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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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