Darren New wrote:
> 
> I think I was more making the point that "distance" isn't a simple 
> definition when you're talking about space that curves in different 
> directions in different places.

I suppose I normally view distance in Euclidean space from the same 
definition that applies to non-Euclidean spaces, rather than the 
sqrt(dx^2 + dy^2) form, so I didn't really consider this.  You're right 
though, if you're starting from the pythorgean theorem view of distance 
it does bear some thinking about how it generalizes to the non-Euclidean 
space.

That said, I'm not sure it's necessary to actually understand the proper 
definition of distance in order to talk about circles in other spaces -- 
particularly if we limit ourselves to spherical and hyperbolic spaces 
which are more or less easy to visualize.

> Indeed, isn't it possible to have spaces where the distance from here to 
> there is different than the distance from there to here?


Sure, but the standard interpretation of "non-Euclidean space" that I'm 
aware, ie "Riemannian manifold", uses a definition of distance which 
doesn't allow such things.  All distances are proper distances in that 
they obey symmetry, the triangle inequality, etc.  I suppose, however, 
that you could argue that "non-Eucliean" should be colloquially taken to 
include pseudo-Riemannian spaces too (so as to incorporate Minkowski 
spaces), but even here you have a pretty nice definition of distance, 
although "circles" become a bit nastier to talk about.

It's possible other people mean different things when they say 
non-Euclidean space though, so I don't want to pretend that it's the 
only way to read that term.  Still, that's what had interpreted to be 
implicit in the conversation.

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