you trace out? A great circle. If you walked on what would be one of those smaller circles you mentioned, you could know immediately you weren't walking in a straight line, because you're always be turning in a particular direction. Since parallel lines must be 'straight', you want to work only with straight lines. Technically, you're actually working with a notion of spacial geodesics since they extremise distances.
Hmm .. I still don't think I understand. If I had two points and connected them in one of those 'smaller circles', wouldn't that be a straight line - even though it doesn't form a great circle ... Thanks for replying :)
No, it wouldn't be. A "great circle" is the shortest distance between two points on a sphere. That's the crucial point that makes a great circle the choice for a "straight line".
What about line a? We could have two points along that line that will yield the shortest distance without producing a great circle per se, right?
NO. if you choose any two points on the latitude circle a, walking to one from the other along a will be a longer path than if you got there via a great circle. A great way to see this is: get yourself a globe and some string. pick two points on the same line of latitude (not the equator) and stretch the string tightly against the globe so that it connects the two points. you will find that the string -- which automatically will stretch to the shortest path between them -- will not follow the line of latitude but will form a portion of a great circle. this is the reason why cross-country airflights look so strange on a map. they are following great circles across the earth (in order to minimize fuel consumption and time).