Circles in Non-Euclidean Geometry

[ƒ(x)]

Are circles considered straight lines in Non-Euclidean Geometry?

 

[yenchin]

Your question is abit vague... *which* non-Euclidean geometry? And what do you mean by "straight line"? Do you mean geodesic? Certainly on a sphere we can define a geometry where every great circle is a geodesic, which is locally distance minimising between two points.

 

[HallsofIvy]

In elliptic geometry, "straight lines" (as yenchin said, geodesics) may be closed paths but, technically, there still exist "circles" that are quite different from those. In hyperbolic geometry, all "lines" are unbounded and so are definitely NOT "circles".

 

[ƒ(x)]

Well, I started wondering about this because my uncle and myself started talking about Nikolai Lobachevsky. I don't know if that will help answer the question.

 

[tiny-tim]

Well, I started wondering about this because my uncle and myself started talking about Nikolai Lobachevsky. I don't know if that will help answer the question.

Hi ƒ(x)!

In Lobachevsky space, all straight lines go off to infinity, so none of them are circles.

(Though there is a "map" of Lobachevsky space, in which all the straight lines are mapped as arcs of circles which meet the enclosing circle at right-angles … but they aren't circles "in" the space, only "in" the map. )

 

[HallsofIvy]

Lobachevsky space being the aforementioned "hyperbolic space". And I corrected the silly typo in my first response.