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I was looking for a simple explanation why only greater circles are considered as straight lines in spherical geometry (in the context of invalidating Euclid's fifth postulate) and not any circles of successively smaller radii that one can find on the surface of a sphere as one moves to poles. Thank you.

prashanth rao
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  • An even more basic question is, why should we consider a great circle to be a "line" in spherical geometry? Give a motivation to define a "line" and from that we may be able to determine what objects should be qualified to be "lines." – David K Dec 10 '21 at 14:26
  • I would say that smaller circles are lines alright. Any curve is a line. But only great circles are straight lines in spherical geometry. – ypercubeᵀᴹ Dec 10 '21 at 14:28
  • "lines" are usually taken as a primitive in geometry. One would have to redefine what line-ish objects "lines" are if the actual lines of the geometry are going to be relabeled to "straight lines." – rschwieb Dec 10 '21 at 16:10
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    Context is very important here: Are you studying Axiomatic Geometry or Differential Geometry? Which book are you reading? Where did you encounter this notion? Without you adding such context, your question is un-answerable since the answer depends on the context. – Moishe Kohan Dec 10 '21 at 16:14
  • In the context of differential geometry, check if the answer given by user65203 here is sufficient for you. Or the answer here by user "40 votes." – Moishe Kohan Dec 10 '21 at 17:50
  • @MoisheKohan Both of those questions seem to answer "why are the minor arcs of great circles the shortest paths between two points?" The question here is "Why are great circles considered lines in spherical geometry?" I think those two questions are not completely irrelevant, but they do not address the question here directly. The first is "why is this certain curve special?" and the other is "Why are these circles the right choice for 'line' rather than these other circles?" – rschwieb Dec 10 '21 at 18:04
  • To claim "they have to be lines because lines always talk about shortest distances" is arguable, although I think it is not totally accurate. – rschwieb Dec 10 '21 at 18:09
  • @rschwieb: These references are meant to supplement my earlier requests for clarification. The answer to the posed question really depends on the context which is badly lacking in OP. – Moishe Kohan Dec 10 '21 at 18:10
  • @MoisheKohan I see: i missed that connection. I think the context is apparent in the tags: just regular geometry. The fact that there are alternative contexts which the question might make sense in alone does not really amount to a lack of context. Probably every question lacks context by that standard. – rschwieb Dec 10 '21 at 18:17
  • @rschwieb: Are you sure? Take a look at the explanation of the "geometry" tag. The latter is a damp of topics including Euclidean geometry, algebraic geometry, differential geometry, even topology. Curiously, spherical geometry and axiomatic geometry are missing. – Moishe Kohan Dec 10 '21 at 18:21
  • @MoisheKohan No, I don't take it for granted the current contents of the tag description are a precise reflection of what they intended to convey. I am just making the obvious interpretation. Arctic char's refinement of the tags seems to suggest much the same sentiment. But I won't play a guessing game on that with you. I'll just leave it at that. – rschwieb Dec 10 '21 at 18:24
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    Sorry for any confusion and thanks to all for the elaborate answers and discussions. The context is axiomatic geometry (I think) as I was trying to understand why Euclid's fifth postulate is false in this geometry. I was referring to youtube as online resource, no particular textbook. It led me to question why only great circles are considered (straight) lines. Thanks All. – prashanth rao Dec 10 '21 at 19:24
  • @prashanthrao: Please update the question to make this clear. I will then be able to remove "close" vote. Another suggestion: It is better to learn math by reading rather than watching movies. – Moishe Kohan Dec 10 '21 at 22:50
  • Thanks@MoisheKohan for your suggestion. – prashanth rao Dec 11 '21 at 11:47

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If you add circles on the surface that are not great circles, then you further degrade a basic condition that we want to be true (where possible) in a geometry:

Two distinct points should uniquely determine a line.

This is an axiom for ordinary geometries (euclidean, hyperbolic, projective), but of course it is not completely true for spherical, since antipodal points do not uniquely define a line. This is resolved if you move from spherical to projective by identifying antipodal points, as the two elements are no longer distinct.

But if you were to throw in these extra "lines" that you described, then every pair of distinct points has infinitely many lines through them. Furthermore, it will create extra lines when you identify antipodal points, and so the result (which used to be a projective geometry) is no longer a projective geometry because it fails the axiom about unique lines through two points.

Maybe this seems like a technicality, but just think about how often in regular geometry we feel free to draw a line through two points. If there isn't a unique line through two points, can you still draw one? Which one should you draw? You see, having too many lines introduces ambiguity.

rschwieb
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    A line is also the shortest path between two points, which is still true even for antipodal points on a sphere if your "lines" are great circles. You lose only the uniqueness of the shortest path in that case. – David K Dec 10 '21 at 14:24
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    @DavidK True: I hesitated to go this path because it did not fit with projective geometry, but if your bag is metric geometry, this is important. I view metrics as less fundamental than the axiomatic condition about two points. – rschwieb Dec 10 '21 at 14:29
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    I think it all comes down to the motivation for looking for a "straight line" in the first place. I could go either way depending on context. – David K Dec 10 '21 at 14:34
  • @DavidK I'm not familiar with any geometries that deal with more than one class of lines. As far as the axioms are concerned, there are only "lines." They don't talk about other subsets that are line-ish, although they may be relevant in physical situations. But I agree that it's sometimes helpful to think of the official lines of a geometry as "straight lines." – rschwieb Dec 10 '21 at 14:43
  • Sorry, I should have just written "line" as I did in my comment under the original question. I didn't mean to muddy the waters with other kinds of "line". – David K Dec 10 '21 at 17:33