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Originally Posted by Slavakion
Well... wait. Are you trying to suggest we draw a circle on a curved surface? Maybe it's technically a circle (I'm not sure) but I wouldn't call it one. Circles are two-dimensional, and what I think you're suggesting is three-dimensional. I can't really stand my ground there, because I haven't done anything with 3-D points, vectors, etc. Only volumes of various solids.
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No, he's not talking about three dimensional geometry, he's talking about two dimensional geometry on a curved surface (e.g. drawing a map of the world on a globe: don't forget, we are talking about the
surface of the globe which is 2D, not the actual globe itself, which is 3D). So they would be two-dimensional, just non-euclidean. They are still referred to as "circles", they just don't have the same properties as circles in a euclidean geometry (most pertinently, the fact that c/d does not equal pi).
EDIT:
From another page:
Is pi constant in relativity?
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Is pi constant in relativity?
Yes. Pi is a mathematical constant usually defined as the ratio of the circumference of a circle to its diameter in Euclidean geometry. It can also be defined in other ways; for example, it can be defined using an infinite series:
pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - . . .
In general relativity, space and spacetime are non-Euclidean geometries. The ratio of the circumference to diameter of a circle in non-Euclidean geometry can be more or less than pi. For the types of non-Euclidean geometry used in physics the ratio is very nearly pi over small distances so we do not notice the difference in ordinary measurements. This does not mean that pi changes because our definition of pi specified Euclidean geometry, not physical geometry. No new theory or experiment in physics can change the value of mathematically defined constants.
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(Emphasis added)